peaks Minimization with Pattern Search
Copyright (c) 2010, The MathWorks, Inc. All rights reserved.
Contents
Objective Function
We wish find the minimum of the peaks function
clear all, close all, clc peaks
z = 3*(1-x).^2.*exp(-(x.^2) - (y+1).^2) ... - 10*(x/5 - x.^3 - y.^5).*exp(-x.^2-y.^2) ... - 1/3*exp(-(x+1).^2 - y.^2)

Nonlinear Constraint Function
Subject to a nonlinear constraint defined by a circular region of radius three around the origin
type circularConstraint
function [c,ceq] = circularConstraint(x) % Nonlinear constraint definition % Copyright (c) 2010, The MathWorks, Inc. % All rights reserved. % Define nonlinear equality constraint (none) ceq = []; % Define nonlinear inequality constraint % circular region with radius 3: x1^2 + x^2 -3^2 <= 0 c = x(:,1).^2 + x(:,2).^2 - 9;
Define Optimization Problem
problem = createOptimProblem('fmincon',... 'objective',@(x) peaks(x(1),x(2)), ... 'nonlcon',@circularConstraint,... 'x0',[-1 -1],... 'lb',[-3 -3],... 'ub',[3 3],... 'options',optimset('OutputFcn',... @peaksPlotIterates))
problem = objective: @(x)peaks(x(1),x(2)) x0: [-1 -1] Aineq: [] bineq: [] Aeq: [] beq: [] lb: [-3 -3] ub: [3 3] nonlcon: @circularConstraint solver: 'fmincon' options: [1x1 struct]
Run the solver fmincon from the inital point
We can see the solution is not the global minimum
[x,f] = fmincon(problem)
x = -1.3473 0.2045 f = -3.0498

Use Pattern Search to Find the Global Minimum
Solve the problem using pattern search.
problem.solver = 'patternsearch'; problem.options = psoptimset('OutputFcn',@peaksPlotIterates,... 'Display','iter',... 'SearchMethod',{@searchlhs}) [x,f] = patternsearch(problem)
problem = objective: @(x)peaks(x(1),x(2)) x0: [-1 -1] Aineq: [] bineq: [] Aeq: [] beq: [] lb: [-3 -3] ub: [3 3] nonlcon: @circularConstraint solver: 'patternsearch' options: [1x1 struct] max Iter f-count f(x) constraint MeshSize Method 0 1 1.85589 0 1 1 7 0.246458 0 0.009772 Update multipliers 2 63 -3.04527 0 0.000955 Update multipliers 3 161 -6.55113 0 9.333e-005 Update multipliers 4 263 -6.55113 0 9.12e-006 Update multipliers 5 376 -6.55113 0 8.913e-007 Update multipliers Optimization terminated: mesh size less than options.TolMesh and constraint violation is less than options.TolCon. x = 0.2283 -1.6255 f = -6.5511
