From 259751dc3ee52eea2d60fd4ff42242025a5f36bc Mon Sep 17 00:00:00 2001 From: Claudio Maggioni Date: Wed, 2 Jun 2021 14:35:22 +0200 Subject: [PATCH] work --- Claudio_Maggioni_5/Claudio_Maggioni_5.md | 123 ++++++++++++++++++++-- Claudio_Maggioni_5/Claudio_Maggioni_5.pdf | Bin 160411 -> 193991 bytes Claudio_Maggioni_5/main.asv | 79 -------------- 3 files changed, 113 insertions(+), 89 deletions(-) delete mode 100644 Claudio_Maggioni_5/main.asv diff --git a/Claudio_Maggioni_5/Claudio_Maggioni_5.md b/Claudio_Maggioni_5/Claudio_Maggioni_5.md index 1ecd25c..e3a7f41 100644 --- a/Claudio_Maggioni_5/Claudio_Maggioni_5.md +++ b/Claudio_Maggioni_5/Claudio_Maggioni_5.md @@ -23,24 +23,127 @@ header-includes: The resulting MATLAB plot of each constraint and of the feasible region is shown below: -![Plot of feasible region and constraints](./ex2-1.png) +![Plot of feasible region and constraints\label{fig:a}](./ex2-1.png) + +## 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: + +- $\beta$ contains exactly $m$ indices, where $m$ is the number of rows of $A$; +- For any $i \notin \beta$, $x_i = 0$, meaning the bound $x_i \geq 0$ can be + inactive only if $i \in \beta$; +- The $m$ x $m$ matrix $B$ defined by $B = [A_i]_{i \in \beta}$ (where $A_i$ is + the i-th column of A) is non-singular, i.e. all columns corresponding to the + indices in $\beta$ are linearly independent from each other. + +The geometric interpretation of basic feasible points is that all of them +are verticies of the polytope that bounds the feasible region. We will use this +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 verticies 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: + +- The lower-left point at the origin, given by the boundaries of the constraints + $x_1 \geq 0$ and $x_2 \geq 0$: + $$x^*_1 = \begin{bmatrix}0\\0\end{bmatrix}$$ +- The top-left point, at the intersection of constraint boundaries $x_1 \geq 0$ and + $-3x_1 + 2x_2 \leq 3$: + $$x_1 = 0 \;\;\; 2x_2 = 3 \Leftrightarrow x_2 = \frac32 \;\;\; x^*_2 = + \frac12 \cdot \begin{bmatrix}0\\3\end{bmatrix}$$ +- The top-center-left point at the intersection of constraint boundaries $-3x_1 + + 2x_2 \leq 3$ and $2x_1 + 3x_2 \leq 6$: + $$-3x_1 + 2x_2 + 3x_1 + \frac92 x_2 = \frac{13}{2} x_2 = 3 + 9 = 12 + \Leftrightarrow x_2 = 12 \cdot \frac2{13} = \frac{24}{13}$$$$ + -3x_1 + 2 \cdot \frac{24}{13} = 3 \Leftrightarrow x_1 = \frac{39 - 48}{13} + \cdot \frac{1}{-3} = \frac{3}{13} \;\;\; x^*_3 = \frac{1}{13} \cdot + \begin{bmatrix}3\\24\end{bmatrix}$$ +- The top-center-right point at the intersection of constraint boundaries $2x_1 + + 3x_2 \leq 6$ and $2x_1 + x_2 \leq 4$: + $$2x_1 + 3x_2 - 2x_1 - x_2 = 2x_2 = 6 - 4 = 2 \Leftrightarrow x_2 = 1 \;\;\; + 2x_1 + 1 = 4 \Leftrightarrow x_1 = \frac32 \;\;\; x^*_4 = \frac12 \cdot + \begin{bmatrix}3\\2\end{bmatrix}$$ +- The right point at the intersection of $2x_1 + x_2 \leq 4$ and $x_2 \geq + 0$: + $$x_2 = 0 \;\;\; 2x_1 + 0 = 4 \Leftrightarrow x_1 = 2 \;\;\; x^*_5 = + \begin{bmatrix}2\\0\end{bmatrix}$$ + +Therefore, $x^*_1$ to $x^*_5$ are all of the basic feasible points for this +constrained minimization problem. + We then compute the objective function value for each basic feasible point found, The smallest objective value will correspond with the constrained minimizer problem solution. $$ -x_1 = \begin{bmatrix}0\\0\end{bmatrix} \;\;\; f(x_1) = 4 \cdot 0 + 3 \cdot 0 = +x^*_1 = \begin{bmatrix}0\\0\end{bmatrix} \;\;\; f(x^*_1) = 4 \cdot 0 + 3 \cdot 0 = 0$$$$ -x_2 = \frac12 \cdot \begin{bmatrix}0\\3\end{bmatrix} \;\;\; -f(x_2) = 4 \cdot 0 + 3 \cdot \frac32 = \frac92$$$$ -x_3 = \frac{1}{13} \cdot \begin{bmatrix}3\\24\end{bmatrix} \;\;\; f(x_3) = 4 +x^*_2 = \frac12 \cdot \begin{bmatrix}0\\3\end{bmatrix} \;\;\; +f(x^*_2) = 4 \cdot 0 + 3 \cdot \frac{3}{2} = \frac92$$$$ +x^*_3 = \frac{1}{13} \cdot \begin{bmatrix}3\\24\end{bmatrix} \;\;\; f(x^*_3) = 4 \cdot \frac{3}{13} + 3 \cdot \frac{24}{13} = \frac{84}{13}$$$$ -x_4 = \frac12 \cdot \begin{bmatrix}3\\2\end{bmatrix} \;\;\; f(x_4) = 4 \cdot -frac32 + 3 \cdot 1 = 9$$$$ -x_5 = \begin{bmatrix}2\\0\end{bmatrix} \;\;\; 4 \cdot 2 + 1 \cdot 0 = 8$$ +x^*_4 = \frac12 \cdot \begin{bmatrix}3\\2\end{bmatrix} \;\;\; f(x^*_4) = 4 \cdot +\frac32 + 3 \cdot 1 = 9$$$$ +x^*_5 = \begin{bmatrix}2\\0\end{bmatrix} \;\;\; f(x^*_5) = 4 \cdot 2 + 1 \cdot 0 = 8$$ + +Therefore, $x^* = x^*_1 = \begin{bmatrix}0 & 0\end{bmatrix}^T$ is the global +constrained minimizer. + +# Exercise 3 + +## Exercise 3.1 + +## Exercise 3.2 + +The MATLAB code used to find the solution can be found under section 3.2 of the +`main.m` script. The solution is: + +$$x=\begin{bmatrix}0.7692\\-2.2308\\2.2308\end{bmatrix} \;\;\; \lambda= +\begin{bmatrix}-10.3846\\2.1538\end{bmatrix}$$ -Therefore, $x^* = x_1$ is the global constrained minimizer with $\lambda^* = \lambda_1 = NaN$ as -the slack variable value. diff --git a/Claudio_Maggioni_5/Claudio_Maggioni_5.pdf b/Claudio_Maggioni_5/Claudio_Maggioni_5.pdf index 62a821ab86b5c53e633a061f65b4f2a0cc48667f..23abb2b1abe0040535937916fea3755d1db1a593 100644 GIT binary patch delta 79759 zcmZs?Q*@wR)TJ9%Sh1Z{Y@=cu72CF*ys>TDM#Xl;wr$(z`@2VvzUXtc_qtzWuQ{JJ zcjh6?$T?ixFECDy#0@(tAXr(}Zk-Xn>salC61RZ7sqEtdd*A5V^Pl=!~zA8FX(<2maqxv?@uc}-E|7eB=Dt|{L??;lqaeW6# zgn)*zN-9Slpn^`-tK;t4ND0_2uiE}GOcJYGw{9D>CWF)(>V>wk#>5Mh4|WO}8iqrR z>92hrVrR$%+t=w|WV#8X)~I3?I~j!rMGK5BC21OSRaZPAwIL3^I?t+IgRXvUHg`Q& zP?cWW(iPLae?1-tXx8{^&Sz@iYYG1;8G97*=*`mCR+3vcHdy+6=4v$M4G@#U>m!OT zB|V>mKP0pcr!^4k1x6W_ltwf^OY|HMWn>-O*oCg@O`G|_jrws>b)J)7qmJKAUOdZS zNpxLLiH~1F3yj(xN8@#x5pwiyEjC`i`~(qxN}C%g{K&onn 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= 4 * px(i) + 3 * py(i); - fprintf("x1=%g x2=%g y=%g\n", px(i), py(i), obj); -end - - -%% Exercise 3.2 - -G = [6 2 1; 2 5 2; 1 2 4]; -c = [-8; -3; -3]; -A = [1 0 1; 0 1 1]; -b = [3; 0]; - -[x, lambda] = uzawa(G, c, A, b, [0;0;0], [0;0], 1e-8, 100); -display(x); -display(lambda); -