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spfminsearch

Optimizes the sparse grid interpolant using MATLAB's fminsearch method.

Syntax

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

Description

X = spfminsearch(Z) Starts the search at the best available sparse grid point and attempts to find a local minimizer of the sparse grid interpolant Z. The entire range of the sparse grid interpolant is searched.

X = spfminsearch(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 = spfminsearch(Z,XBOX,OPTIONS) Minimizes with the default optimization parameters replaced by values in the structure OPTIONS, created with the spoptimset function. See spoptimset for details.

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

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

[X,FVAL,EXITFLAG,OUTPUT] = spfminsearch(...) Returns a structure OUTPUT with the number of function evaluations in OUTPUT.nFEvals and the computing time in .time. The OUTPUT result from the fminsearch call are returned as OUTPUT.fminsearchOutput.

Examples

spfminsearch internally calls MATLAB's fminsearch function to perform the search. The sparse grid interpolant is modified by a penalty function such that the search is restricted to the provided search box.

spfminsearch is a derivative-free method that is suitable for all sparse grid types. However, it is usually outperformed by spcompsearch for the grid types Maximum, NoBoundary, or Clenshaw-Curtis, and by spcgsearch for the grid type Chebyshev.

As with the example presented for spcgsearch, we consider the six-hump camel-back function (see that example for further details). 

f = @(x,y) (4-2.1.*x.^2+x.^4./3).*x.^2+x.*y+(-4+4.*y.^2).*y.^2;

Interpolant creation and setting optional parameters:

options = spset('keepFunctionValues','on', 'GridType', 'Chebyshev', ...
  'DimensionAdaptive', 'on', 'DimAdaptDegree', 1, 'MinPoints', 10);
range = [-3 3; -2 2];
z = spvals(f, 2, range, options);
optoptions = spoptimset('Display', 'iter');

Performing the optimization:

[xopt, fval] = spfminsearch(z, [], optoptions)

 
 Iteration   Func-count     min f(x)         Procedure
     0            1        -0.970563         
     1            3        -0.970563         initial simplex
     2            5        -0.997137         expand
     3            7         -0.99731         reflect
     4            9         -0.99731         contract inside
     5           11        -0.999861         contract inside
     6           13         -1.00004         reflect
     7           15         -1.00004         contract inside
     8           17         -1.00004         contract inside
     9           19         -1.00004         contract inside
    10           21          -1.0002         expand
    11           23         -1.00055         expand
    12           25         -1.00087         expand
    13           27         -1.00192         expand
    14           29         -1.00227         expand
    15           31         -1.00483         expand
    16           32         -1.00483         reflect
    17           34         -1.00771         expand
    18           36         -1.01172         expand
    19           38         -1.01615         expand
    20           40         -1.02567         expand
    21           41         -1.02567         reflect
    22           43         -1.03063         reflect
    23           44         -1.03063         reflect
    24           46         -1.03083         reflect
    25           48         -1.03119         contract inside
    26           50         -1.03155         contract inside
    27           52         -1.03155         contract inside
    28           54         -1.03155         contract inside
    29           56         -1.03162         contract inside
    30           58         -1.03162         contract inside
    31           60         -1.03162         contract inside
    32           62         -1.03162         reflect
    33           64         -1.03163         contract inside
    34           66         -1.03163         contract inside
    35           68         -1.03163         contract inside
    36           70         -1.03163         contract inside
    37           72         -1.03163         contract inside
    38           74         -1.03163         contract inside
    39           76         -1.03163         contract inside
    40           78         -1.03163         contract inside
    41           80         -1.03163         contract inside
    42           82         -1.03163         reflect
    43           84         -1.03163         contract inside
 
Optimization terminated:
 the current x satisfies the termination criteria using OPTIONS.TolX of 1.000000e-04 
 and F(X) satisfies the convergence criteria using OPTIONS.TolFun of 1.000000e-04 

xopt =
   -0.0899
    0.7127
fval =
   -1.0316

See Also

spoptimset.
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