Richard Craig – EngD Systems Thinking in Cyberdefence

Problem

Find minimum of   x^2 + 4y^2 -3xy -3x -7y

Subject to   0<x<7    0<y<7

Change constraints till you hit both

Matlab Solution

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Programme  : EngD Maths for Systems
% Date       : 2011-06-01
% Session    : Optimisation
% Exercise   : Material Blending
% By         : Richard Craig
% % Description: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%% quadprog4.m
%Find minimum of  x^2 + 4y^2 -3xy -3x -7y
%Subject to   0<x<7    0<y<7

clc; clear;
%% Set up function for optimization
% H will give expression of form:
%     h11.x1^2 +h12.x1.x2 +h21.x2.x1 +h22.x2^2
% but programme forces symmetric Hessian: h12=h21 etc.,
% so might as well make it symmetric now.

% from the eq we can see H = [x^2 + 4y^2 -3xy] f = [ -3x -7y]
H = [2 -3; -3 8];       %H = [x^2 + 4y^2 -3xy]

% f will give inner product f.x
f = [-3; -7];           % f = [ -3x -7y]
%% Constraints
%x = quadprog(H,f,A,b,Aeq,beq) solves the preceding problem while
%additionally satisfying the equality constraints Aeq*x = beq.
%If no inequalities exist, set A = [] and b = [].
A = [];b = [];
%lower bounds on x:
lb = zeros(2,1);
%upper bound on x:
ub = [16 7];    %[7 7];     %Subject to   0<x<7    0<y<7
%% View Function
% make a "meshgrid" to cover allowed ranges of the variables.
u=linspace(lb(1),ub(1),11); v=linspace(lb(2),ub(2),11);
[X, Y]=meshgrid(u,v);
% put a value to each point-pair of the grid:
Z=0.5*(H(1,1)*X.^2 + H(2,2)*Y.^2 + 2*H(1,2).*X.*Y) + f(1).*X + f(2).*Y;
%plot the surface with some colour and a contour plot underneath:
h = figure(1);
surfc(X,Y,Z)
saveas(h,'QuadSurface.jpg');
%% Minimize using 'quadprog'
% Call for quadratic programming method:
[x,fval] = quadprog(H,f,A,b,[],[],lb,ub)