hw5: work

Claudio_Maggioni_5/Claudio_Maggioni_5.md

@@ -0,0 +1,46 @@
+<!-- vim: set ts=2 sw=2 et tw=80: -->
+
+---
+title: Homework 5 -- Optimization Methods
+author: Claudio Maggioni
+header-includes:
+- \usepackage{amsmath}
+- \usepackage{hyperref}
+- \usepackage[utf8]{inputenc}
+- \usepackage[margin=2.5cm]{geometry}
+- \usepackage[ruled,vlined]{algorithm2e}
+- \usepackage{float}
+- \floatplacement{figure}{H}
+- \hypersetup{colorlinks=true,linkcolor=blue}
+
+---
+\maketitle
+
+# Exercise 2
+
+## Exercise 2.1
+
+The resulting MATLAB plot of each constraint and of the feasible region is shown
+below:
+
+![Plot of feasible region and constraints](./ex2-1.png)
+
+## Exercise 2.3
+
+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.
+
+$$
+x_1 = \begin{bmatrix}0\\0\end{bmatrix} \;\;\; f(x_1) = 4 \cdot 0 + 3 \cdot 0 =
+0$$$$
+x_2 = \frac12 \cdot \begin{bmatrix}0\\3\end{bmatrix} \;\;\;
+f(x_2) = 4 \cdot 0 + 3 \cdot \frac32 = \frac92$$$$
+x_3 = \frac{1}{13} \cdot \begin{bmatrix}3\\24\end{bmatrix} \;\;\; f(x_3) = 4
+\cdot \frac{3}{13} + 3 \cdot \frac{24}{13} = \frac{84}{13}$$$$
+x_4 = \frac12 \cdot \begin{bmatrix}3\\2\end{bmatrix} \;\;\; f(x_4) = 4 \cdot
+frac32 + 3 \cdot 1 = 9$$$$
+x_5 = \begin{bmatrix}2\\0\end{bmatrix} \;\;\; 4 \cdot 2 + 1 \cdot 0 = 8$$
+
+Therefore, $x^* = x_1$ is the global constrained minimizer with $\lambda^* = \lambda_1 = NaN$ as
+the slack variable value.

Claudio_Maggioni_5/main.asv

@@ -20,36 +20,52 @@ i2 = solve(c1 == c4, x1, 'Real', true);
 i3 = solve(c4 == 0, x1, 'Real', true);
 
 px = double([0 0 i1 i2 i3]);
-py = double([0 subs(c2, x1, 0) subs(c1, x1, i1) subs(c4, x1, i2) subs(c4, x1, i3)]); 
+pysym = [0 subs(c2, x1, 0) subs(c1, x1, i1) subs(c4, x1, i2) subs(c4, x1, i3)];
+py = double(pysym); 
 
 xl = -0.05;
 xh = 2.05;
 
-colors = [1 0 1; 0 0 1; 1 0 0; 0 1 0];
+orange = [232/255 128/255 18/255];
+grey = [0.5 0.5 0.5];
+colors = [orange; 0 0 1; 1 0 0; 0 1 0];
+dirs = [30 30 30 30];
 
 axis([-0.05 2.05 -0.15 5.15])
 i = 1;
 hold on
 for c = [c1 c2 c3 c4]
-    %plot([xl, xh], [subs(c, x1, xl), subs(c, x1, xh)]);
-    hatchfill(patch([xl, xl, xh, xh], ...
-              double([0, subs(c, x1, xl), subs(c, x1, xh), 0]), ...
-              colors(i, :)), ...
-        'HatchColor', colors(i, :), 'HatchOffset', (i-1)/5);
+    pl = patch([xl, xl, xh, xh], ...
+      double([-0.15, subs(c, x1, xl), subs(c, x1, xh), -0.15]), ...
+      colors(i, :));
+    pl.EdgeColor = colors(i, :);
+    pl.FaceAlpha = .2;
+    pl.EdgeAlpha = .2;
     i = i + 1;
 end
-xline(0);
-hatchfill(patch([0, 0, xh, xh], ...
+
+pl = patch([0, 0, xh, xh], ...
               double([0, 5.15, 5.15, 0]), ...
-              'white'), ...
-        'HatchColor', [1 0.5 0], 'HatchOffset', 4/5);
-plot([xl, xh], [0, 0]);
-patch(px, py, 'black');
-alpha(.05)
-legend('','2x1 + 3x2 <= 6', '', '-3x1 + 2x2 <= 3', '', '2x2 <= 5', ...
-    '', '2x1 + x2 <= 4', '', 'x1 > 0 and x2 > 0', 'feasible region');
+              grey);
+pl.EdgeColor = grey;
+pl.FaceAlpha = .2;
+pl.EdgeAlpha = .2;
+
+pl = patch(px, py, 'green');
+pl.EdgeColor = 'green';
+
+legend('2x1 + 3x2 <= 6', '-3x1 + 2x2 <= 3', '2x2 <= 5', ...
+    '2x1 + x2 <= 4', 'x1 > 0 and x2 > 0', 'feasible region');
 hold off
 
+%% gsppn
+
+for i=1:5
+    obj = 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];

Claudio_Maggioni_5/main.m

@@ -20,36 +20,52 @@ i2 = solve(c1 == c4, x1, 'Real', true);
 i3 = solve(c4 == 0, x1, 'Real', true);
 
 px = double([0 0 i1 i2 i3]);
-py = double([0 subs(c2, x1, 0) subs(c1, x1, i1) subs(c4, x1, i2) subs(c4, x1, i3)]); 
+pysym = [0 subs(c2, x1, 0) subs(c1, x1, i1) subs(c4, x1, i2) subs(c4, x1, i3)];
+py = double(pysym); 
 
 xl = -0.05;
 xh = 2.05;
 
-colors = [224/255 6/255 191/255; 0 0 1; 1 0 0; 0 1 0];
+orange = [232/255 128/255 18/255];
+grey = [0.5 0.5 0.5];
+colors = [orange; 0 0 1; 1 0 0; 0 1 0];
+dirs = [30 30 30 30];
 
 axis([-0.05 2.05 -0.15 5.15])
 i = 1;
 hold on
 for c = [c1 c2 c3 c4]
-    %plot([xl, xh], [subs(c, x1, xl), subs(c, x1, xh)]);
-    hatchfill(patch([xl, xl, xh, xh], ...
+    pl = patch([xl, xl, xh, xh], ...
       double([-0.15, subs(c, x1, xl), subs(c, x1, xh), -0.15]), ...
-              colors(i, :)), ...
-        'HatchColor', colors(i, :), 'HatchOffset', (i-1)/5, 'HatchAngle', 45);
+      colors(i, :));
+    pl.EdgeColor = colors(i, :);
+    pl.FaceAlpha = .2;
+    pl.EdgeAlpha = .2;
     i = i + 1;
 end
-%xline(0);
-hatchfill(patch([0, 0, xh, xh], ...
+
+pl = patch([0, 0, xh, xh], ...
               double([0, 5.15, 5.15, 0]), ...
-              'white'), ...
-        'HatchColor', [249/255 216/255 49/255], 'HatchOffset', 4/5, 'HatchAngle', 45);
-%plot([xl, xh], [0, 0]);
-hatchfill(patch(px, py, 'black'),'HatchColor', 'black', 'HatchAngle', 90);
-alpha(.02)
-legend('','2x1 + 3x2 <= 6', '', '-3x1 + 2x2 <= 3', '', '2x2 <= 5', ...
-    '', '2x1 + x2 <= 4', '', 'x1 > 0 and x2 > 0', '', 'feasible region');
+              grey);
+pl.EdgeColor = grey;
+pl.FaceAlpha = .2;
+pl.EdgeAlpha = .2;
+
+pl = patch(px, py, 'green');
+pl.EdgeColor = 'green';
+
+legend('2x1 + 3x2 <= 6', '-3x1 + 2x2 <= 3', '2x2 <= 5', ...
+    '2x1 + x2 <= 4', 'x1 > 0 and x2 > 0', 'feasible region');
 hold off
 
+%% gsppn
+
+for i=1:5
+    obj = 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];