6_search_in_continuous_space #11
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| Search in Continuous Space | ||
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| # **Table of Contents** | ||
| ` `**TOC \o "1-3" \h \z \u [Introduction PAGEREF _Toc86421092 \h 2**](#_Toc86421092)** |
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Your PAGEREFs seem to have some problems when I view the maekdown.
| # **Table of Contents** | ||
| ` `**TOC \o "1-3" \h \z \u [Introduction PAGEREF _Toc86421092 \h 2**](#_Toc86421092)** | ||
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| [**Type of optimization techniques PAGEREF _Toc86421093 \h 2**](#_Toc86421093) |
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| # Introduction |
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You can provide some examples of optimization problems that can help readers to understand the significance of this topic. (check out slides)
| ## 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, | ||
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| minimize fx |
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Try to use LaTeX for equations (or if it's hard to use LaTeX in markdown, use screenshots of LaTeX equations)
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| 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) | ||
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| ***Unconstrained optimization problems*** the answers are constrained into being subject of set C as the picture bellow shows: |
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Try to avoid using different formats in a section. For example "Constrained optimization and Unconstrained optimization" is just bold while "Unconstrained optimization problems" is in bold and italics.
| ### 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. | ||
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| 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. |
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Try to reformat this sentence and provide at least a reference for further information.
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| fαx+1-αy≤αfx+1-αfy (\*) | ||
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| *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.* |
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This figure went into your other texts which is not desirable
| 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. | ||
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| ## Local and global optimization: | ||
| ### Local optimization |
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It's better to add examples or some figures which can show the difference between these two concepts
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| 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. | ||
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| # Convexity |
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Try to use examples of convex/non-convex sets/functions and proof why they are convex/non-convex.
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Some examples are provided, but use other examples and prove them by definition.
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| Search in Continuous Space | ||
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| # **Table of Contents** |
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Try to use the same format throughout this part, if you start your words with uppercase letters then do the same for other words.
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| *Figure SEQ Figure \\* ARABIC 1. reconstructed image after solving the optimization* |
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| In unconstrained optimization problems the answers are constrained into being subject of set C as the picture bellow shows: | ||
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| *Figure SEQ Figure \\* ARABIC 2. constrained vs unconstrained 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. | ||
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| *Figure SEQ Figure \\* ARABIC 3. Non-differentiable function* |
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