2. Existence of optimal solutions and optimality conditions

Exercises Chapter 2

2.1 Existence of global optima for quadratic programming problems

Exercise 1.1. Prove that the quadratic programming problem

$$\left\lbrace \begin{array}{ll} \mathrm{minimize} & \frac{1}{2}x_1^2 -x_2^2 +x_1 -{2x}_2 \\ \mathrm{s.t} & {-x}_1 {+x}_2 \le -1\\ \ & {-x}_2 \le 0 \end{array}\right.$$

close all;
clear;
clc;

matlab.lang.OnOffSwitchState = 1;

Q = [1 0
     0 -1];


% The Quadratic Problem is indefinite
eig(Q)

% Objective function
f = @(x1,x2) (1/2).*x1.^2 - (1/2).*x2.^2 + x1 - 2.*x2;
g1 = @(x1,x2) -x1 + x2 <= -1;
g2 = @(x1,x2) -x2 <= 0;

% 3D Surface plot

fsurf(f);
hold on
fsurf(g1);
fsurf(g2);
hold off

Output

% 2D Contour plot 

fcontour(f, 'LevelList',1);
hold on
fcontour(g1,'LevelList',1);
fcontour(g2, 'LevelList',1);
hold off

Output