From 9ce4b7be43c1d7dcf4cb34ca883975fd90faf6e6 Mon Sep 17 00:00:00 2001 From: royaghavami <59202308+royaghavami@users.noreply.github.com> Date: Fri, 29 Oct 2021 22:49:57 +0330 Subject: [PATCH 01/57] Create index.md --- notebook/6_search_in_continuous_spaces/index.md | 1 + 1 file changed, 1 insertion(+) create mode 100644 notebook/6_search_in_continuous_spaces/index.md diff --git a/notebook/6_search_in_continuous_spaces/index.md b/notebook/6_search_in_continuous_spaces/index.md new file mode 100644 index 00000000..8b137891 --- /dev/null +++ b/notebook/6_search_in_continuous_spaces/index.md @@ -0,0 +1 @@ + From 58c3adc0aaaaaff6bce7f5b2104ab56d23323772 Mon Sep 17 00:00:00 2001 From: royaghavami <59202308+royaghavami@users.noreply.github.com> Date: Fri, 29 Oct 2021 22:57:30 +0330 Subject: [PATCH 02/57] Create matadata.yml --- .../matadata.yml | 29 +++++++++++++++++++ 1 file changed, 29 insertions(+) create mode 100644 notebook/6_search_in_continuous_spaces/matadata.yml diff --git a/notebook/6_search_in_continuous_spaces/matadata.yml b/notebook/6_search_in_continuous_spaces/matadata.yml new file mode 100644 index 00000000..9c1984f7 --- /dev/null +++ b/notebook/6_search_in_continuous_spaces/matadata.yml @@ -0,0 +1,29 @@ +title: # shown on browser tab + +header: + title: Search in Continuous Spaces # title of your notebook + description: # short description of your notebook + +authors: + label: + position: top + content: + # list of notebook authors + - name: Rosta Roghani + Bardia Khodadadi + Roya Ghavami Adel # name of author + role: Author # change this if you want + contact: + # list of contact information + - link: https://github.com/rrosta + https://github.com/royaghavami + https://github.com/bardia-hub-boop + icon: fab fa-github + # optionally add other contact information like + # - link: # contact link + # icon: # awsomefont tag for link (check: https://fontawesome.com/v5.15/icons) + +comments: + # enable comments for your post + label: false + kind: 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-authors: - label: - position: top - content: - # list of notebook authors - - name: Rosta Roghani - Bardia Khodadadi - Roya Ghavami Adel # name of author - role: Author # change this if you want - contact: - # list of contact information - - link: https://github.com/rrosta - https://github.com/royaghavami - https://github.com/bardia-hub-boop - icon: fab fa-github - # optionally add other contact information like - # - link: # contact link - # icon: # awsomefont tag for link (check: https://fontawesome.com/v5.15/icons) - -comments: - # enable comments for your post - label: false - kind: comments From 4a78fab9775efda609235b9339c0f5011f3e760b Mon Sep 17 00:00:00 2001 From: royaghavami <59202308+royaghavami@users.noreply.github.com> Date: Fri, 29 Oct 2021 23:09:07 +0330 Subject: [PATCH 05/57] Create index.md --- notebooks/6_search_in_continuous_spaces/index.md | 1 + 1 file changed, 1 insertion(+) create mode 100644 notebooks/6_search_in_continuous_spaces/index.md diff --git a/notebooks/6_search_in_continuous_spaces/index.md b/notebooks/6_search_in_continuous_spaces/index.md new file mode 100644 index 00000000..8b137891 --- /dev/null +++ b/notebooks/6_search_in_continuous_spaces/index.md @@ -0,0 +1 @@ + From 8351fdf54110ec9be3a85bc7e5b6294aae870567 Mon Sep 17 00:00:00 2001 From: royaghavami <59202308+royaghavami@users.noreply.github.com> Date: Fri, 29 Oct 2021 23:09:30 +0330 Subject: [PATCH 06/57] Create matadata.yml --- .../matadata.yml | 29 +++++++++++++++++++ 1 file changed, 29 insertions(+) create mode 100644 notebooks/6_search_in_continuous_spaces/matadata.yml diff --git a/notebooks/6_search_in_continuous_spaces/matadata.yml b/notebooks/6_search_in_continuous_spaces/matadata.yml new file mode 100644 index 00000000..9c1984f7 --- /dev/null +++ b/notebooks/6_search_in_continuous_spaces/matadata.yml @@ -0,0 +1,29 @@ +title: # shown on browser tab + +header: + title: Search in Continuous Spaces # title of your notebook + description: # short description of 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notes/notebooks/6_search_in_continuous_space/Search in Continuous Space (1).zip + kind: zip file text: Temporal Probability Models - Part 1 - notebook: notebooks/11_temporal_probability_models/index2.ipynb kind: S2021, LN, Notebook @@ -54,4 +56,4 @@ notebooks: kind: S2021, LN, Notebook #- notebook: notebooks/17_markov_decision_processes/ - notebook: notebooks/18_reinforcement_learning/ - kind: S2021, LN, Notebook \ No newline at end of file + kind: S2021, LN, Notebook From 3b474d02a0b7266645dd872ea3bfb33b3a18349c Mon Sep 17 00:00:00 2001 From: royaghavami <59202308+royaghavami@users.noreply.github.com> Date: Sun, 31 Oct 2021 13:20:27 +0330 Subject: [PATCH 09/57] Delete Search in Continuous Space (1).zip --- .../Search in Continuous Space (1).zip | Bin 314147 -> 0 bytes 1 file changed, 0 insertions(+), 0 deletions(-) delete mode 100644 notebooks/6_search_in_continuous_spaces/Search in Continuous Space (1).zip diff --git a/notebooks/6_search_in_continuous_spaces/Search in Continuous 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Sep 17 00:00:00 2001 From: royaghavami <59202308+royaghavami@users.noreply.github.com> Date: Sun, 31 Oct 2021 13:23:47 +0330 Subject: [PATCH 11/57] Add files via upload --- .../6_search_in_continuous_spaces/index.md | 282 ++++++++++++++++++ 1 file changed, 282 insertions(+) create mode 100644 notebooks/6_search_in_continuous_spaces/index.md diff --git a/notebooks/6_search_in_continuous_spaces/index.md b/notebooks/6_search_in_continuous_spaces/index.md new file mode 100644 index 00000000..310efa03 --- /dev/null +++ b/notebooks/6_search_in_continuous_spaces/index.md @@ -0,0 +1,282 @@ +Search in Continuous Space +# +# **Table of Contents** +` `**TOC \o "1-3" \h \z \u [Introduction PAGEREF _Toc86421092 \h 2**](#_Toc86421092)** + +[**Type of optimization techniques PAGEREF _Toc86421093 \h 2**](#_Toc86421093) + +[*Linear programming (LP) and Nonlinear programming (NLP) PAGEREF _Toc86421094 \h 2*](#_Toc86421094) + +[*Continuous optimization and Integer programming (IP) PAGEREF _Toc86421095 \h 2*](#_Toc86421095) + +[*Constrained optimization and Unconstrained optimization PAGEREF _Toc86421096 \h 2*](#_Toc86421096) + +[*Differentiable optimization and Non-differentiable optimization PAGEREF _Toc86421097 \h 3*](#_Toc86421097) + +[Cost functions PAGEREF _Toc86421098 \h 3](#_Toc86421098) + +[**Convexity PAGEREF _Toc86421099 \h 4**](#_Toc86421099) + +[*Convex versus non-Convex Optimization Problems PAGEREF _Toc86421100 \h 5*](#_Toc86421100) + +[*Local and global optimization: PAGEREF _Toc86421101 \h 5*](#_Toc86421101) + +[Local optimization PAGEREF _Toc86421102 \h 5](#_Toc86421102) + +[Global optimization PAGEREF _Toc86421103 \h 6](#_Toc86421103) + +[*Locally optimal for a convex optimization problem PAGEREF _Toc86421104 \h 6*](#_Toc86421104) + +[*Hessian being positive semi-definite iff convexity (Theorem one: One dimensional) PAGEREF _Toc86421105 \h 6*](#_Toc86421105) + +[*Hessian being positive semi-definite iff convexity (Theorem two: Multi-dimensional) PAGEREF _Toc86421106 \h 7*](#_Toc86421106) + +[**Gradient PAGEREF _Toc86421107 \h 8**](#_Toc86421107) + +[*Gradient Descent PAGEREF _Toc86421108 \h 8*](#_Toc86421108) + +[*Challenges with gradient descent PAGEREF _Toc86421109 \h 10*](#_Toc86421109) + +[Local minima and saddle points PAGEREF _Toc86421110 \h 10](#_Toc86421110) + +[Vanishing and Exploding Gradients PAGEREF _Toc86421111 \h 10](#_Toc86421111) + +[**Conclusion PAGEREF _Toc86421112 \h 11**](#_Toc86421112) + +[**References PAGEREF _Toc86421113 \h 11**](#_Toc86421113) + + +# +# + + +# Introduction +In real life optimization problems, we mostly face continuous environments which are made of continuous variables such as time and location. Thus, in this lecture note we will discuss some useful search methods in continuous spaces. The typical problems are linear programming (LP) and diverse kinds of nonlinear programming (such as convex programming, quadratic programming or semidefinite programming). In this lecture note we will discuss different types of optimization techniques in continuous spaces, the differences between different types of optimization techniques, how convexity will help us solve optimization problems easier, difference between local and global optimization, gradient descent method and the challenges with finding the best learning rate. + +# Type of optimization techniques +An essential step to optimization technique is to categorize the optimization model since the algorithms used for solving optimization problems are customized as per the nature of the problem. Various optimization problem types are: +## Linear programming (LP) and Nonlinear programming (NLP) +Linear programming is a method to achieve the best outcome in a mathematical model whose requirements are represented by linear relationships whereas nonlinear programming is a process of solving an optimization problem where the constraints or the objective functions are nonlinear. Thus, this is the main difference between linear and nonlinear programming. +## Continuous optimization and Integer programming (IP) +In order to conceptualize the difference, linear programming can solve problems about minimizing (or maximizing) an objective function by continuous variables. For instance, maybe the optimal solution for a problem to be x1=5,46 and x2=2,65. But in integer programming we can only use integers as variables and we can’t have fractional numbers. +## Constrained optimization and Unconstrained optimization +**Constrained optimization problems** consider the problem of optimizing an objective function subject to constraints on the variables. In general terms, + +minimize fx + +subject to cix=0 ∀i ϵ ε + +cix≤0 ∀i ϵ τ + +Where *f and* the functions ci(x) are all smooth, real-valued functions on a subset of Rn and ε and τ are index sets for equality and inequality constraints, respectively. The feasible set is the set of points x that satisfy the constraints. + +We denote the set of points for which all the constraints are satisfied as C, and say that any x ∈ C (resp. x ∈/ C) is feasible (resp. infeasible) + +***Unconstrained optimization problems*** the answers are constrained into being subject of set C as the picture bellow shows: + +![](Aspose.Words.f2571b5b-5e5e-42a8-b357-02f03b593c59.001.png) + + + + + +*Figure SEQ Figure \\* ARABIC 1. constrained vs unconstrained optimization* +![](Aspose.Words.f2571b5b-5e5e-42a8-b357-02f03b593c59.002.png) + +## Differentiable optimization and Non-differentiable optimization +*Figure SEQ Figure \\* ARABIC 2. Non-differentiable function* +![](Aspose.Words.f2571b5b-5e5e-42a8-b357-02f03b593c59.003.png)Non-differentiable optimization is a category of optimization that deals with objective that for a variety of reasons is non-differentiable and thus non-convex. The functions in this class of optimization are generally non-smooth. These functions although continuous often contain sharp points or corners that do not allow for the solution of a tangent and are thus non-differentiable. In practice non-differentiable optimization encompasses a large variety of problems and a single one-size fits all solution is not applicable however solution is often reached through implementation of the sub gradient method. Non-differentiable functions often arise in real world applications and commonly in the field of economics where cost functions often include sharp points. + +![](Aspose.Words.f2571b5b-5e5e-42a8-b357-02f03b593c59.004.png) +### Cost functions +In many cases, particularly economics the cost function which is the objective function of an optimization problem is non-differentiable. These non-smooth cost functions may include discontinuities and discontinuous gradients and are often seen in discontinuous physical processes. Optimal solution of these cost functions is a matter of importance to economists but presents a variety of issues when using numerical methods thus leading to the need for special solution methods. + +In this lecture we don’t discuss non-differential optimization and non-smooth functions and the text above was for introduction and further information on this topic. + +# Convexity +A set C is convex if for any two points x,y ∈C ,αx+1-αy∈C for all α∈0,1, i.e., all points on the line between x and y also lie in C. A function f is convex if its domain C is convex and + +fαx+1-αy≤αfx+1-αfy (\*) + +*Figure SEQ Figure \\* ARABIC 3. In convex function f, for every two point x,y∈domainf, the line segment between them lies above the graph of f.* +![](Aspose.Words.f2571b5b-5e5e-42a8-b357-02f03b593c59.005.png)![](Aspose.Words.f2571b5b-5e5e-42a8-b357-02f03b593c59.006.jpeg)for all α∈0,1, i.e., the line segment betweenx,fxandy,fy, which is the chord from x to y, lies above the graph of f. + + +For an affine function we always have equality in (\*), so all affine (and therefore also linear) functions are both convex and concave. Conversely, any function that is convex and concave is affine. + +A function is convex if and only if it is convex when restricted to any line that intersects its domain. In other words, f is convex if and only if for all x ∈ domain (f) and all v, the function gt=fx+tv is convex (on its domain, {t | x + t v ∈ domain (f) }). This property is very useful, since it allows us to check whether a function is convex by restricting it to a line. + +*Figure SEQ Figure \\* ARABIC 4. A Non-convex function vs a Convex function* + +![](Aspose.Words.f2571b5b-5e5e-42a8-b357-02f03b593c59.007.png)![](Aspose.Words.f2571b5b-5e5e-42a8-b357-02f03b593c59.008.png) +## Convex versus non-Convex Optimization Problems +*Figure SEQ Figure \\* ARABIC 5. convex region* +![](Aspose.Words.f2571b5b-5e5e-42a8-b357-02f03b593c59.009.png)![](Aspose.Words.f2571b5b-5e5e-42a8-b357-02f03b593c59.010.png)A *convex optimization problem* is a problem where all of the constraints are convex functions, and the objective is a convex function if minimizing, or a concave function if maximizing. In a convex optimization problem, the feasible region is a convex region, as pictured below. + +With a convex objective and a convex feasible region, there can be only one optimal solution, which is globally optimal.  Several methods will either find the globally optimal solution, or prove that there is no feasible solution to the problem.  Convex problems can be solved efficiently up to very large size. + +*Figure SEQ Figure \\* ARABIC 6. Non-convex region* + +![](Aspose.Words.f2571b5b-5e5e-42a8-b357-02f03b593c59.011.png)![](Aspose.Words.f2571b5b-5e5e-42a8-b357-02f03b593c59.012.png)A *non-convex optimization problem* is any problem where the objective or any of the constraints are non-convex, as pictured below. + +Such a problem may have multiple feasible regions and multiple locally optimal points within each region.  It can take time exponential in the number of variables and constraints to determine that a non-convex problem is infeasible, that the objective function is unbounded, or that an optimal solution is the "global optimum" across all feasible regions. +## +## Local and global optimization: +### Local optimization +In local optimization, the compromise is to give up seeking the optimal x, which minimizes the objective over all feasible points. Instead we seek a point that is only locally optimal, which means that it minimizes the objective function among feasible points that are near it, but is not guaranteed to have a lower objective value than all other feasible points. + +Local optimization methods can be fast, can handle large-scale problems, and are widely applicable, since they only require differentiability of the objective and constraint functions. + +There are several disadvantages of local optimization methods, beyond (possibly) not finding the true, globally optimal solution. The methods require an initial guess for the optimization variable. This initial guess or starting point is critical, and can greatly affect the objective value of the local solution obtained. +### Global optimization +In global optimization, the true global solution of the optimization problem is found; the compromise is efficiency. The worst-case complexity of global optimization methods grows exponentially with the problem size. + +The hope is that in practice, for the particular problem instances encountered, the method is far faster. While this favorable situation does occur, it is not typical. Even small problems, with a few tens of variables, can take a very long time (e.g., hours or days) to solve. Global optimization is used for problems with a small number of variables, where computing time is not critical, and the value of finding the true global solution is very high. +## +## Locally optimal for a convex optimization problem +A fundamental property of convex optimization problems is that any locally optimal point is also (globally) optimal. To see this, suppose that x is locally optimal for a convex optimization problem, i.e., x is feasible and + +f\_0(x) = inf{f\_0(z) | z   feasible, kz - xk^2 ≤ R} , (@) + +for some R>0. Now suppose that x is not globally optimal, i.e., there is a feasible y such that f0yR, since otherwise f0y + +[2] Vandenberghe. + +[3] Hladík, Milan. *Discrete and Continuous Optimization*. + +[4] Stein, Oliver. *What is continuous optimization?* Institute of Operations Research Karlsruhe Institute of Technology (KIT) + +[5] Patriksson, Michael. Evgrafov, Anton. Andreasson, Niclas. *An Introduction to Continuous Optimization: Foundations and Fundamental Algorithms*. + +[6] Wolkowicz, Henry. *Continuous Optimization and its Applications.* + + From 36e2084119af1a33af4c0756d395f26fb18c7ecd Mon Sep 17 00:00:00 2001 From: royaghavami <59202308+royaghavami@users.noreply.github.com> Date: Sun, 31 Oct 2021 13:26:35 +0330 Subject: [PATCH 12/57] Add files via upload --- ...f2571b5b-5e5e-42a8-b357-02f03b593c59.001.png | Bin 0 -> 29430 bytes ...f2571b5b-5e5e-42a8-b357-02f03b593c59.002.png | Bin 0 -> 1570 bytes ...f2571b5b-5e5e-42a8-b357-02f03b593c59.003.png | Bin 0 -> 1527 bytes ...f2571b5b-5e5e-42a8-b357-02f03b593c59.004.png | Bin 0 -> 8171 bytes ...f2571b5b-5e5e-42a8-b357-02f03b593c59.005.png | Bin 0 -> 3733 bytes ...2571b5b-5e5e-42a8-b357-02f03b593c59.006.jpeg | Bin 0 -> 7908 bytes ...f2571b5b-5e5e-42a8-b357-02f03b593c59.007.png | Bin 0 -> 1698 bytes ...f2571b5b-5e5e-42a8-b357-02f03b593c59.008.png | Bin 0 -> 13061 bytes 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Date: Fri, 12 Nov 2021 20:54:53 +0330 Subject: [PATCH 15/57] Delete Aspose.Words.f2571b5b-5e5e-42a8-b357-02f03b593c59.021.png --- ....f2571b5b-5e5e-42a8-b357-02f03b593c59.021.png | Bin 2269 -> 0 bytes 1 file changed, 0 insertions(+), 0 deletions(-) delete mode 100644 notebooks/6_search_in_continuous_spaces/Aspose.Words.f2571b5b-5e5e-42a8-b357-02f03b593c59.021.png diff --git a/notebooks/6_search_in_continuous_spaces/Aspose.Words.f2571b5b-5e5e-42a8-b357-02f03b593c59.021.png b/notebooks/6_search_in_continuous_spaces/Aspose.Words.f2571b5b-5e5e-42a8-b357-02f03b593c59.021.png deleted file mode 100644 index e95cd3f22a1bbdf48b2647c9cdeaa8f875a67ad4..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 2269 zcmb_ei#yYO8~;gfQBlevR)_~V)RK%u9>P#0wwmLTBAYXF$f<{NKDI;@favjYdyTH_-XHXgD&kxIL*N9n3 z=d1s4R8{|jOn{cB+MIf3eoUfkpPQ8PRdGD^48&0Z5?(t>o8p}+@mysUQX23bkns%> z&xqBM(EgZ#LM4L;-DxWEV1lw(LS&MKN{%&h#O%mkvm?K*-xpO8OHej1KLDiJi!*9U 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Nonlinear programming (NLP) PAGEREF _Toc86421094 \h 2*](#_Toc86421094) - -[*Continuous optimization and Integer programming (IP) PAGEREF _Toc86421095 \h 2*](#_Toc86421095) - -[*Constrained optimization and Unconstrained optimization PAGEREF _Toc86421096 \h 2*](#_Toc86421096) - -[*Differentiable optimization and Non-differentiable optimization PAGEREF _Toc86421097 \h 3*](#_Toc86421097) - -[Cost functions PAGEREF _Toc86421098 \h 3](#_Toc86421098) - -[**Convexity PAGEREF _Toc86421099 \h 4**](#_Toc86421099) - -[*Convex versus non-Convex Optimization Problems PAGEREF _Toc86421100 \h 5*](#_Toc86421100) - -[*Local and global optimization: PAGEREF _Toc86421101 \h 5*](#_Toc86421101) - -[Local optimization PAGEREF _Toc86421102 \h 5](#_Toc86421102) - -[Global optimization PAGEREF _Toc86421103 \h 6](#_Toc86421103) - -[*Locally optimal for a convex optimization problem PAGEREF _Toc86421104 \h 6*](#_Toc86421104) - -[*Hessian being positive semi-definite iff convexity (Theorem one: One dimensional) PAGEREF _Toc86421105 \h 6*](#_Toc86421105) - -[*Hessian being positive semi-definite iff convexity (Theorem two: Multi-dimensional) PAGEREF _Toc86421106 \h 7*](#_Toc86421106) - -[**Gradient PAGEREF _Toc86421107 \h 8**](#_Toc86421107) - -[*Gradient Descent PAGEREF _Toc86421108 \h 8*](#_Toc86421108) - -[*Challenges with gradient descent PAGEREF _Toc86421109 \h 10*](#_Toc86421109) - -[Local minima and saddle points PAGEREF _Toc86421110 \h 10](#_Toc86421110) - -[Vanishing and Exploding Gradients PAGEREF _Toc86421111 \h 10](#_Toc86421111) - -[**Conclusion PAGEREF _Toc86421112 \h 11**](#_Toc86421112) - -[**References PAGEREF _Toc86421113 \h 11**](#_Toc86421113) - - -# -# - - -# Introduction -In real life optimization problems, we mostly face continuous environments which are made of continuous variables such as time and location. Thus, in this lecture note we will discuss some useful search methods in continuous spaces. The typical problems are linear programming (LP) and diverse kinds of nonlinear programming (such as convex programming, quadratic programming or semidefinite programming). In this lecture note we will discuss different types of optimization techniques in continuous spaces, the differences between different types of optimization techniques, how convexity will help us solve optimization problems easier, difference between local and global optimization, gradient descent method and the challenges with finding the best learning rate. - -# Type of optimization techniques -An essential step to optimization technique is to categorize the optimization model since the algorithms used for solving optimization problems are customized as per the nature of the problem. Various optimization problem types are: -## Linear programming (LP) and Nonlinear programming (NLP) -Linear programming is a method to achieve the best outcome in a mathematical model whose requirements are represented by linear relationships whereas nonlinear programming is a process of solving an optimization problem where the constraints or the objective functions are nonlinear. Thus, this is the main difference between linear and nonlinear programming. -## Continuous optimization and Integer programming (IP) -In order to conceptualize the difference, linear programming can solve problems about minimizing (or maximizing) an objective function by continuous variables. For instance, maybe the optimal solution for a problem to be x1=5,46 and x2=2,65. But in integer programming we can only use integers as variables and we can’t have fractional numbers. -## Constrained optimization and Unconstrained optimization -**Constrained optimization problems** consider the problem of optimizing an objective function subject to constraints on the variables. In general terms, - -minimize fx - -subject to cix=0 ∀i ϵ ε - -cix≤0 ∀i ϵ τ - -Where *f and* the functions ci(x) are all smooth, real-valued functions on a subset of Rn and ε and τ are index sets for equality and inequality constraints, respectively. The feasible set is the set of points x that satisfy the constraints. - -We denote the set of points for which all the constraints are satisfied as C, and say that any x ∈ C (resp. x ∈/ C) is feasible (resp. infeasible) - -***Unconstrained optimization problems*** the answers are constrained into being subject of set C as the picture bellow shows: - -![](Aspose.Words.f2571b5b-5e5e-42a8-b357-02f03b593c59.001.png) - - - - - -*Figure SEQ Figure \\* ARABIC 1. constrained vs unconstrained optimization* -![](Aspose.Words.f2571b5b-5e5e-42a8-b357-02f03b593c59.002.png) - -## Differentiable optimization and Non-differentiable optimization -*Figure SEQ Figure \\* ARABIC 2. Non-differentiable function* -![](Aspose.Words.f2571b5b-5e5e-42a8-b357-02f03b593c59.003.png)Non-differentiable optimization is a category of optimization that deals with objective that for a variety of reasons is non-differentiable and thus non-convex. The functions in this class of optimization are generally non-smooth. These functions although continuous often contain sharp points or corners that do not allow for the solution of a tangent and are thus non-differentiable. In practice non-differentiable optimization encompasses a large variety of problems and a single one-size fits all solution is not applicable however solution is often reached through implementation of the sub gradient method. Non-differentiable functions often arise in real world applications and commonly in the field of economics where cost functions often include sharp points. - -![](Aspose.Words.f2571b5b-5e5e-42a8-b357-02f03b593c59.004.png) -### Cost functions -In many cases, particularly economics the cost function which is the objective function of an optimization problem is non-differentiable. These non-smooth cost functions may include discontinuities and discontinuous gradients and are often seen in discontinuous physical processes. Optimal solution of these cost functions is a matter of importance to economists but presents a variety of issues when using numerical methods thus leading to the need for special solution methods. - -In this lecture we don’t discuss non-differential optimization and non-smooth functions and the text above was for introduction and further information on this topic. - -# Convexity -A set C is convex if for any two points x,y ∈C ,αx+1-αy∈C for all α∈0,1, i.e., all points on the line between x and y also lie in C. A function f is convex if its domain C is convex and - -fαx+1-αy≤αfx+1-αfy (\*) - -*Figure SEQ Figure \\* ARABIC 3. In convex function f, for every two point x,y∈domainf, the line segment between them lies above the graph of f.* -![](Aspose.Words.f2571b5b-5e5e-42a8-b357-02f03b593c59.005.png)![](Aspose.Words.f2571b5b-5e5e-42a8-b357-02f03b593c59.006.jpeg)for all α∈0,1, i.e., the line segment betweenx,fxandy,fy, which is the chord from x to y, lies above the graph of f. - - -For an affine function we always have equality in (\*), so all affine (and therefore also linear) functions are both convex and concave. Conversely, any function that is convex and concave is affine. - -A function is convex if and only if it is convex when restricted to any line that intersects its domain. In other words, f is convex if and only if for all x ∈ domain (f) and all v, the function gt=fx+tv is convex (on its domain, {t | x + t v ∈ domain (f) }). This property is very useful, since it allows us to check whether a function is convex by restricting it to a line. - -*Figure SEQ Figure \\* ARABIC 4. A Non-convex function vs a Convex function* - -![](Aspose.Words.f2571b5b-5e5e-42a8-b357-02f03b593c59.007.png)![](Aspose.Words.f2571b5b-5e5e-42a8-b357-02f03b593c59.008.png) -## Convex versus non-Convex Optimization Problems -*Figure SEQ Figure \\* ARABIC 5. convex region* -![](Aspose.Words.f2571b5b-5e5e-42a8-b357-02f03b593c59.009.png)![](Aspose.Words.f2571b5b-5e5e-42a8-b357-02f03b593c59.010.png)A *convex optimization problem* is a problem where all of the constraints are convex functions, and the objective is a convex function if minimizing, or a concave function if maximizing. In a convex optimization problem, the feasible region is a convex region, as pictured below. - -With a convex objective and a convex feasible region, there can be only one optimal solution, which is globally optimal.  Several methods will either find the globally optimal solution, or prove that there is no feasible solution to the problem.  Convex problems can be solved efficiently up to very large size. - -*Figure SEQ Figure \\* ARABIC 6. Non-convex region* - -![](Aspose.Words.f2571b5b-5e5e-42a8-b357-02f03b593c59.011.png)![](Aspose.Words.f2571b5b-5e5e-42a8-b357-02f03b593c59.012.png)A *non-convex optimization problem* is any problem where the objective or any of the constraints are non-convex, as pictured below. - -Such a problem may have multiple feasible regions and multiple locally optimal points within each region.  It can take time exponential in the number of variables and constraints to determine that a non-convex problem is infeasible, that the objective function is unbounded, or that an optimal solution is the "global optimum" across all feasible regions. -## -## Local and global optimization: -### Local optimization -In local optimization, the compromise is to give up seeking the optimal x, which minimizes the objective over all feasible points. Instead we seek a point that is only locally optimal, which means that it minimizes the objective function among feasible points that are near it, but is not guaranteed to have a lower objective value than all other feasible points. - -Local optimization methods can be fast, can handle large-scale problems, and are widely applicable, since they only require differentiability of the objective and constraint functions. - -There are several disadvantages of local optimization methods, beyond (possibly) not finding the true, globally optimal solution. The methods require an initial guess for the optimization variable. This initial guess or starting point is critical, and can greatly affect the objective value of the local solution obtained. -### Global optimization -In global optimization, the true global solution of the optimization problem is found; the compromise is efficiency. The worst-case complexity of global optimization methods grows exponentially with the problem size. - -The hope is that in practice, for the particular problem instances encountered, the method is far faster. While this favorable situation does occur, it is not typical. Even small problems, with a few tens of variables, can take a very long time (e.g., hours or days) to solve. Global optimization is used for problems with a small number of variables, where computing time is not critical, and the value of finding the true global solution is very high. -## -## Locally optimal for a convex optimization problem -A fundamental property of convex optimization problems is that any locally optimal point is also (globally) optimal. To see this, suppose that x is locally optimal for a convex optimization problem, i.e., x is feasible and - -f\_0(x) = inf{f\_0(z) | z   feasible, kz - xk^2 ≤ R} , (@) - -for some R>0. Now suppose that x is not globally optimal, i.e., there is a feasible y such that f0yR, since otherwise f0y - -[2] Vandenberghe. - -[3] Hladík, Milan. *Discrete and Continuous Optimization*. - -[4] Stein, Oliver. *What is continuous optimization?* Institute of Operations Research Karlsruhe Institute of Technology (KIT) - -[5] Patriksson, Michael. Evgrafov, Anton. Andreasson, Niclas. *An Introduction to Continuous Optimization: Foundations and Fundamental Algorithms*. - -[6] Wolkowicz, Henry. *Continuous Optimization and its Applications.* - - From b2b256d18ef964c3288b4db0f676b1bb00b3a26b Mon Sep 17 00:00:00 2001 From: royaghavami <59202308+royaghavami@users.noreply.github.com> Date: Fri, 12 Nov 2021 20:59:10 +0330 Subject: [PATCH 36/57] Add files via upload --- .../6_search_in_continuous_spaces/index.md | 332 ++++++++++++++++++ 1 file changed, 332 insertions(+) create mode 100644 notebooks/6_search_in_continuous_spaces/index.md diff --git a/notebooks/6_search_in_continuous_spaces/index.md b/notebooks/6_search_in_continuous_spaces/index.md new file mode 100644 index 00000000..4458dd04 --- /dev/null +++ b/notebooks/6_search_in_continuous_spaces/index.md @@ -0,0 +1,332 @@ + +Search in Continuous Space +# +# **Table of Contents** +` `**TOC \o "1-3" \h \z \u [Introduction PAGEREF _Toc87638545 \h 2**](#_Toc87638545)** + +[**Types of optimization techniques PAGEREF _Toc87638546 \h 3**](#_Toc87638546) + +[*Linear programming (LP) and Nonlinear programming (NLP) PAGEREF _Toc87638547 \h 3*](#_Toc87638547) + +[*Continuous optimization and Integer programming (IP) PAGEREF _Toc87638548 \h 3*](#_Toc87638548) + +[*Constrained optimization and Unconstrained optimization PAGEREF _Toc87638549 \h 3*](#_Toc87638549) + +[*Differentiable optimization and Non-differentiable optimization PAGEREF _Toc87638550 \h 4*](#_Toc87638550) + +[Cost functions PAGEREF _Toc87638551 \h 4](#_Toc87638551) + +[**Convexity PAGEREF _Toc87638552 \h 4**](#_Toc87638552) + +[*Convex versus non-Convex Optimization Problems PAGEREF _Toc87638553 \h 6*](#_Toc87638553) + +[*Local and global optimization: PAGEREF _Toc87638554 \h 7*](#_Toc87638554) + +[Local optimization PAGEREF _Toc87638555 \h 7](#_Toc87638555) + +[Global optimization PAGEREF _Toc87638556 \h 7](#_Toc87638556) + +[*Locally optimal for a convex optimization problem PAGEREF _Toc87638557 \h 8*](#_Toc87638557) + +[*Hessian being positive semi-definite iff convexity (Theorem one: One dimensional) PAGEREF _Toc87638558 \h 9*](#_Toc87638558) + +[*Hessian being positive semi-definite iff convexity (Theorem two: Multi-dimensional) PAGEREF _Toc87638559 \h 9*](#_Toc87638559) + +[*Examples of Convex Functions: PAGEREF _Toc87638560 \h 10*](#_Toc87638560) + +[**Gradient PAGEREF _Toc87638561 \h 11**](#_Toc87638561) + +[*Gradient Descent PAGEREF _Toc87638562 \h 11*](#_Toc87638562) + +[*Challenges with gradient descent PAGEREF _Toc87638563 \h 14*](#_Toc87638563) + +[Local minima and saddle points PAGEREF _Toc87638564 \h 14](#_Toc87638564) + +[Vanishing and Exploding Gradients PAGEREF _Toc87638565 \h 14](#_Toc87638565) + +[**Conclusion PAGEREF _Toc87638566 \h 15**](#_Toc87638566) + +[**References PAGEREF _Toc87638567 \h 15**](#_Toc87638567) + + +# +# + + +# Introduction +In real life optimization problems, we mostly face continuous environments which are made of continuous variables such as time and location. Thus, in this lecture note we will discuss some useful search methods in continuous spaces. The typical problems are linear programming (LP) and diverse kinds of nonlinear programming (such as convex programming, quadratic programming or semidefinite programming). In this lecture note we will discuss different types of optimization techniques in continuous spaces, the differences between different types of optimization techniques, how convexity will help us solve optimization problems easier, difference between local and global optimization, gradient descent method and the challenges with finding the best learning rate. + +**Example.1** (Weber Point) Given a collection of cities (assume on 2D plane) how can we find the location that minimizes the sum of distances to all cities? We model this problem by denoting the locations of the cities as y(1), …,y(m) and write the optimization problem as: + +![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.001.jpeg) + +**Example.2** (Image deblurring and denoising) Given corrupted imageY∈Rm×n, reconstruct the image by solving the optimization: + +![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.002.jpeg) + +where K \* denotes convolution with a blurring filter. + +![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.003.jpeg) + +*Figure SEQ Figure \\* ARABIC 1. reconstructed image after solving the optimization* + +# Types of optimization techniques +An essential step to optimization technique is to categorize the optimization model since the algorithms used for solving optimization problems are customized as per the nature of the problem. Various optimization problem types are: +## Linear programming (LP) and Nonlinear programming (NLP) +Linear programming is a method to achieve the best outcome in a mathematical model whose requirements are represented by linear relationships whereas nonlinear programming is a process of solving an optimization problem where the constraints or the objective functions are nonlinear. Thus, this is the main difference between linear and nonlinear programming. +## Continuous optimization and Integer programming (IP) +In order to conceptualize the difference, linear programming can solve problems about minimizing (or maximizing) an objective function by continuous variables. For instance, maybe the optimal solution for a problem to be x1=5,46 and x2=2,65. But in integer programming we can only use integers as variables and we can’t have fractional numbers. +## Constrained optimization and Unconstrained optimization +A constrained optimization problem is the problem of optimizing an objective function subject to constraints on the variables. In general terms, + +![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.004.png) + +Where f* and the functions ci(x) are all smooth, real-valued functions on a subset of Rn and ε and τ are index sets for equality and inequality constraints, respectively. The feasible set is the set of points x that satisfy the constraints. + +We denote the set of points for which all the constraints are satisfied as C, and say that any x ∈ C (resp. x ∈/ C) is feasible (resp. infeasible) + +In unconstrained optimization problems the answers are constrained into being subject of set C as the picture bellow shows: + +*Figure SEQ Figure \\* ARABIC 2. constrained vs unconstrained optimization* + +![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.005.png) + + + +## +## Differentiable optimization and Non-differentiable optimization +Non-differentiable optimization is a category of optimization that deals with objective that for a variety of reasons is non-differentiable and thus non-convex. The functions in this class of optimization are generally non-smooth. These functions although continuous often contain sharp points or corners that do not allow for the solution of a tangent and are thus non-differentiable. In practice non-differentiable optimization encompasses a large variety of problems and a single one-size fits all solution is not applicable however solution is often reached through implementation of the sub gradient method. Non-differentiable functions often arise in real world applications and commonly in the field of economics where cost functions often include sharp points. + +*Figure SEQ Figure \\* ARABIC 3. Non-differentiable function* +![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.006.png) +### Cost functions +In many cases, particularly economics the cost function which is the objective function of an optimization problem is non-differentiable. These non-smooth cost functions may include discontinuities and discontinuous gradients and are often seen in discontinuous physical processes. Optimal solution of these cost functions is a matter of importance to economists but presents a variety of issues when using numerical methods thus leading to the need for special solution methods. + +In this lecture we don’t discuss non-differential optimization and non-smooth functions and the text above was for introduction and further information on this topic. If you want to know more about it, see these links: [nondifferentiable optimization via approximation](https://www.mit.edu/~dimitrib/Nondiff_Approx.pdf) and [non-convex optimization](https://www.cs.cornell.edu/courses/cs6787/2017fa/Lecture7.pdf). + +# Convexity +A set C is convex if for any two points x,y ∈C , αx + (1 – α)y ∈ C for all α∈ [0,1], i.e., all points on the line between x and y also lie in C. Some examples of convex sets are: + +- All points C = Rn where n is a non-negative integer +- Intersection of variable number of convex sets +- Intervals C = {x ∈ Rn | l ≤ x ≤ u} + +A function f is convex if its domain C is convex and + +![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.007.png) + +for all α∈ [0,1], i.e., the line segment between (x, f(x)) and (y, f(y))* , which is the chord from x to y, lies above the graph of f. + + + +*Figure SEQ Figure \\* ARABIC 4. In convex function f, for every two point x,y∈domainf, the line segment between them lies above the graph of f.* +![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.008.png) + +For an affine function we always have equality in (\*), so all affine (and therefore also linear) functions are both convex and concave. Conversely, any function that is convex and concave is affine. + +A function is convex if and only if it is convex when restricted to any line that intersects its domain. In other words, f is convex if and only if for all x ∈ domain (f) and all v, the function g(t) = f(x + tv) is convex (on its domain, {t| x + tv ∈ domain(f) }* ). This property is very useful, since it allows us to check whether a function is convex by restricting it to a line. + +*Figure SEQ Figure \\* ARABIC 5. A Non-convex function vs a Convex function* +![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.009.png) + +Some examples of convex functions are: + +- Exponential functions: f(x) = exp(ax), a ∈ R +- Negative logarithm: f(x) = -log(x), x > 0 +- Euclidean norm: f(x) = ||x||2 + +## Convex versus non-Convex Optimization Problems +A convex optimization *problem* is a problem where all of the constraints are convex functions, and the objective is a convex function if minimizing, or a concave function if maximizing. In a convex optimization problem, the feasible region is a convex region, as pictured below. + + +*Figure SEQ Figure \\* ARABIC 6. convex region* +![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.010.png) + + + +With a convex objective and a convex feasible region, there can be only one optimal solution, which is globally optimal.  Several methods will either find the globally optimal solution, or prove that there is no feasible solution to the problem.  Convex problems can be solved efficiently up to very large size. + +A *non-convex optimization problem* is any problem where the objective or any of the constraints are non-convex, as pictured below. + +*Figure SEQ Figure \\* ARABIC 7. Non-convex region* + +![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.011.png) + +Such a problem may have multiple feasible regions and multiple locally optimal points within each region.  It can take time exponential in the number of variables and constraints to determine that a non-convex problem is infeasible, that the objective function is unbounded, or that an optimal solution is the "global optimum" across all feasible regions. +## +## Local and global optimization: +### Local optimization +In local optimization, the compromise is to give up seeking the optimal x, which minimizes the objective over all feasible points. Instead we seek a point that is only locally optimal, which means that it minimizes the objective function among feasible points that are near it, but is not guaranteed to have a lower objective value than all other feasible points. + +Local optimization methods can be fast, can handle large-scale problems, and are widely applicable, since they only require differentiability of the objective and constraint functions. + +There are several disadvantages of local optimization methods, beyond (possibly) not finding the true, globally optimal solution. The methods require an initial guess for the optimization variable. This initial guess or starting point is critical, and can greatly affect the objective value of the local solution obtained. +### Global optimization +In global optimization, the true global solution of the optimization problem is found; the compromise is efficiency. The worst-case complexity of global optimization methods grows exponentially with the problem size. + +The hope is that in practice, for the particular problem instances encountered, the method is far faster. While this favorable situation does occur, it is not typical. Even small problems, with a few tens of variables, can take a very long time (e.g., hours or days) to solve. Global optimization is used for problems with a small number of variables, where computing time is not critical, and the value of finding the true global solution is very high. + +So, to summarize, A maximum or minimum is said to be local if it is the largest or smallest value of the function, respectively, within a given range. However, a maximum or minimum is said to be global if it is the largest or smallest value of the function, respectively, on the entire domain of a function. The image shows it clearly. + +![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.012.png) + +*Figure SEQ Figure \\* ARABIC 8. local and global maximum and minimum* +## +## Locally optimal for a convex optimization problem +A fundamental property of convex optimization problems is that any locally optimal point is also (globally) optimal. To see this, suppose that x is locally optimal for a convex optimization problem, i.e., x is feasible and + +![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.013.png) + +for some R > 0. Now suppose that x is not globally optimal, i.e., there is a feasible y such that f0(y) < f0(x). Evidently ||y – x||2 > R, since otherwise f0(y) < f0(x). Consider the point z given by + +![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.014.png) + +Then we have ||z – x|| = R/2 < R, and by convexity of the feasible set, z is feasible. By convexity of f0 we have + +![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.015.png) + +which contradicts (@). Hence there exists no feasible y with f0(y) < f0(x), i.e., x is globally optimal. +## Hessian being positive semi-definite iff convexity (Theorem one: One dimensional) +One way of checking a function’s convexity is by checking its Hessian matrix. let Ø ≠ M ⊆ Rn be a convex set and let f(x) be a function differentiable on an open superset of M then f(x) is convex on M if and only if for every x1,x2 ∈ M + +![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.016.png) + +Proof: ⇒ let x1,x2 ∈ M and λ∈ (0,1) be arbitrary. Then + +![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.017.png) + +By the limit transition λ ⟶0 we get (\*) utilizing the chain rule for the derivative of a composite function g(λ) = g(x1 + λ(x2 – x1)) with respect to λ. + +⇐ Let x1,x2 ∈ M and consider a convex combination x = λ1x1 + λ2x2 by (\*) we have: + +![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.018.png) + +Multiply the first inequality by λ1, the one by λ2 and summing up we get + +![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.019.png) +## Hessian being positive semi-definite iff convexity (Theorem two: Multi-dimensional) +Now we prove the same property for n-dimensional functions. let Ø ≠ M ⊆ Rn be an open convex set of dimension n, and suppose that a function f: M→R is twice continuously differentiable on M. Then f(x) is convex on M if and only the Hessian ∇2f(x) is positive semi-definite for every x∈M. + +Proof:let x\* ∈ M be arbitrary. Due to continuity of the second partial derivatives we have that for every λ∈R and y ∈ Rn, x\* + λy ∈ M, there is θ∈ (0,1) such that + +![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.020.png) + +⇒ from theorem one we get: + +![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.021.png) + +so, we can conclude that + +![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.022.png) + +By the limit transition λ ⟶0 we have + +![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.023.png) + +⇐ due to positive semi-definitiveness of the Hessian, we have yT∇2 f(x\* + θλy)y ≥ 0. Hence: + +which shows the convexity of f(x) in the view of Theorem one. + +![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.024.png) + +## Examples of Convex Functions: +In this section we check the convexity of two functions using the theorems we learned. + +- f(x) = 2x3 – 18x2 : + +differentiating this function, we have f’’(x) = 12x – 36. The second derivative is negative if x<3 and positive if x>3. Hence, the function is convex on the interval (3,+∞). + +- f(x1,x2, x3) = x12 + 2x22 + 3x32 + 2x1x2 + 2x1x3 + +The Hessian is: + +![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.025.png) + +The leading principal minors of the Hessian are 2 > 0, 4 > 0, and 8 > 0. Therefore the Hessian is positive semi-definite and the function is convex. +# Gradient +A gradient simply measures the change in all weights with regard to the change in error. We can also think of a gradient as the slope of a function that has more than one input variable. The higher the gradient, the steeper the slope and the faster a model can learn. But if the slope is zero, the model stops learning. + +## Gradient Descent +Gradient descent is an optimization algorithm used for training a machine learning model. It works on a convex function and changes the function’s parameters iteratively to find the local minimum (Since the function is convex, by finding local minimum we also find the global minimum). + +*Figure SEQ Figure \\* ARABIC 9. The gradient field of function f = x2 – 4x + y2 + 2y* + +![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.026.png) + +To find the local minimum, we must take steps proportional to the negative of the gradient (move away from the gradient) of the function at the current point. If we take steps proportional to the positive of the gradient, we will approach a local maximum of the function, and the procedure is called Gradient Ascent. + +*Figure SEQ Figure \\* ARABIC 10. The way Gradient Descent works* + +![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.027.png) + +Gradient descent performs two steps iteratively: + +1. **Computes the gradient** (slope), the first order derivative of the function at that point +1. **Moves in the direction opposite to the gradient** + +This algorithm can be written as followed: + +![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.028.png) + +Here Alpha is called Learning rate and determines the length of the steps. For gradient descent to reach the local minimum we must set the learning rate to an appropriate value. If alpha is too high, algorithm may not reach the local minimum because it keeps changing. If we set the learning rate to a very small value, gradient descent will eventually reach the local minimum but it may take a while because the function changes very slowly. + +Note: While moving towards the local minima, both gradient and the size of the step decreases. So, the learning rate can be constant over the optimization. + +*Figure SEQ Figure \\* ARABIC 11. The effect of learning in Gradient Descent* + +![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.029.png) + + +In Figure 10, four different learning rates have been used for a function f. We can make the following observations: + +a) Learning rate is optimal, model converges to the minimum + +b) Learning rate is too small, it takes more time but converges to the minimum + +c) Learning rate is higher than the optimal value, it overshoots but converges ( 1/C < η <2/C) + +*Figure SEQ Figure \\* ARABIC 12. Four different learning rates working on function f* + +![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.030.png)d) Learning rate is very large, it overshoots and diverges, moves away from the minima, performance **decreases on learning** + + +## Challenges with gradient descent +Although gradient descent is the most common approach for optimization problems, it has its own challenges. Some of them include: +### Local minima and saddle points +In nonconvex problems, gradient descent can have difficulties in finding the global minimum. + +In the previous section we mentioned that when the slope of the cost function is at or close to zero, the model stops learning. Other than global minimum, local minima and saddle points can also reach this slope. Local minima have the same shape as the global minimum, where the slope of the cost function increases on either side of the current point. Saddle points, have negative gradient on one side and a non-negative gradient on the other side, causing them to reaching a local maximum on one side and a local minimum on the other. We can use noisy gradients to escape local minimums and saddle points. + +*Figure SEQ Figure \\* ARABIC 13. Local minima and Saddle point can face problems using Gradient Descent* + +![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.031.png) + +### Vanishing and Exploding Gradients +In deeper neural networks we can also encounter two other problems: + +Vanishing gradients: This occurs when the gradient is too small. As we move backwards during backpropagation, the gradient continues to become smaller, so the earlier layers in the network learn more slowly than later layers. When this happens, the weight parameters update until they become insignificant and the algorithm stops learning. + +Exploding gradients: This happens when the gradient is too large and creates an unstable model. In this case, the model weights will grow too large, and they will eventually be represented as NaN. + +# Conclusion +In this lecture we talked about the different types of optimizations in continuous spaces. It has been proven that finding an epsilon-optimal minimizer that has the following relationship with the optimal solution x\* + +![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.032.png) + +is impossible. However, we introduced a feature called convexity which can help us solve some of these problems. We proved that if a function’s Hessian matrix is positive semi-definitive, it is convex and vice versa. Now if a function is convex, its local minimums are also global. So, by using a decent iterative algorithm, like Gradient Descent, we can find its solutions. But keep in mind that this algorithm can face problems if used on non-convex functions or deeper neural networks + +# References +[1] Vandenberghe. + +[2] Vandenberghe. + +[3] Hladík, Milan. *Discrete and Continuous Optimization*. + +[4] Stein, Oliver. *What is continuous optimization?* Institute of Operations Research Karlsruhe Institute of Technology (KIT) + +[5] Patriksson, Michael. Evgrafov, Anton. Andreasson, Niclas. *An Introduction to Continuous Optimization: Foundations and Fundamental Algorithms*. + +[6] Wolkowicz, Henry. *Continuous Optimization and its Applications.* + + + From 2c19ad0b3e34403eb5d764779acf001916e7b258 Mon Sep 17 00:00:00 2001 From: royaghavami <59202308+royaghavami@users.noreply.github.com> Date: Fri, 12 Nov 2021 21:01:46 +0330 Subject: [PATCH 37/57] Add files via upload --- ...3ba005-881a-4c4b-97e3-6ea90eff943d.001.jpeg | Bin 0 -> 3269 bytes ...3ba005-881a-4c4b-97e3-6ea90eff943d.002.jpeg | Bin 0 -> 7871 bytes ...3ba005-881a-4c4b-97e3-6ea90eff943d.003.jpeg | Bin 0 -> 58392 bytes ...53ba005-881a-4c4b-97e3-6ea90eff943d.004.png | Bin 0 -> 9654 bytes ...53ba005-881a-4c4b-97e3-6ea90eff943d.005.png | Bin 0 -> 31471 bytes ...53ba005-881a-4c4b-97e3-6ea90eff943d.006.png | Bin 0 -> 9924 bytes ...53ba005-881a-4c4b-97e3-6ea90eff943d.007.png | Bin 0 -> 7443 bytes ...53ba005-881a-4c4b-97e3-6ea90eff943d.008.png | Bin 0 -> 17318 bytes 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-2,51 +2,51 @@ Search in Continuous Space # # **Table of Contents** -` `**TOC \o "1-3" \h \z \u [Introduction PAGEREF _Toc87638545 \h 2**](#_Toc87638545)** +[Introduction **](#_Toc87638545)** -[**Types of optimization techniques PAGEREF _Toc87638546 \h 3**](#_Toc87638546) +[**Types of optimization techniques **](#_Toc87638546) -[*Linear programming (LP) and Nonlinear programming (NLP) PAGEREF _Toc87638547 \h 3*](#_Toc87638547) +[*Linear programming (LP) and Nonlinear programming (NLP) *](#_Toc87638547) -[*Continuous optimization and Integer programming (IP) PAGEREF _Toc87638548 \h 3*](#_Toc87638548) +[*Continuous optimization and Integer programming (IP) *](#_Toc87638548) -[*Constrained optimization and Unconstrained optimization PAGEREF _Toc87638549 \h 3*](#_Toc87638549) +[*Constrained optimization and Unconstrained optimization *](#_Toc87638549) -[*Differentiable optimization and Non-differentiable optimization PAGEREF _Toc87638550 \h 4*](#_Toc87638550) +[*Differentiable optimization and Non-differentiable optimization *](#_Toc87638550) -[Cost functions PAGEREF _Toc87638551 \h 4](#_Toc87638551) +[Cost functions ](#_Toc87638551) -[**Convexity PAGEREF _Toc87638552 \h 4**](#_Toc87638552) +[**Convexity **](#_Toc87638552) -[*Convex versus non-Convex Optimization Problems PAGEREF _Toc87638553 \h 6*](#_Toc87638553) +[*Convex versus non-Convex Optimization Problems *](#_Toc87638553) -[*Local and global optimization: PAGEREF _Toc87638554 \h 7*](#_Toc87638554) +[*Local and global optimization: *](#_Toc87638554) -[Local optimization PAGEREF _Toc87638555 \h 7](#_Toc87638555) +[Local optimization ](#_Toc87638555) -[Global optimization PAGEREF _Toc87638556 \h 7](#_Toc87638556) +[Global optimization ](#_Toc87638556) -[*Locally optimal for a convex optimization problem PAGEREF _Toc87638557 \h 8*](#_Toc87638557) +[*Locally optimal for a convex optimization problem *](#_Toc87638557) -[*Hessian being positive semi-definite iff convexity (Theorem one: One dimensional) PAGEREF _Toc87638558 \h 9*](#_Toc87638558) +[*Hessian being positive semi-definite iff convexity (Theorem one: One dimensional) *](#_Toc87638558) -[*Hessian being positive semi-definite iff convexity (Theorem two: Multi-dimensional) PAGEREF _Toc87638559 \h 9*](#_Toc87638559) +[*Hessian being positive semi-definite iff convexity (Theorem two: Multi-dimensional) *](#_Toc87638559) -[*Examples of Convex Functions: PAGEREF _Toc87638560 \h 10*](#_Toc87638560) +[*Examples of Convex Functions: *](#_Toc87638560) -[**Gradient PAGEREF _Toc87638561 \h 11**](#_Toc87638561) +[**Gradient **](#_Toc87638561) -[*Gradient Descent PAGEREF _Toc87638562 \h 11*](#_Toc87638562) +[*Gradient Descent *](#_Toc87638562) -[*Challenges with gradient descent PAGEREF _Toc87638563 \h 14*](#_Toc87638563) +[*Challenges with gradient descent *](#_Toc87638563) -[Local minima and saddle points PAGEREF _Toc87638564 \h 14](#_Toc87638564) +[Local minima and saddle points ](#_Toc87638564) -[Vanishing and Exploding Gradients PAGEREF _Toc87638565 \h 14](#_Toc87638565) +[Vanishing and Exploding Gradients ](#_Toc87638565) -[**Conclusion PAGEREF _Toc87638566 \h 15**](#_Toc87638566) +[**Conclusion **](#_Toc87638566) -[**References PAGEREF _Toc87638567 \h 15**](#_Toc87638567) +[**References **](#_Toc87638567) # From dc0328a51ce5e72ecb173a390b1c60d47eeb7f8e Mon Sep 17 00:00:00 2001 From: royaghavami <59202308+royaghavami@users.noreply.github.com> Date: Fri, 12 Nov 2021 22:44:59 +0330 Subject: [PATCH 39/57] Update index.md --- .../6_search_in_continuous_spaces/index.md | 36 +++++++++---------- 1 file changed, 18 insertions(+), 18 deletions(-) diff --git a/notebooks/6_search_in_continuous_spaces/index.md b/notebooks/6_search_in_continuous_spaces/index.md index bf46db3f..d4729fe0 100644 --- a/notebooks/6_search_in_continuous_spaces/index.md +++ b/notebooks/6_search_in_continuous_spaces/index.md @@ -2,51 +2,51 @@ Search in Continuous Space # # **Table of Contents** -[Introduction **](#_Toc87638545)** +[Introduction ](#_Toc87638545)** -[**Types of optimization techniques **](#_Toc87638546) +[Types of optimization techniques ](#_Toc87638546) -[*Linear programming (LP) and Nonlinear programming (NLP) *](#_Toc87638547) +[Linear programming (LP) and Nonlinear programming (NLP) ](#_Toc87638547) -[*Continuous optimization and Integer programming (IP) *](#_Toc87638548) +[Continuous optimization and Integer programming (IP) ](#_Toc87638548) -[*Constrained optimization and Unconstrained optimization *](#_Toc87638549) +[Constrained optimization and Unconstrained optimization ](#_Toc87638549) -[*Differentiable optimization and Non-differentiable optimization *](#_Toc87638550) +[Differentiable optimization and Non-differentiable optimization ](#_Toc87638550) [Cost functions ](#_Toc87638551) -[**Convexity **](#_Toc87638552) +[Convexity ](#_Toc87638552) -[*Convex versus non-Convex Optimization Problems *](#_Toc87638553) +[Convex versus non-Convex Optimization Problems ](#_Toc87638553) -[*Local and global optimization: *](#_Toc87638554) +[Local and global optimization: ](#_Toc87638554) [Local optimization ](#_Toc87638555) [Global optimization ](#_Toc87638556) -[*Locally optimal for a convex optimization problem *](#_Toc87638557) +[Locally optimal for a convex optimization problem ](#_Toc87638557) -[*Hessian being positive semi-definite iff convexity (Theorem one: One dimensional) *](#_Toc87638558) +[Hessian being positive semi-definite iff convexity (Theorem one: One dimensional) ](#_Toc87638558) -[*Hessian being positive semi-definite iff convexity (Theorem two: Multi-dimensional) *](#_Toc87638559) +[Hessian being positive semi-definite iff convexity (Theorem two: Multi-dimensional) ](#_Toc87638559) -[*Examples of Convex Functions: *](#_Toc87638560) +[Examples of Convex Functions: ](#_Toc87638560) -[**Gradient **](#_Toc87638561) +[Gradient ](#_Toc87638561) -[*Gradient Descent *](#_Toc87638562) +[Gradient Descent ](#_Toc87638562) -[*Challenges with gradient descent *](#_Toc87638563) +[Challenges with gradient descent](#_Toc87638563) [Local minima and saddle points ](#_Toc87638564) [Vanishing and Exploding Gradients ](#_Toc87638565) -[**Conclusion **](#_Toc87638566) +[Conclusion ](#_Toc87638566) -[**References **](#_Toc87638567) +[References ](#_Toc87638567) # From 46d1c6eff2fabd0b9d9c6a6b9828736fa7e6da68 Mon Sep 17 00:00:00 2001 From: royaghavami <59202308+royaghavami@users.noreply.github.com> Date: Fri, 12 Nov 2021 22:45:19 +0330 Subject: [PATCH 40/57] Update index.md --- notebooks/6_search_in_continuous_spaces/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/6_search_in_continuous_spaces/index.md b/notebooks/6_search_in_continuous_spaces/index.md index d4729fe0..47494d9f 100644 --- a/notebooks/6_search_in_continuous_spaces/index.md +++ b/notebooks/6_search_in_continuous_spaces/index.md @@ -2,7 +2,7 @@ Search in Continuous Space # # **Table of Contents** -[Introduction ](#_Toc87638545)** +[Introduction ](#_Toc87638545) [Types of optimization techniques ](#_Toc87638546) From 47b1731e900829812d68815170f773ae46f3730b Mon Sep 17 00:00:00 2001 From: royaghavami <59202308+royaghavami@users.noreply.github.com> Date: Fri, 12 Nov 2021 22:48:03 +0330 Subject: [PATCH 41/57] Update index.md --- notebooks/6_search_in_continuous_spaces/index.md | 5 ++++- 1 file changed, 4 insertions(+), 1 deletion(-) diff --git a/notebooks/6_search_in_continuous_spaces/index.md b/notebooks/6_search_in_continuous_spaces/index.md index 47494d9f..38b8e2fe 100644 --- a/notebooks/6_search_in_continuous_spaces/index.md +++ b/notebooks/6_search_in_continuous_spaces/index.md @@ -285,9 +285,12 @@ b) Learning rate is too small, it takes more time but converges to the minimum c) Learning rate is higher than the optimal value, it overshoots but converges ( 1/C < η <2/C) +d) Learning rate is very large, it overshoots and diverges, moves away from the minima, performance **decreases on learning** + + *Figure SEQ Figure \\* ARABIC 12. Four different learning rates working on function f* -![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.030.png)d) Learning rate is very large, it overshoots and diverges, moves away from the minima, performance **decreases on learning** +![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.030.png) ## Challenges with gradient descent From d74caadd8a1b3b3cac5361c141498910d8245962 Mon Sep 17 00:00:00 2001 From: royaghavami <59202308+royaghavami@users.noreply.github.com> Date: Fri, 12 Nov 2021 22:54:48 +0330 Subject: [PATCH 42/57] Update index.md --- notebooks/6_search_in_continuous_spaces/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/6_search_in_continuous_spaces/index.md b/notebooks/6_search_in_continuous_spaces/index.md index 38b8e2fe..8e6bb477 100644 --- a/notebooks/6_search_in_continuous_spaces/index.md +++ b/notebooks/6_search_in_continuous_spaces/index.md @@ -2,7 +2,7 @@ Search in Continuous Space # # **Table of Contents** -[Introduction ](#_Toc87638545) +* [Introduction ](#_Toc87638545) [Types of optimization techniques ](#_Toc87638546) From 69d12acce479530d2e0b611d4d065086c2feb594 Mon Sep 17 00:00:00 2001 From: royaghavami <59202308+royaghavami@users.noreply.github.com> Date: Fri, 12 Nov 2021 22:59:57 +0330 Subject: [PATCH 43/57] Update index.md --- .../6_search_in_continuous_spaces/index.md | 20 +++++++------------ 1 file changed, 7 insertions(+), 13 deletions(-) diff --git a/notebooks/6_search_in_continuous_spaces/index.md b/notebooks/6_search_in_continuous_spaces/index.md index 8e6bb477..28986aa8 100644 --- a/notebooks/6_search_in_continuous_spaces/index.md +++ b/notebooks/6_search_in_continuous_spaces/index.md @@ -2,19 +2,13 @@ Search in Continuous Space # # **Table of Contents** -* [Introduction ](#_Toc87638545) - -[Types of optimization techniques ](#_Toc87638546) - -[Linear programming (LP) and Nonlinear programming (NLP) ](#_Toc87638547) - -[Continuous optimization and Integer programming (IP) ](#_Toc87638548) - -[Constrained optimization and Unconstrained optimization ](#_Toc87638549) - -[Differentiable optimization and Non-differentiable optimization ](#_Toc87638550) - -[Cost functions ](#_Toc87638551) +- [Introduction](#introduction) +- [Types of optimization techniques](#types-of-optimization-techniques) + * [Linear programming (LP) and Nonlinear programming (NLP)](#linear-programming--lp--and-nonlinear-programming--nlp-) + * [Continuous optimization and Integer programming (IP)](#continuous-optimization-and-integer-programming--ip-) + * [Constrained optimization and Unconstrained optimization](#constrained-optimization-and-unconstrained-optimization) + * [Differentiable optimization and Non-differentiable optimization](#differentiable-optimization-and-non-differentiable-optimization) + + [Cost functions](#cost-functions) [Convexity ](#_Toc87638552) From a9b1e498131279da1af8fe3bc014e8de1ccc2b78 Mon Sep 17 00:00:00 2001 From: royaghavami <59202308+royaghavami@users.noreply.github.com> Date: Fri, 12 Nov 2021 23:07:07 +0330 Subject: [PATCH 44/57] Update index.md --- .../6_search_in_continuous_spaces/index.md | 48 +++++++------------ 1 file changed, 16 insertions(+), 32 deletions(-) diff --git a/notebooks/6_search_in_continuous_spaces/index.md b/notebooks/6_search_in_continuous_spaces/index.md index 28986aa8..51df8bcc 100644 --- a/notebooks/6_search_in_continuous_spaces/index.md +++ b/notebooks/6_search_in_continuous_spaces/index.md @@ -9,38 +9,22 @@ Search in Continuous Space * [Constrained optimization and Unconstrained optimization](#constrained-optimization-and-unconstrained-optimization) * [Differentiable optimization and Non-differentiable optimization](#differentiable-optimization-and-non-differentiable-optimization) + [Cost functions](#cost-functions) - -[Convexity ](#_Toc87638552) - -[Convex versus non-Convex Optimization Problems ](#_Toc87638553) - -[Local and global optimization: ](#_Toc87638554) - -[Local optimization ](#_Toc87638555) - -[Global optimization ](#_Toc87638556) - -[Locally optimal for a convex optimization problem ](#_Toc87638557) - -[Hessian being positive semi-definite iff convexity (Theorem one: One dimensional) ](#_Toc87638558) - -[Hessian being positive semi-definite iff convexity (Theorem two: Multi-dimensional) ](#_Toc87638559) - -[Examples of Convex Functions: ](#_Toc87638560) - -[Gradient ](#_Toc87638561) - -[Gradient Descent ](#_Toc87638562) - -[Challenges with gradient descent](#_Toc87638563) - -[Local minima and saddle points ](#_Toc87638564) - -[Vanishing and Exploding Gradients ](#_Toc87638565) - -[Conclusion ](#_Toc87638566) - -[References ](#_Toc87638567) +- [Convexity](#convexity) + * [Convex versus non-Convex Optimization Problems](#convex-versus-non-convex-optimization-problems) + * [Local and global optimization](#local-and-global-optimization) + + [Local optimization](#local-optimization) + + [Global optimization](#global-optimization) + * [Locally optimal for a convex optimization problem](#locally-optimal-for-a-convex-optimization-problem) + * [Hessian being positive semi-definite iff convexity (Theorem one: One dimensional)](#hessian-being-positive-semi-definite-iff-convexity--theorem-one--one-dimensional-) + * [Hessian being positive semi-definite iff convexity (Theorem two: Multi-dimensional)](#hessian-being-positive-semi-definite-iff-convexity--theorem-two--multi-dimensional-) + * [Examples of Convex Functions](#examples-of-convex-functions) +- [Gradient](#gradient) + * [Gradient Descent](#gradient-descent) + * [Challenges with gradient descent](#challenges-with-gradient-descent) + + [Local minima and saddle points](#local-minima-and-saddle-points) + + [Vanishing and Exploding Gradients](#vanishing-and-exploding-gradients) +- [Conclusion](#conclusion) +- [References](#references) # From aaccbc2a48706d43a4842b5073b496eba986bc50 Mon Sep 17 00:00:00 2001 From: royaghavami <59202308+royaghavami@users.noreply.github.com> Date: Mon, 29 Nov 2021 14:35:21 +0330 Subject: [PATCH 45/57] Update index.md --- .../6_search_in_continuous_spaces/index.md | 26 +++++++++---------- 1 file changed, 13 insertions(+), 13 deletions(-) diff --git a/notebooks/6_search_in_continuous_spaces/index.md b/notebooks/6_search_in_continuous_spaces/index.md index 51df8bcc..f346b3b6 100644 --- a/notebooks/6_search_in_continuous_spaces/index.md +++ b/notebooks/6_search_in_continuous_spaces/index.md @@ -46,7 +46,7 @@ where K \* denotes convolution with a blurring filter. ![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.003.jpeg) -*Figure SEQ Figure \\* ARABIC 1. reconstructed image after solving the optimization* +*Figure1. reconstructed image after solving the optimization* # Types of optimization techniques An essential step to optimization technique is to categorize the optimization model since the algorithms used for solving optimization problems are customized as per the nature of the problem. Various optimization problem types are: @@ -65,7 +65,7 @@ We denote the set of points for which all the constraints are satisfied as C, an In unconstrained optimization problems the answers are constrained into being subject of set C as the picture bellow shows: -*Figure SEQ Figure \\* ARABIC 2. constrained vs unconstrained optimization* +*Figure2. constrained vs unconstrained optimization* ![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.005.png) @@ -75,7 +75,7 @@ In unconstrained optimization problems the answers are constrained into being s ## Differentiable optimization and Non-differentiable optimization Non-differentiable optimization is a category of optimization that deals with objective that for a variety of reasons is non-differentiable and thus non-convex. The functions in this class of optimization are generally non-smooth. These functions although continuous often contain sharp points or corners that do not allow for the solution of a tangent and are thus non-differentiable. In practice non-differentiable optimization encompasses a large variety of problems and a single one-size fits all solution is not applicable however solution is often reached through implementation of the sub gradient method. Non-differentiable functions often arise in real world applications and commonly in the field of economics where cost functions often include sharp points. -*Figure SEQ Figure \\* ARABIC 3. Non-differentiable function* +*Figure3. Non-differentiable function* ![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.006.png) ### Cost functions In many cases, particularly economics the cost function which is the objective function of an optimization problem is non-differentiable. These non-smooth cost functions may include discontinuities and discontinuous gradients and are often seen in discontinuous physical processes. Optimal solution of these cost functions is a matter of importance to economists but presents a variety of issues when using numerical methods thus leading to the need for special solution methods. @@ -97,14 +97,14 @@ for all α∈ [0,1], i.e., the line segment between (x, f(x)) and (y, f(y))* , -*Figure SEQ Figure \\* ARABIC 4. In convex function f, for every two point x,y∈domainf, the line segment between them lies above the graph of f.* +*Figure4. In convex function f, for every two point x,y∈domainf, the line segment between them lies above the graph of f.* ![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.008.png) For an affine function we always have equality in (\*), so all affine (and therefore also linear) functions are both convex and concave. Conversely, any function that is convex and concave is affine. A function is convex if and only if it is convex when restricted to any line that intersects its domain. In other words, f is convex if and only if for all x ∈ domain (f) and all v, the function g(t) = f(x + tv) is convex (on its domain, {t| x + tv ∈ domain(f) }* ). This property is very useful, since it allows us to check whether a function is convex by restricting it to a line. -*Figure SEQ Figure \\* ARABIC 5. A Non-convex function vs a Convex function* +*Figure5. A Non-convex function vs a Convex function* ![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.009.png) Some examples of convex functions are: @@ -117,7 +117,7 @@ Some examples of convex functions are: A convex optimization *problem* is a problem where all of the constraints are convex functions, and the objective is a convex function if minimizing, or a concave function if maximizing. In a convex optimization problem, the feasible region is a convex region, as pictured below. -*Figure SEQ Figure \\* ARABIC 6. convex region* +*Figure6. convex region* ![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.010.png) @@ -126,7 +126,7 @@ With a convex objective and a convex feasible region, there can be only one opti A *non-convex optimization problem* is any problem where the objective or any of the constraints are non-convex, as pictured below. -*Figure SEQ Figure \\* ARABIC 7. Non-convex region* +*Figure7. Non-convex region* ![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.011.png) @@ -148,7 +148,7 @@ So, to summarize, A maximum or minimum is said to be local if it is the larges ![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.012.png) -*Figure SEQ Figure \\* ARABIC 8. local and global maximum and minimum* +*Figure8. local and global maximum and minimum* ## ## Locally optimal for a convex optimization problem A fundamental property of convex optimization problems is that any locally optimal point is also (globally) optimal. To see this, suppose that x is locally optimal for a convex optimization problem, i.e., x is feasible and @@ -227,13 +227,13 @@ A gradient simply measures the change in all weights with regard to the change i ## Gradient Descent Gradient descent is an optimization algorithm used for training a machine learning model. It works on a convex function and changes the function’s parameters iteratively to find the local minimum (Since the function is convex, by finding local minimum we also find the global minimum). -*Figure SEQ Figure \\* ARABIC 9. The gradient field of function f = x2 – 4x + y2 + 2y* +*Figure9. The gradient field of function f = x2 – 4x + y2 + 2y* ![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.026.png) To find the local minimum, we must take steps proportional to the negative of the gradient (move away from the gradient) of the function at the current point. If we take steps proportional to the positive of the gradient, we will approach a local maximum of the function, and the procedure is called Gradient Ascent. -*Figure SEQ Figure \\* ARABIC 10. The way Gradient Descent works* +*Figure10. The way Gradient Descent works* ![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.027.png) @@ -250,7 +250,7 @@ Here Alpha is called Learning rate and determines the length of the steps. For Note: While moving towards the local minima, both gradient and the size of the step decreases. So, the learning rate can be constant over the optimization. -*Figure SEQ Figure \\* ARABIC 11. The effect of learning in Gradient Descent* +*Figure11. The effect of learning in Gradient Descent* ![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.029.png) @@ -266,7 +266,7 @@ c) Learning rate is higher than the optimal value, it overshoots but converges ( d) Learning rate is very large, it overshoots and diverges, moves away from the minima, performance **decreases on learning** -*Figure SEQ Figure \\* ARABIC 12. Four different learning rates working on function f* +*Figure12. Four different learning rates working on function f* ![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.030.png) @@ -278,7 +278,7 @@ In nonconvex problems, gradient descent can have difficulties in finding the glo In the previous section we mentioned that when the slope of the cost function is at or close to zero, the model stops learning. Other than global minimum, local minima and saddle points can also reach this slope. Local minima have the same shape as the global minimum, where the slope of the cost function increases on either side of the current point. Saddle points, have negative gradient on one side and a non-negative gradient on the other side, causing them to reaching a local maximum on one side and a local minimum on the other. We can use noisy gradients to escape local minimums and saddle points. -*Figure SEQ Figure \\* ARABIC 13. Local minima and Saddle point can face problems using Gradient Descent* +*Figure13. Local minima and Saddle point can face problems using Gradient Descent* ![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.031.png) From df00c3d795aa7c7984e3739a7429ecbc54f38a51 Mon Sep 17 00:00:00 2001 From: royaghavami <59202308+royaghavami@users.noreply.github.com> Date: Mon, 29 Nov 2021 14:44:15 +0330 Subject: [PATCH 46/57] Update index.md --- .../6_search_in_continuous_spaces/index.md | 38 +++++++++---------- 1 file changed, 19 insertions(+), 19 deletions(-) diff --git a/notebooks/6_search_in_continuous_spaces/index.md b/notebooks/6_search_in_continuous_spaces/index.md index f346b3b6..bb495f11 100644 --- a/notebooks/6_search_in_continuous_spaces/index.md +++ b/notebooks/6_search_in_continuous_spaces/index.md @@ -4,25 +4,25 @@ Search in Continuous Space # **Table of Contents** - [Introduction](#introduction) - [Types of optimization techniques](#types-of-optimization-techniques) - * [Linear programming (LP) and Nonlinear programming (NLP)](#linear-programming--lp--and-nonlinear-programming--nlp-) - * [Continuous optimization and Integer programming (IP)](#continuous-optimization-and-integer-programming--ip-) - * [Constrained optimization and Unconstrained optimization](#constrained-optimization-and-unconstrained-optimization) - * [Differentiable optimization and Non-differentiable optimization](#differentiable-optimization-and-non-differentiable-optimization) + * [Linear programming (LP) and nonlinear programming (NLP)](#linear-programming--lp--and-nonlinear-programming--nlp-) + * [Continuous optimization and integer programming (IP)](#continuous-optimization-and-integer-programming--ip-) + * [Constrained optimization and unconstrained optimization](#constrained-optimization-and-unconstrained-optimization) + * [Differentiable optimization and non-differentiable optimization](#differentiable-optimization-and-non-differentiable-optimization) + [Cost functions](#cost-functions) - [Convexity](#convexity) - * [Convex versus non-Convex Optimization Problems](#convex-versus-non-convex-optimization-problems) + * [Convex versus non-convex optimization problems](#convex-versus-non-convex-optimization-problems) * [Local and global optimization](#local-and-global-optimization) + [Local optimization](#local-optimization) + [Global optimization](#global-optimization) * [Locally optimal for a convex optimization problem](#locally-optimal-for-a-convex-optimization-problem) - * [Hessian being positive semi-definite iff convexity (Theorem one: One dimensional)](#hessian-being-positive-semi-definite-iff-convexity--theorem-one--one-dimensional-) - * [Hessian being positive semi-definite iff convexity (Theorem two: Multi-dimensional)](#hessian-being-positive-semi-definite-iff-convexity--theorem-two--multi-dimensional-) - * [Examples of Convex Functions](#examples-of-convex-functions) + * [Hessian being positive semi-definite iff convexity (theorem one: one dimensional)](#hessian-being-positive-semi-definite-iff-convexity--theorem-one--one-dimensional-) + * [Hessian being positive semi-definite iff convexity (theorem two: multi-dimensional)](#hessian-being-positive-semi-definite-iff-convexity--theorem-two--multi-dimensional-) + * [Examples of convex functions](#examples-of-convex-functions) - [Gradient](#gradient) - * [Gradient Descent](#gradient-descent) + * [Gradient descent](#gradient-descent) * [Challenges with gradient descent](#challenges-with-gradient-descent) + [Local minima and saddle points](#local-minima-and-saddle-points) - + [Vanishing and Exploding Gradients](#vanishing-and-exploding-gradients) + + [Vanishing and exploding gradients](#vanishing-and-exploding-gradients) - [Conclusion](#conclusion) - [References](#references) @@ -52,9 +52,9 @@ where K \* denotes convolution with a blurring filter. An essential step to optimization technique is to categorize the optimization model since the algorithms used for solving optimization problems are customized as per the nature of the problem. Various optimization problem types are: ## Linear programming (LP) and Nonlinear programming (NLP) Linear programming is a method to achieve the best outcome in a mathematical model whose requirements are represented by linear relationships whereas nonlinear programming is a process of solving an optimization problem where the constraints or the objective functions are nonlinear. Thus, this is the main difference between linear and nonlinear programming. -## Continuous optimization and Integer programming (IP) +## Continuous optimization and integer programming (IP) In order to conceptualize the difference, linear programming can solve problems about minimizing (or maximizing) an objective function by continuous variables. For instance, maybe the optimal solution for a problem to be x1=5,46 and x2=2,65. But in integer programming we can only use integers as variables and we can’t have fractional numbers. -## Constrained optimization and Unconstrained optimization +## Constrained optimization and unconstrained optimization A constrained optimization problem is the problem of optimizing an objective function subject to constraints on the variables. In general terms, ![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.004.png) @@ -72,7 +72,7 @@ In unconstrained optimization problems the answers are constrained into being s ## -## Differentiable optimization and Non-differentiable optimization +## Differentiable optimization and non-differentiable optimization Non-differentiable optimization is a category of optimization that deals with objective that for a variety of reasons is non-differentiable and thus non-convex. The functions in this class of optimization are generally non-smooth. These functions although continuous often contain sharp points or corners that do not allow for the solution of a tangent and are thus non-differentiable. In practice non-differentiable optimization encompasses a large variety of problems and a single one-size fits all solution is not applicable however solution is often reached through implementation of the sub gradient method. Non-differentiable functions often arise in real world applications and commonly in the field of economics where cost functions often include sharp points. *Figure3. Non-differentiable function* @@ -113,7 +113,7 @@ Some examples of convex functions are: - Negative logarithm: f(x) = -log(x), x > 0 - Euclidean norm: f(x) = ||x||2 -## Convex versus non-Convex Optimization Problems +## Convex versus non-convex optimization problems A convex optimization *problem* is a problem where all of the constraints are convex functions, and the objective is a convex function if minimizing, or a concave function if maximizing. In a convex optimization problem, the feasible region is a convex region, as pictured below. @@ -164,7 +164,7 @@ Then we have ||z – x|| = R/2 < R, and by convexity of the feasible set, z is f ![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.015.png) which contradicts (@). Hence there exists no feasible y with f0(y) < f0(x), i.e., x is globally optimal. -## Hessian being positive semi-definite iff convexity (Theorem one: One dimensional) +## Hessian being positive semi-definite iff convexity (theorem one: one dimensional) One way of checking a function’s convexity is by checking its Hessian matrix. let Ø ≠ M ⊆ Rn be a convex set and let f(x) be a function differentiable on an open superset of M then f(x) is convex on M if and only if for every x1,x2 ∈ M ![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.016.png) @@ -182,7 +182,7 @@ By the limit transition λ ⟶0 we get (\*) utilizing the chain rule for the der Multiply the first inequality by λ1, the one by λ2 and summing up we get ![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.019.png) -## Hessian being positive semi-definite iff convexity (Theorem two: Multi-dimensional) +## Hessian being positive semi-definite iff convexity (theorem two: multi-dimensional) Now we prove the same property for n-dimensional functions. let Ø ≠ M ⊆ Rn be an open convex set of dimension n, and suppose that a function f: M→R is twice continuously differentiable on M. Then f(x) is convex on M if and only the Hessian ∇2f(x) is positive semi-definite for every x∈M. Proof:let x\* ∈ M be arbitrary. Due to continuity of the second partial derivatives we have that for every λ∈R and y ∈ Rn, x\* + λy ∈ M, there is θ∈ (0,1) such that @@ -207,7 +207,7 @@ which shows the convexity of f(x) in the view of Theorem one. ![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.024.png) -## Examples of Convex Functions: +## Examples of convex functions: In this section we check the convexity of two functions using the theorems we learned. - f(x) = 2x3 – 18x2 : @@ -224,7 +224,7 @@ The leading principal minors of the Hessian are 2 > 0, 4 > 0, and 8 > 0. Therefo # Gradient A gradient simply measures the change in all weights with regard to the change in error. We can also think of a gradient as the slope of a function that has more than one input variable. The higher the gradient, the steeper the slope and the faster a model can learn. But if the slope is zero, the model stops learning. -## Gradient Descent +## Gradient descent Gradient descent is an optimization algorithm used for training a machine learning model. It works on a convex function and changes the function’s parameters iteratively to find the local minimum (Since the function is convex, by finding local minimum we also find the global minimum). *Figure9. The gradient field of function f = x2 – 4x + y2 + 2y* @@ -282,7 +282,7 @@ In the previous section we mentioned that when the slope of the cost function is ![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.031.png) -### Vanishing and Exploding Gradients +### Vanishing and exploding gradients In deeper neural networks we can also encounter two other problems: Vanishing gradients: This occurs when the gradient is too small. As we move backwards during backpropagation, the gradient continues to become smaller, so the earlier layers in the network learn more slowly than later layers. When this happens, the weight parameters update until they become insignificant and the algorithm stops learning. From a6f1e68c5a4ee9bc247cecd1cf9d302c3d3071d5 Mon Sep 17 00:00:00 2001 From: royaghavami <59202308+royaghavami@users.noreply.github.com> Date: Fri, 17 Dec 2021 22:33:17 +0330 Subject: [PATCH 47/57] Update index.md --- .../6_search_in_continuous_spaces/index.md | 34 +++++++++---------- 1 file changed, 17 insertions(+), 17 deletions(-) diff --git a/notebooks/6_search_in_continuous_spaces/index.md b/notebooks/6_search_in_continuous_spaces/index.md index bb495f11..536fb3a4 100644 --- a/notebooks/6_search_in_continuous_spaces/index.md +++ b/notebooks/6_search_in_continuous_spaces/index.md @@ -1,5 +1,5 @@  -Search in Continuous Space +**Search in Continuous Space** # # **Table of Contents** - [Introduction](#introduction) @@ -31,7 +31,7 @@ Search in Continuous Space # -# Introduction +# **Introduction** In real life optimization problems, we mostly face continuous environments which are made of continuous variables such as time and location. Thus, in this lecture note we will discuss some useful search methods in continuous spaces. The typical problems are linear programming (LP) and diverse kinds of nonlinear programming (such as convex programming, quadratic programming or semidefinite programming). In this lecture note we will discuss different types of optimization techniques in continuous spaces, the differences between different types of optimization techniques, how convexity will help us solve optimization problems easier, difference between local and global optimization, gradient descent method and the challenges with finding the best learning rate. **Example.1** (Weber Point) Given a collection of cities (assume on 2D plane) how can we find the location that minimizes the sum of distances to all cities? We model this problem by denoting the locations of the cities as y(1), …,y(m) and write the optimization problem as: @@ -48,11 +48,11 @@ where K \* denotes convolution with a blurring filter. *Figure1. reconstructed image after solving the optimization* -# Types of optimization techniques +# **Types of optimization techniques** An essential step to optimization technique is to categorize the optimization model since the algorithms used for solving optimization problems are customized as per the nature of the problem. Various optimization problem types are: ## Linear programming (LP) and Nonlinear programming (NLP) Linear programming is a method to achieve the best outcome in a mathematical model whose requirements are represented by linear relationships whereas nonlinear programming is a process of solving an optimization problem where the constraints or the objective functions are nonlinear. Thus, this is the main difference between linear and nonlinear programming. -## Continuous optimization and integer programming (IP) +## **Continuous optimization and integer programming (IP)** In order to conceptualize the difference, linear programming can solve problems about minimizing (or maximizing) an objective function by continuous variables. For instance, maybe the optimal solution for a problem to be x1=5,46 and x2=2,65. But in integer programming we can only use integers as variables and we can’t have fractional numbers. ## Constrained optimization and unconstrained optimization A constrained optimization problem is the problem of optimizing an objective function subject to constraints on the variables. In general terms, @@ -72,7 +72,7 @@ In unconstrained optimization problems the answers are constrained into being s ## -## Differentiable optimization and non-differentiable optimization +## **Differentiable optimization and non-differentiable optimization** Non-differentiable optimization is a category of optimization that deals with objective that for a variety of reasons is non-differentiable and thus non-convex. The functions in this class of optimization are generally non-smooth. These functions although continuous often contain sharp points or corners that do not allow for the solution of a tangent and are thus non-differentiable. In practice non-differentiable optimization encompasses a large variety of problems and a single one-size fits all solution is not applicable however solution is often reached through implementation of the sub gradient method. Non-differentiable functions often arise in real world applications and commonly in the field of economics where cost functions often include sharp points. *Figure3. Non-differentiable function* @@ -82,7 +82,7 @@ In many cases, particularly economics the cost function which is the objective f In this lecture we don’t discuss non-differential optimization and non-smooth functions and the text above was for introduction and further information on this topic. If you want to know more about it, see these links: [nondifferentiable optimization via approximation](https://www.mit.edu/~dimitrib/Nondiff_Approx.pdf) and [non-convex optimization](https://www.cs.cornell.edu/courses/cs6787/2017fa/Lecture7.pdf). -# Convexity +# **Convexity** A set C is convex if for any two points x,y ∈C , αx + (1 – α)y ∈ C for all α∈ [0,1], i.e., all points on the line between x and y also lie in C. Some examples of convex sets are: - All points C = Rn where n is a non-negative integer @@ -113,7 +113,7 @@ Some examples of convex functions are: - Negative logarithm: f(x) = -log(x), x > 0 - Euclidean norm: f(x) = ||x||2 -## Convex versus non-convex optimization problems +## **Convex versus non-convex optimization problems** A convex optimization *problem* is a problem where all of the constraints are convex functions, and the objective is a convex function if minimizing, or a concave function if maximizing. In a convex optimization problem, the feasible region is a convex region, as pictured below. @@ -132,14 +132,14 @@ A *non-convex optimization problem* is any problem where the objective or any Such a problem may have multiple feasible regions and multiple locally optimal points within each region.  It can take time exponential in the number of variables and constraints to determine that a non-convex problem is infeasible, that the objective function is unbounded, or that an optimal solution is the "global optimum" across all feasible regions. ## -## Local and global optimization: -### Local optimization +## **Local and global optimization:** +### **Local optimization** In local optimization, the compromise is to give up seeking the optimal x, which minimizes the objective over all feasible points. Instead we seek a point that is only locally optimal, which means that it minimizes the objective function among feasible points that are near it, but is not guaranteed to have a lower objective value than all other feasible points. Local optimization methods can be fast, can handle large-scale problems, and are widely applicable, since they only require differentiability of the objective and constraint functions. There are several disadvantages of local optimization methods, beyond (possibly) not finding the true, globally optimal solution. The methods require an initial guess for the optimization variable. This initial guess or starting point is critical, and can greatly affect the objective value of the local solution obtained. -### Global optimization +### **Global optimization** In global optimization, the true global solution of the optimization problem is found; the compromise is efficiency. The worst-case complexity of global optimization methods grows exponentially with the problem size. The hope is that in practice, for the particular problem instances encountered, the method is far faster. While this favorable situation does occur, it is not typical. Even small problems, with a few tens of variables, can take a very long time (e.g., hours or days) to solve. Global optimization is used for problems with a small number of variables, where computing time is not critical, and the value of finding the true global solution is very high. @@ -150,7 +150,7 @@ So, to summarize, A maximum or minimum is said to be local if it is the larges *Figure8. local and global maximum and minimum* ## -## Locally optimal for a convex optimization problem +## **Locally optimal for a convex optimization problem** A fundamental property of convex optimization problems is that any locally optimal point is also (globally) optimal. To see this, suppose that x is locally optimal for a convex optimization problem, i.e., x is feasible and ![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.013.png) @@ -221,10 +221,10 @@ The Hessian is: ![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.025.png) The leading principal minors of the Hessian are 2 > 0, 4 > 0, and 8 > 0. Therefore the Hessian is positive semi-definite and the function is convex. -# Gradient +# **Gradient** A gradient simply measures the change in all weights with regard to the change in error. We can also think of a gradient as the slope of a function that has more than one input variable. The higher the gradient, the steeper the slope and the faster a model can learn. But if the slope is zero, the model stops learning. -## Gradient descent +## **Gradient descent** Gradient descent is an optimization algorithm used for training a machine learning model. It works on a convex function and changes the function’s parameters iteratively to find the local minimum (Since the function is convex, by finding local minimum we also find the global minimum). *Figure9. The gradient field of function f = x2 – 4x + y2 + 2y* @@ -271,7 +271,7 @@ d) Learning rate is very large, it overshoots and diverges, moves away from the ![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.030.png) -## Challenges with gradient descent +## **Challenges with gradient descent** Although gradient descent is the most common approach for optimization problems, it has its own challenges. Some of them include: ### Local minima and saddle points In nonconvex problems, gradient descent can have difficulties in finding the global minimum. @@ -282,21 +282,21 @@ In the previous section we mentioned that when the slope of the cost function is ![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.031.png) -### Vanishing and exploding gradients +### **Vanishing and exploding gradients** In deeper neural networks we can also encounter two other problems: Vanishing gradients: This occurs when the gradient is too small. As we move backwards during backpropagation, the gradient continues to become smaller, so the earlier layers in the network learn more slowly than later layers. When this happens, the weight parameters update until they become insignificant and the algorithm stops learning. Exploding gradients: This happens when the gradient is too large and creates an unstable model. In this case, the model weights will grow too large, and they will eventually be represented as NaN. -# Conclusion +# **Conclusion** In this lecture we talked about the different types of optimizations in continuous spaces. It has been proven that finding an epsilon-optimal minimizer that has the following relationship with the optimal solution x\* ![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.032.png) is impossible. However, we introduced a feature called convexity which can help us solve some of these problems. We proved that if a function’s Hessian matrix is positive semi-definitive, it is convex and vice versa. Now if a function is convex, its local minimums are also global. So, by using a decent iterative algorithm, like Gradient Descent, we can find its solutions. But keep in mind that this algorithm can face problems if used on non-convex functions or deeper neural networks -# References +# **References** [1] Vandenberghe. [2] Vandenberghe. From 116747444a131850eb8ae7b4270c423beb3115c1 Mon Sep 17 00:00:00 2001 From: royaghavami <59202308+royaghavami@users.noreply.github.com> Date: Fri, 17 Dec 2021 22:35:33 +0330 Subject: [PATCH 48/57] Update index.md --- notebooks/6_search_in_continuous_spaces/index.md | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/notebooks/6_search_in_continuous_spaces/index.md b/notebooks/6_search_in_continuous_spaces/index.md index 536fb3a4..dd962914 100644 --- a/notebooks/6_search_in_continuous_spaces/index.md +++ b/notebooks/6_search_in_continuous_spaces/index.md @@ -77,7 +77,7 @@ Non-differentiable optimization is a category of optimization that deals with ob *Figure3. Non-differentiable function* ![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.006.png) -### Cost functions +### **Cost functions** In many cases, particularly economics the cost function which is the objective function of an optimization problem is non-differentiable. These non-smooth cost functions may include discontinuities and discontinuous gradients and are often seen in discontinuous physical processes. Optimal solution of these cost functions is a matter of importance to economists but presents a variety of issues when using numerical methods thus leading to the need for special solution methods. In this lecture we don’t discuss non-differential optimization and non-smooth functions and the text above was for introduction and further information on this topic. If you want to know more about it, see these links: [nondifferentiable optimization via approximation](https://www.mit.edu/~dimitrib/Nondiff_Approx.pdf) and [non-convex optimization](https://www.cs.cornell.edu/courses/cs6787/2017fa/Lecture7.pdf). @@ -164,7 +164,7 @@ Then we have ||z – x|| = R/2 < R, and by convexity of the feasible set, z is f ![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.015.png) which contradicts (@). Hence there exists no feasible y with f0(y) < f0(x), i.e., x is globally optimal. -## Hessian being positive semi-definite iff convexity (theorem one: one dimensional) +## **Hessian being positive semi-definite iff convexity (theorem one: one dimensional)** One way of checking a function’s convexity is by checking its Hessian matrix. let Ø ≠ M ⊆ Rn be a convex set and let f(x) be a function differentiable on an open superset of M then f(x) is convex on M if and only if for every x1,x2 ∈ M ![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.016.png) @@ -182,7 +182,7 @@ By the limit transition λ ⟶0 we get (\*) utilizing the chain rule for the der Multiply the first inequality by λ1, the one by λ2 and summing up we get ![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.019.png) -## Hessian being positive semi-definite iff convexity (theorem two: multi-dimensional) +## **Hessian being positive semi-definite iff convexity (theorem two: multi-dimensional)** Now we prove the same property for n-dimensional functions. let Ø ≠ M ⊆ Rn be an open convex set of dimension n, and suppose that a function f: M→R is twice continuously differentiable on M. Then f(x) is convex on M if and only the Hessian ∇2f(x) is positive semi-definite for every x∈M. Proof:let x\* ∈ M be arbitrary. Due to continuity of the second partial derivatives we have that for every λ∈R and y ∈ Rn, x\* + λy ∈ M, there is θ∈ (0,1) such that @@ -207,7 +207,7 @@ which shows the convexity of f(x) in the view of Theorem one. ![](Aspose.Words.153ba005-881a-4c4b-97e3-6ea90eff943d.024.png) -## Examples of convex functions: +## **Examples of convex functions:** In this section we check the convexity of two functions using the theorems we learned. - f(x) = 2x3 – 18x2 : From 7a748124781be2a4d1a6c2f2530099bfcb2a8933 Mon Sep 17 00:00:00 2001 From: royaghavami <59202308+royaghavami@users.noreply.github.com> Date: Fri, 17 Dec 2021 23:29:03 +0330 Subject: [PATCH 49/57] Update index.md --- notebooks/6_search_in_continuous_spaces/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/6_search_in_continuous_spaces/index.md b/notebooks/6_search_in_continuous_spaces/index.md index dd962914..9993e93f 100644 --- a/notebooks/6_search_in_continuous_spaces/index.md +++ b/notebooks/6_search_in_continuous_spaces/index.md @@ -1,5 +1,5 @@  -**Search in Continuous Space** +#**Search in Continuous Space** # # **Table of Contents** - [Introduction](#introduction) From eeb177cec49f5ce844159a1c9f29124092cf3d3d Mon Sep 17 00:00:00 2001 From: royaghavami <59202308+royaghavami@users.noreply.github.com> Date: Fri, 17 Dec 2021 23:30:33 +0330 Subject: [PATCH 50/57] Update index.md --- notebooks/6_search_in_continuous_spaces/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/6_search_in_continuous_spaces/index.md b/notebooks/6_search_in_continuous_spaces/index.md index 9993e93f..283a4ec1 100644 --- a/notebooks/6_search_in_continuous_spaces/index.md +++ b/notebooks/6_search_in_continuous_spaces/index.md @@ -1,5 +1,5 @@  -#**Search in Continuous Space** +%**Search in Continuous Space** # # **Table of Contents** - [Introduction](#introduction) From 89fc003edf7e1e63d741946b7165a5f48e16f26e Mon Sep 17 00:00:00 2001 From: royaghavami <59202308+royaghavami@users.noreply.github.com> Date: Fri, 17 Dec 2021 23:31:02 +0330 Subject: [PATCH 51/57] Update index.md --- notebooks/6_search_in_continuous_spaces/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/6_search_in_continuous_spaces/index.md b/notebooks/6_search_in_continuous_spaces/index.md index 283a4ec1..68ed3cc2 100644 --- a/notebooks/6_search_in_continuous_spaces/index.md +++ b/notebooks/6_search_in_continuous_spaces/index.md @@ -1,5 +1,5 @@  -%**Search in Continuous Space** +% **Search in Continuous Space** # # **Table of Contents** - [Introduction](#introduction) From 5d8bffc09df478fe38a99c894cecb7dc8b623a1f Mon Sep 17 00:00:00 2001 From: royaghavami <59202308+royaghavami@users.noreply.github.com> Date: Fri, 17 Dec 2021 23:33:37 +0330 Subject: [PATCH 52/57] Update index.md --- notebooks/6_search_in_continuous_spaces/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/6_search_in_continuous_spaces/index.md b/notebooks/6_search_in_continuous_spaces/index.md index 68ed3cc2..83ad7815 100644 --- a/notebooks/6_search_in_continuous_spaces/index.md +++ b/notebooks/6_search_in_continuous_spaces/index.md @@ -1,5 +1,5 @@  -% **Search in Continuous Space** +Search in Continuous Space # # **Table of Contents** - [Introduction](#introduction) From 1a08c6a973619f5db67faf363ae712baea1d78c9 Mon Sep 17 00:00:00 2001 From: royaghavami <59202308+royaghavami@users.noreply.github.com> Date: Fri, 17 Dec 2021 23:34:01 +0330 Subject: [PATCH 53/57] Update index.md --- notebooks/6_search_in_continuous_spaces/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/6_search_in_continuous_spaces/index.md b/notebooks/6_search_in_continuous_spaces/index.md index 83ad7815..d151fd39 100644 --- a/notebooks/6_search_in_continuous_spaces/index.md +++ b/notebooks/6_search_in_continuous_spaces/index.md @@ -1,5 +1,5 @@  -Search in Continuous Space +Search in Continuous Space # # **Table of Contents** - [Introduction](#introduction) From 55def5466f1e769991fc66f297d1930460698b4a Mon Sep 17 00:00:00 2001 From: royaghavami <59202308+royaghavami@users.noreply.github.com> Date: Fri, 17 Dec 2021 23:34:37 +0330 Subject: [PATCH 54/57] Update index.md --- notebooks/6_search_in_continuous_spaces/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/6_search_in_continuous_spaces/index.md b/notebooks/6_search_in_continuous_spaces/index.md index d151fd39..a7d00698 100644 --- a/notebooks/6_search_in_continuous_spaces/index.md +++ b/notebooks/6_search_in_continuous_spaces/index.md @@ -1,5 +1,5 @@  -Search in Continuous Space +# **Search in Continuous Space** # # **Table of Contents** - [Introduction](#introduction) From 008d328d08907f4d1237981434238b76c9e8e591 Mon Sep 17 00:00:00 2001 From: royaghavami <59202308+royaghavami@users.noreply.github.com> Date: Fri, 17 Dec 2021 23:36:35 +0330 Subject: [PATCH 55/57] Update index.md --- notebooks/6_search_in_continuous_spaces/index.md | 4 +--- 1 file changed, 1 insertion(+), 3 deletions(-) diff --git a/notebooks/6_search_in_continuous_spaces/index.md b/notebooks/6_search_in_continuous_spaces/index.md index a7d00698..84e67686 100644 --- a/notebooks/6_search_in_continuous_spaces/index.md +++ b/notebooks/6_search_in_continuous_spaces/index.md @@ -1,6 +1,4 @@ - -# **Search in Continuous Space** -# +# ** Search in Continuous Space # **Table of Contents** - [Introduction](#introduction) - [Types of optimization techniques](#types-of-optimization-techniques) From 3723ddb0e72727a50a6f3eb2bf7a8900fae13abd Mon Sep 17 00:00:00 2001 From: royaghavami <59202308+royaghavami@users.noreply.github.com> Date: Fri, 17 Dec 2021 23:36:54 +0330 Subject: [PATCH 56/57] Update index.md --- notebooks/6_search_in_continuous_spaces/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/6_search_in_continuous_spaces/index.md b/notebooks/6_search_in_continuous_spaces/index.md index 84e67686..ed60016e 100644 --- a/notebooks/6_search_in_continuous_spaces/index.md +++ b/notebooks/6_search_in_continuous_spaces/index.md @@ -1,4 +1,4 @@ -# ** Search in Continuous Space +# **Search in Continuous Space # **Table of Contents** - [Introduction](#introduction) - [Types of optimization techniques](#types-of-optimization-techniques) From e5a91d76f4fb22813fd589f8cadb4ffe97f63b8f Mon Sep 17 00:00:00 2001 From: royaghavami <59202308+royaghavami@users.noreply.github.com> Date: Fri, 17 Dec 2021 23:37:30 +0330 Subject: [PATCH 57/57] Update index.md --- notebooks/6_search_in_continuous_spaces/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/6_search_in_continuous_spaces/index.md b/notebooks/6_search_in_continuous_spaces/index.md index ed60016e..d4f6c1be 100644 --- a/notebooks/6_search_in_continuous_spaces/index.md +++ b/notebooks/6_search_in_continuous_spaces/index.md @@ -1,4 +1,4 @@ -# **Search in Continuous Space +# **Search in Continuous Space** # **Table of Contents** - [Introduction](#introduction) - [Types of optimization techniques](#types-of-optimization-techniques)