Run the Solver

Optimize by Calling run

Running a solver is nearly identical for GlobalSearch and MultiStart. The only difference in syntax is MultiStart takes an additional input describing the start points.

For example, suppose you want to find several local minima of the sixmin function

sixmin = 4x² – 2.1x⁴ + x⁶/3 + xy – 4y² + 4y⁴.

 Code for generating the figure

[X,Y] = meshgrid(-2:.1:2,-1:.05:1);
Z = 4*X.^2 - 2.1*X.^4 + X.^6./3 + X.*Y - 4*Y.^2 + 4*Y.^4;
surf(X,Y,Z);view(-44,14)
xlabel('x');ylabel('y')
title('sixmin(x,y)')

This function is also called the six-hump camel back function [3]. All the local minima lie in the region –3 ≤ x,y ≤ 3.

Example of Run with GlobalSearch

To find several local minima of the sixmin function using GlobalSearch, enter:

% % Set the random stream to get exactly the same output
% rng(14,'twister')
gs = GlobalSearch;
opts = optimoptions(@fmincon,'Algorithm','interior-point');
sixmin = @(x)(4*x(1)^2 - 2.1*x(1)^4 + x(1)^6/3 ...
    + x(1)*x(2) - 4*x(2)^2 + 4*x(2)^4);
problem = createOptimProblem('fmincon','x0',[-1,2],...
    'objective',sixmin,'lb',[-3,-3],'ub',[3,3],...
    'options',opts);
[xming,fming,flagg,outptg,manyminsg] = run(gs,problem);

The output of the run (which varies, based on the random seed):

xming,fming,flagg,outptg,manyminsg

xming =
    0.0898   -0.7127

fming =
   -1.0316

flagg =
     1

outptg =

  struct with fields:

                funcCount: 2115
         localSolverTotal: 3
       localSolverSuccess: 3
    localSolverIncomplete: 0
    localSolverNoSolution: 0
                  message: 'GlobalSearch stopped because it analyzed all the trial po...'

manyminsg = 
  1x2 GlobalOptimSolution array with properties:

    X
    Fval
    Exitflag
    Output
    X0

Example of Run with MultiStart

To find several local minima of the sixmin function using 50 runs of fmincon with MultiStart, enter:

% % Set the random stream to get exactly the same output
% rng(14,'twister')
ms = MultiStart;
opts = optimoptions(@fmincon,'Algorithm','interior-point');
sixmin = @(x)(4*x(1)^2 - 2.1*x(1)^4 + x(1)^6/3 ...
    + x(1)*x(2) - 4*x(2)^2 + 4*x(2)^4);
problem = createOptimProblem('fmincon','x0',[-1,2],...
    'objective',sixmin,'lb',[-3,-3],'ub',[3,3],...
    'options',opts);
[xminm,fminm,flagm,outptm,manyminsm] = run(ms,problem,50);

The output of the run (which varies based on the random seed):

xminm,fminm,flagm,outptm,manyminsm

xminm =
    0.0898    -0.7127

fminm =
   -1.0316

flagm =
     1

outptm =

  struct with fields:

                funcCount: 2034
         localSolverTotal: 50
       localSolverSuccess: 50
    localSolverIncomplete: 0
    localSolverNoSolution: 0
                  message: 'MultiStart completed the runs from all start points.…'

manyminsm = 
  1x6 GlobalOptimSolution array with properties:

    X
    Fval
    Exitflag
    Output
    X0

In this case, MultiStart located all six local minima, while GlobalSearch located two. For pictures of the MultiStart solutions, see Visualize the Basins of Attraction.