From d1182de9f86b37645e98caeb8e7d1777b14b5859 Mon Sep 17 00:00:00 2001 From: Claudio Maggioni Date: Thu, 3 Jun 2021 17:42:27 +0200 Subject: [PATCH] hw5: done (check 1.3 antiplagiarism) --- Claudio_Maggioni_5/Claudio_Maggioni_5.md | 66 +++++++++++++++++++++- Claudio_Maggioni_5/Claudio_Maggioni_5.pdf | Bin 234677 -> 241141 bytes 2 files changed, 65 insertions(+), 1 deletion(-) diff --git a/Claudio_Maggioni_5/Claudio_Maggioni_5.md b/Claudio_Maggioni_5/Claudio_Maggioni_5.md index 4437273..9305b98 100644 --- a/Claudio_Maggioni_5/Claudio_Maggioni_5.md +++ b/Claudio_Maggioni_5/Claudio_Maggioni_5.md @@ -5,6 +5,7 @@ title: Homework 5 -- Optimization Methods author: Claudio Maggioni header-includes: - \usepackage{amsmath} +- \usepackage{amssymb} - \usepackage{hyperref} - \usepackage[utf8]{inputenc} - \usepackage[margin=2.5cm]{geometry} @@ -57,13 +58,76 @@ objective function with is the summation of: point $x$ violates that constraint and zero otherwise. With some fine tuning of the coefficients for these new "penalty" terms, it is -possible to build an equivalend unconstrained minimization problem whose +possible to build an equivalent unconstrained minimization problem whose minimizer is also constrained minimizer for the original problem. ## Exercise 1.2 +The simplex method, as said in the previous section, works by iterating along +basic feasible points and minimizing the cost function along them. In terms of +strict linear algebra terms, the simplex method works by finding an initial set +of indices $\mathcal{B}$ which represent column indices of $A$. At each +iteration, the lagrangian inequality set of constraints $s$ is computed and +checked for negative components, since in order to satisfy the KKT conditions +the components must be all $\geq 0$. The iteration consists of removing one of +said components by changing the $\mathcal{B}$ index set, by effectively swapping +one of the basis vectors with one of the non-basic ones. The method terminates +once all components of $s$ are non-negative. + +Geometrically speaking the meaning of each iteration is a simple transition from +a basic feasible point to another neighboring one, and the search step is +effectively equivalent to one of the edges of the polytope. If the $s$ component +is always chosen to be the smallest (i.e. we choose the "most negative" +component), then the method behaves effectively a gradient descent operation of +the cost function hopping between basic feasible points. + ## Exercise 1.3 +In order to discuss in detail the interior point method, we first define two +sets named "feasible set" ($\mathcal{F}$) and "strictly feasible set" +($\mathcal{F}^{o}$) respectively: + +$$ +\begin{array}{l} \mathcal{F}=\left\{(x, \lambda, s) \mid A x=b, A^{T} +\lambda+s=c,(x, s) \geq 0\right\} \\ \mathcal{F}^{o}=\left\{(x, \lambda, s) \mid +A x=b, A^{T} \lambda+s=c,(x, s)>0\right\} \end{array} +$$ + +A central path $\mathcal{C}$ is defined as an arc strictly composed of feasible +points which is parametrized by a scalar $\tau>0$ where each point +$\left(x_{\tau}, \lambda_{\tau}, s_{\tau}\right) \in \mathcal{C}$ satisfies the +following conditions: + +$$ +\begin{array}{c} +A^{T} \lambda+s=c \\ +A x=b \\ +x_{i} s_{i}=\tau \quad i=1,2, \ldots, n \\ +(x, s)>0 +\end{array} +$$ + +We can observe how these conditions are very much similar to the KKT conditions +and, in fact, they only differ by the factor $\tau$ and by requiring the +pairwise products to be strictly greater than zero. With this, we can define the +central path as follows: + +$$ +\mathcal{C}=\left\{\left(x_{\tau}, \lambda_{\tau}, s_{\tau}\right) \mid +\tau>0\right\} +$$ + +Given these definitions, we can also observe that, as $\tau$ approaches zero, +the conditions we have defined become a closer and closer approximation to the +original KKT conditions. Therefore, if the central path $\mathcal{C}$ converges +to anything as $\tau$ approaches zero, then we know that it will converge to a +solution of the linear program. Meaning that the central path is leading us to a +solution by maintaining $x$ and $s$ positive while reducing the pairwise +products to zero at the same time. Usually, the Newton method is used to take +steps following $\mathcal{C}$ rather than by following the set of feasible +points $\mathcal{F}$ because it allows for longer steps before violating the +positivity constraint. + # Exercise 2 ## Exercise 2.1 diff --git a/Claudio_Maggioni_5/Claudio_Maggioni_5.pdf b/Claudio_Maggioni_5/Claudio_Maggioni_5.pdf index 74923e693d118ffe89cd7e6b8eaa8be297b7c906..c0a065af2cc92ae1819bf1d86c5c21334c2c6e41 100644 GIT binary patch delta 66541 zcmZs?LzFIBux^>Qt(~@Q+qP}nf7-US)3$Bfww*hzckZiEqZ+qbD;jG=d&V~-S}G8p zw-Ct56h+19nCRJH$mW+v)?wHP83`SXtzme1VHjl1>@8d^37I+A*peThC;_=zGRa4+ z$i1H$=ag!(mWM(?vhrP2JWjgq$|n_*MA@JcgObztg$c{E_Wph!;Jw9}ZYQE3q8y!p zw{-+D9A$Fd?@t~N9C-u;eV;!Pf7vI+w8`9`-r(w_OjVksmUx-Dw|Mb6ez$*0YQC@J z%eGp5Hg4pq+n@#bN3?f;+yU5=onH=2DR#ju)}PS?vp4p2J1^^ZKhJL(>l5}VQIDhx z?J8-alN~d=>y$DsyXOjOPJC&CW`<1$QA^P6sj@kjv5kohRLFN~X%e#=GrQ(WrFhA| 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