spmultistart

Attemps to find a global optimizer of the sparse grid interpolant by performing multiple local searches starting from random start points.

Syntax

X = spmultistart(Z)
X = spmultistart(Z,XBOX)
X = spmultistart(Z,XBOX,OPTIONS)
[X,FVAL] = spmultistart(...)
[X,FVAL,EXITFLAG] = spmultistart(...)
[X,FVAL,EXITFLAG,OUTPUT] = spmultistart(...)

Description

X = spmultistart(Z) Attemps to find a global optimizer X of the sparse grid interpolant Z by performing multiple local searches starting from random start points. The entire range of the sparse grid interpolant is searched. Using the default settings, the first start point is not random but the best available sparse grid point. By default, spcompsearch is used for the local searches if the grid type is not of typeChebyshev. If it is, spcgsearch is used.

X = spmultistart(Z,XBOX) Uses the search box XBOX = [a1, b1; a2, b2; ...]. The size of search box XBOX must be smaller than or equal to the range of the interpolant.

X = spmultistart(Z,XBOX,OPTIONS) Minimizes with the default optimization parameters replaced by values in the structure OPTIONS, created with the spoptimset function. For instance, the local optimization method can be selected. See spoptimset for details.

[X,FVAL] = spmultistart(...) Returns the value of the sparse grid interpolant at X.

[X,FVAL,EXITFLAG] = spmultistart(...) Returns an EXITFLAG that describes the exit condition of spmultistart. Possible values of EXITFLAG and the corresponding exit conditions are

• 1   spmultistart converged to at least one solution X.
• 0   Maximum number of function evaluations or iterations reached for all local searches.

[X,FVAL,EXITFLAG,OUTPUT] = spmultistart(...) Returns a structure OUTPUT with the total computing time in .time, and a cell array of all local search results .allResults.

Examples

The following example demonstrates the usage of spmultistart with its default options. Many parameters can be modified using spoptimset.

Here, we optimize Branin's function, a function with three global optimizers.

f = inline(['(5/pi*x-5.1/(4*pi^2)*x.^2+y-6).^2 +' ...
            '10*(1-1/(8*pi))*cos(x)+10']);

Interpolant creation:

range = [-5 10; 0 15];
options = spset('keepFunctionValues','on', 'GridType', 'Chebyshev', ...
                'DimensionAdaptive', 'on', 'DimAdaptDegree', 1, ...
                'MinPoints', 10);
z = spvals(f, 2, range, options);

Performing the optimization:

[xopt, fval, exitflag, output] = spmultistart(z)

xopt =
   -3.1416
   12.2751
fval =
    0.3978
exitflag =
     1
output = 
          time: 4.0003
    allResults: [1x1 struct]

All local optimization results are available from the output structure:

output.allResults.x
output.allResults.fval

ans =
    3.1416    9.4248    3.1416    9.4248    3.1416    9.4248   -3.1416    9.4248   -3.1416    9.4248
    2.2750    2.4750    2.2750    2.4750    2.2750    2.4750   12.2751    2.4750   12.2751    2.4750
ans =
    0.3980    0.3979    0.3980    0.3979    0.3980    0.3979    0.3978    0.3979    0.3978    0.3979

See Also

spoptimset.