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Compare Surrogate Optimization with Other Solvers

This example compares surrogateopt to two other solvers: fmincon, the recommended solver for smooth problems, and patternsearch, the recommended solver for nonsmooth problems. The example uses a nonsmooth function on a two-dimensional region.

type nonSmoothFcn
function [f, g] = nonSmoothFcn(x)
%NONSMOOTHFCN is a non-smooth objective function

%   Copyright 2005 The MathWorks, Inc.

for i = 1:size(x,1)
    if  x(i,1) < -7
        f(i) = (x(i,1))^2 + (x(i,2))^2 ;
    elseif x(i,1) < -3
        f(i) = -2*sin(x(i,1)) - (x(i,1)*x(i,2)^2)/10 + 15 ;
    elseif x(i,1) < 0
        f(i) = 0.5*x(i,1)^2 + 20 + abs(x(i,2))+ patho(x(i,:));
    elseif x(i,1) >= 0
        f(i) = .3*sqrt(x(i,1)) + 25 +abs(x(i,2)) + patho(x(i,:));
    end
end

%Calculate gradient
g = NaN;
if x(i,1) < -7
    g = 2*[x(i,1); x(i,2)];
elseif x(i,1) < -3
    g = [-2*cos(x(i,1))-(x(i,2)^2)/10; -x(i,1)*x(i,2)/5];
elseif x(i,1) < 0
    [fp,gp] = patho(x(i,:));
    if x(i,2) > 0
        g = [x(i,1)+gp(1); 1+gp(2)];
    elseif x(i,2) < 0
        g =  [x(i,1)+gp(1); -1+gp(2)];
    end
elseif x(i,1) >0
    [fp,gp] = patho(x(i,:));
    if x(i,2) > 0
        g = [.15/sqrt(x(i,1))+gp(1); 1+ gp(2)];
    elseif x(i,2) < 0
        g = [.15/sqrt(x(i,1))+gp(1); -1+ gp(2)];
    end
end

function [f,g] = patho(x)
Max = 500;
f = zeros(size(x,1),1);
g = zeros(size(x));
for k = 1:Max  %k 
   arg = sin(pi*k^2*x)/(pi*k^2);
   f = f + sum(arg,2);
   g = g + cos(pi*k^2*x);
end
mplier = 0.1; % Scale the control variable
Objfcn = @(x)nonSmoothFcn(mplier*x); % Handle to the objective function
range = [-6 6;-6 6]/mplier; % Range used to plot the objective function
rng default % Reset the global random number generator
showNonSmoothFcn(Objfcn,range);
title('Nonsmooth Objective Function')
view(-151,44)

Figure contains an axes. The axes with title Nonsmooth Objective Function contains 4 objects of type surface, contour.

drawnow

See how well surrogateopt does in locating the global minimum within the default number of iterations.

lb = -6*ones(1,2)/mplier;
ub = -lb;
[xs,fvals,eflags,outputs] = surrogateopt(Objfcn,lb,ub);

Figure Optimization Plot Function contains an axes. The axes with title Best Function Value: 13 contains an object of type line. This object represents Best function value.

surrogateopt stopped because it exceeded the function evaluation limit set by 
'options.MaxFunctionEvaluations'.
fprintf("Lowest found value = %g.\r",fvals)
Lowest found value = 13.
figure
showNonSmoothFcn(Objfcn,range);
view(-151,44)
hold on
p1 = plot3(xs(1),xs(2),fvals,'om','MarkerSize',15,'MarkerFaceColor','m');
legend(p1,{'Solution'})
hold off

Figure contains an axes. The axes contains 5 objects of type surface, contour, line. This object represents Solution.

Compare with patternsearch

Set patternsearch options to use the same number of function evaluations, starting from a random point within the bounds.

rng default
x0 = lb + rand(size(lb)).*(ub - lb);
optsps = optimoptions('patternsearch','MaxFunctionEvaluations',200,'PlotFcn','psplotbestf');
[xps,fvalps,eflagps,outputps] = patternsearch(Objfcn,x0,[],[],[],[],lb,ub,[],optsps);
Optimization terminated: mesh size less than options.MeshTolerance.

Figure Pattern Search contains an axes. The axes with title Best Function Value: 13 contains an object of type line.

figure
showNonSmoothFcn(Objfcn,range);
view(-151,44)
hold on
p1 = plot3(x0(1),x0(2),Objfcn(x0),'ob','MarkerSize',12,'MarkerFaceColor','b');
p2 = plot3(xps(1),xps(2),fvalps,'om','MarkerSize',15,'MarkerFaceColor','m');
legend([p1,p2],{'Start Point','Solution'})
hold off

Figure contains an axes. The axes contains 6 objects of type surface, contour, line. These objects represent Start Point, Solution.

patternsearch found the same solution as surrogateopt.

Restrict the number of function evaluations and try again.

optsurr = optimoptions('surrogateopt','MaxFunctionEvaluations',40);
[xs,fvals,eflags,outputs] = surrogateopt(Objfcn,lb,ub,optsurr);

Figure Optimization Plot Function contains an axes. The axes with title Best Function Value: 13.0238 contains an object of type line. This object represents Best function value.

surrogateopt stopped because it exceeded the function evaluation limit set by 
'options.MaxFunctionEvaluations'.
optsps.MaxFunctionEvaluations = 40;
[xps,fvalps,eflagps,outputps] = patternsearch(Objfcn,x0,[],[],[],[],lb,ub,[],optsps);
Maximum number of function evaluations exceeded: increase options.MaxFunctionEvaluations.

Figure Pattern Search contains an axes. The axes with title Best Function Value: 13.0983 contains an object of type line.

Again, both solvers found the global solution quickly.

Compare with fmincon

fmincon is efficient at finding a local solution near the start point. However, it can easily get stuck far from the global solution in a nonconvex or nonsmooth problem.

Set fmincon options to use a plot function, the same number of function evaluations as the previous solvers, and the same start point as patternsearch.

opts = optimoptions('fmincon','PlotFcn','optimplotfval','MaxFunctionEvaluations',200);
[fmsol,fmfval,eflag,fmoutput] = fmincon(Objfcn,x0,[],[],[],[],lb,ub,[],opts);

Figure Optimization Plot Function contains an axes. The axes with title Current Function Value: 30.1703 contains an object of type line.

Local minimum possible. Constraints satisfied.

fmincon stopped because the size of the current step is less than
the value of the step size tolerance and constraints are 
satisfied to within the value of the constraint tolerance.
figure
showNonSmoothFcn(Objfcn,range);
view(-151,44)
hold on
p1 = plot3(x0(1),x0(2),Objfcn(x0),'ob','MarkerSize',12,'MarkerFaceColor','b');
p2 = plot3(fmsol(1),fmsol(2),fmfval,'om','MarkerSize',15,'MarkerFaceColor','m');
legend([p1,p2],{'Start Point','Solution'})
hold off

Figure contains an axes. The axes contains 6 objects of type surface, contour, line. These objects represent Start Point, Solution.

fmincon is stuck in a local minimum near the start point.

See Also

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