From bdb6e70934519465a763de478b39132fe14eaf2b Mon Sep 17 00:00:00 2001 From: Claudio Maggioni Date: Fri, 4 Jun 2021 11:11:17 +0200 Subject: [PATCH] fanculo giuseppona --- Claudio_Maggioni_5/Claudio_Maggioni_5.md | 82 +- Claudio_Maggioni_5/Claudio_Maggioni_5.pdf | Bin 241141 -> 241359 bytes Claudio_Maggioni_5/hatchfill.m | 967 ---------------------- 3 files changed, 22 insertions(+), 1027 deletions(-) delete mode 100644 Claudio_Maggioni_5/hatchfill.m diff --git a/Claudio_Maggioni_5/Claudio_Maggioni_5.md b/Claudio_Maggioni_5/Claudio_Maggioni_5.md index 9305b98..31262e4 100644 --- a/Claudio_Maggioni_5/Claudio_Maggioni_5.md +++ b/Claudio_Maggioni_5/Claudio_Maggioni_5.md @@ -13,7 +13,6 @@ header-includes: - \usepackage{float} - \floatplacement{figure}{H} - \hypersetup{colorlinks=true,linkcolor=blue} - --- \maketitle @@ -93,40 +92,40 @@ $$ A x=b, A^{T} \lambda+s=c,(x, s)>0\right\} \end{array} $$ -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: +The definition of the central path $\mathcal{C}$ is an arc, strictly composed of +feasible points. The path is parametrized by a scalar $\tau>0$ where each point +in the path $\left(x_{\tau}, \lambda_{\tau}, s_{\tau}\right) \in \mathcal{C}$ +satisfies the following conditions: $$ \begin{array}{c} A^{T} \lambda+s=c \\ A x=b \\ -x_{i} s_{i}=\tau \quad i=1,2, \ldots, n \\ +x_{i} s_{i}=\tau > 0 \quad i=1,2, \ldots, n \\ (x, s)>0 \end{array} $$ -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: +These conditions are very similar to the KKT conditions, differning only +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. +We can then observe that as $\tau$ approaches 0, the set of conditions defined +above get closer and closer to the true KKT conditions. Therefore, if the +central path C converges to anything then $\tau$ approaches zero, therefore +leading the path to the constrained minimizer while keeping $x$ and $s$ +positive but at the same time minimizing their pairwise products to zero. + +In practice, most central path implementation use Newton steps toward points on +$\mathcal{C}$ for which $\tau > 0$, rather than pure Newton steps for $F$. This +is usually possible since those newton steps are biased to stay in the +hyperspace region where $x > 0$ and $s > 0$ even without enforcing these +constraints. # Exercise 2 @@ -139,22 +138,6 @@ below: ## Exercise 2.2 - - According to Nocedal, a vector $x$ is a basic feasible point if it is in the feasible region and if there exists a subset $\beta$ of the index set $1, 2, \ldots, n$ such that: @@ -172,35 +155,14 @@ proven property to manually solve the constrained minimization problem presented in this section by aiding us with the graphical plot of the feasible region in figure \ref{fig:a}. - - ## Exercise 2.3 Since the geometrical interpretation of the definition of basic feasible point states that these point are non-other than the vertices of the feasible region, we first look at the plot above and to these points (i.e. the vertices of the -bright green non-trasparent region). Then, we look which constraint boundaries cross these -edges, and we formulate an algebraic expression to find these points. In -clockwise order, we have: +bright green non-trasparent region). Then, we look which constraint boundaries +cross these edges, and we formulate an algebraic expression to find these +points. 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