Two-Dimensional Semi-Infinite Constraint

Find values ofxthat minimize

f(x) = (x₁– 0.2)²+ (x₂– 0.2)²+ (x₃– 0.2)²,

where

$\begin{array}{l} K_{1} (x, w) = \sin (w_{1} x_{1}) \cos (w_{2} x_{2}) - \frac{1}{1000} {(w_{1} - 50)}^{2} - \sin (w_{1} x_{3}) - x_{3} + ... \\ \sin (w_{2} x_{2}) \cos (w_{1} x_{1}) - \frac{1}{1000} {(w_{2} - 50)}^{2} - \sin (w_{2} x_{3}) - x_{3} \leq 1.5, \end{array}$

for all values ofw₁andw₂over the ranges

1 ≤w₁≤ 100,
1 ≤w₂≤ 100,

starting at the pointx= [0.25,0.25,0.25].

Note that the semi-infinite constraint is two-dimensional, that is, a matrix.

First, write a file that computes the objective function.

function f = myfun(x,s) % Objective function f = sum((x-0.2).^2);

Second, write a file for the constraints, calledmycon.m. Include code to draw the surface plot of the semi-infinite constraint each timemyconis called. This enables you to see how the constraint changes asXis being minimized.

function [c,ceq,K1,s] = mycon(X,s) % Initial sampling interval if isnan(s(1,1)), s = [2 2]; end % Sampling set w1x = 1:s(1,1):100; w1y = 1:s(1,2):100; [wx,wy] = meshgrid(w1x,w1y); % Semi-infinite constraint K1 = sin(wx*X(1)).*cos(wx*X(2))-1/1000*(wx-50).^2 -... sin(wx*X(3))-X(3)+sin(wy*X(2)).*cos(wx*X(1))-... 1/1000*(wy-50).^2-sin(wy*X(3))-X(3)-1.5; % No finite nonlinear constraints c = []; ceq=[]; % Mesh plot m = surf(wx,wy,K1,'edgecolor','none','facecolor','interp'); camlight headlight title('Semi-infinite constraint') drawnow

Next, invoke an optimization routine.

x0 = [0.25, 0.25, 0.25]; % Starting guess [x,fval] = fseminf(@myfun,x0,1,@mycon)

After nine iterations, the solution is

x x = 0.2523 0.1715 0.1938

and the function value at the solution is

fval fval = 0.0036

The goal was to minimize the objectivef(x) such that the semi-infinite constraint satisfiedK₁(x,w) ≤ 1.5. Evaluatingmyconat the solutionxand looking at the maximum element of the matrixK1shows the constraint is easily satisfied.

[c,ceq,K1] = mycon(x,[0.5,0.5]); % Sampling interval 0.5 max(max(K1)) ans = -0.0370

This call tomyconproduces the following surf plot, which shows the semi-infinite constraint atx.

Two-Dimensional Semi-Infinite Constraint

See Also

Related Topics

Optimization Toolbox Documentation

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