Simple Quadratic Programing

Synopsis

Note that x^T denotes the transpose of x, and Ax < b means that the inequality is taken element-wise over the vectors Ax and b. The above objective function is convex if and only if H is positive- semidefinite. The quadprog function expects a problem of the above form, defined by the parameters fH; f; A; b; Aeq; beq; l; u; x0; H and f are required, the others are optional (empty matrix [])

General Formul

Example

 rewrite pattern above

Code in matlab

clc;clear all;close all;
H = diag([1; 0]);
f = [3; 4];
A = [-1 -3; 2 5; 3 4];
b = [-15; 100; 80];
l = zeros(2,1);
Aeq = [];
Beq = [];
u = [];
x0 = [];
options = optimset('Algorithm','interior-point-convex');
[x,fval] = quadprog(H,f,A,b,Aeq,Beq,l,u,x0,options);

Output

>> x

x =

    0.0000
    5.0000

>> fval

fval =

   20.0000

>>

We can verify  that x∗ = 0; y∗ = 5, with an optimal value 20.