peaks Minimization with MultiStart
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 MultiStart to Find the Global Minimum
Define the multistart solver
close all
ms = MultiStart
ms = MultiStart Properties: UseParallel: 'never' Display: 'final' TolFun: 1.0000e-006 TolX: 1.0000e-006 MaxTime: Inf StartPointsToRun: 'all'
Run Multistart
Well use 5 starting points
[x,f,exitflag,output,solutions] = run(ms, problem, 5)
MultiStart completed the runs from all start points. All 5 local solver runs converged with a positive local solver exit flag. x = 0.2283 -1.6255 f = -6.5511 exitflag = 1 output = funcCount: 273 localSolverTotal: 5 localSolverSuccess: 5 localSolverIncomplete: 0 localSolverNoSolution: 0 message: [1x127 char] solutions = 1x5 GlobalOptimSolution Properties: X Fval Exitflag Output X0
