Surrogate Optimization of Multidimensional Function

This example shows the behavior of three recommended solvers on a minimization problem. The objective function is the multirosenbrock function:

type multirosenbrock

function F = multirosenbrock(x)
% This function is a multidimensional generalization of Rosenbrock's
% function. It operates in a vectorized manner, assuming that x is a matrix
% whose rows are the individuals.

% Copyright 2014 by The MathWorks, Inc.

N = size(x,2); % assumes x is a row vector or 2-D matrix
if mod(N,2) % if N is odd
    error('Input rows must have an even number of elements')
end

odds  = 1:2:N-1;
evens = 2:2:N;
F = zeros(size(x));
F(:,odds)  = 1-x(:,odds);
F(:,evens) = 10*(x(:,evens)-x(:,odds).^2);
F = sum(F.^2,2);

The multirosenbrock function has a single local minimum of 0 at the point [1,1,...,1]. See how well the three best solvers for general nonlinear problems work on this function in 20 dimensions with a challenging maximum function count of only 200.

Set up the problem.

N = 20; % any even number
mf = 200; % max fun evals
fun = @multirosenbrock;
lb = -3*ones(1,N);
ub = -lb;
rng default
x0 = -3*rand(1,N);

Set options for surrogateopt to use only 200 function evaluations, and then run the solver.

options = optimoptions('surrogateopt','MaxFunctionEvaluations',mf);
[xm,fvalm,~,~,pop] = surrogateopt(fun,lb,ub,options);

surrogateopt stopped because it exceeded the function evaluation limit set by 
'options.MaxFunctionEvaluations'.

Set similar options for patternsearch, including a plot function to monitor the optimization.

psopts = optimoptions('patternsearch','PlotFcn','psplotbestf','MaxFunctionEvaluations',mf);
[psol,pfval] = patternsearch(fun,x0,[],[],[],[],lb,ub,[],psopts);

Maximum number of function evaluations exceeded: increase options.MaxFunctionEvaluations.

Set similar options for fmincon.

opts = optimoptions('fmincon','PlotFcn','optimplotfval','MaxFunctionEvaluations',mf);
[fmsol,fmfval,eflag,fmoutput] = fmincon(fun,x0,[],[],[],[],lb,ub,[],opts);

Solver stopped prematurely.

fmincon stopped because it exceeded the function evaluation limit,
options.MaxFunctionEvaluations = 2.000000e+02.

For this extremely restricted number of function evaluations, the surrogateopt solution is closest to the true minimum value of 0.

table(fvalm,pfval,fmfval,'VariableNames',{'surrogateopt','patternsearch','fmincon'})

ans=1×3 table
    surrogateopt    patternsearch    fmincon
    ____________    _____________    _______

       8.9965           774.8         493.7 

Allowing another 200 function evaluations shows that the other solvers rapidly approach the true solution, while surrogateopt does not improve significantly. Restart the solvers from their previous solutions, which adds 200 function evaluations to each optimization.

options = optimoptions(options,'InitialPoints',pop);
[xm,fvalm,~,~,pop] = surrogateopt(fun,lb,ub,options);

surrogateopt stopped because it exceeded the function evaluation limit set by 
'options.MaxFunctionEvaluations'.

[psol,pfval] = patternsearch(fun,psol,[],[],[],[],lb,ub,[],psopts);

Maximum number of function evaluations exceeded: increase options.MaxFunctionEvaluations.

[fmsol,fmfval,eflag,fmoutput] = fmincon(fun,fmsol,[],[],[],[],lb,ub,[],opts);

Solver stopped prematurely.

fmincon stopped because it exceeded the function evaluation limit,
options.MaxFunctionEvaluations = 2.000000e+02.

table(fvalm,pfval,fmfval,'VariableNames',{'surrogateopt','patternsearch','fmincon'})

ans=1×3 table
    surrogateopt    patternsearch    fmincon
    ____________    _____________    _______

       8.2655          326.73        8.5989