Multiple Solutions

About Multiple Solutions

You obtain multiple solutions in an object by calling run with the syntax

[x fval eflag output manymins] = run(...);

manymins is a vector of solution objects; see GlobalOptimSolution. The manymins vector is in order of objective function value, from lowest (best) to highest (worst). Each solution object contains the following properties (fields):

  • X — a local minimum

  • Fval — the value of the objective function at X

  • Exitflag — the exit flag for the local solver (described in the local solver function reference page: fmincon, fminunc, lsqcurvefit, or lsqnonlin)

  • Output — an output structure for the local solver (described in the local solver function reference page: fmincon, fminunc, lsqcurvefit, or lsqnonlin)

  • X0 — a cell array of start points that led to the solution point X

There are several ways to examine the vector of solution objects:

  • In the MATLAB® Workspace Browser. Double-click the solution object, and then double-click the resulting display in the Variables editor.

  • Using dot notation. GlobalOptimSolution properties are capitalized. Use proper capitalization to access the properties.

    For example, to find the vector of function values, enter:

    fcnvals = [manymins.Fval]
    
    fcnvals =
       -1.0316   -0.2155         0

    To get a cell array of all the start points that led to the lowest function value (the first element of manymins), enter:

    smallX0 = manymins(1).X0
  • Plot some field values. For example, to see the range of resulting Fval, enter:

    hist([manymins.Fval])

    This results in a histogram of the computed function values. (The figure shows a histogram from a different example than the previous few figures.)

Change the Definition of Distinct Solutions

You might find out, after obtaining multiple local solutions, that your tolerances were not appropriate. You can have many more local solutions than you want, spaced too closely together. Or you can have fewer solutions than you want, with GlobalSearch or MultiStart clumping together too many solutions.

To deal with this situation, run the solver again with different tolerances. The TolX and TolFun tolerances determine how the solvers group their outputs into the GlobalOptimSolution vector. These tolerances are properties of the GlobalSearch or MultiStart object.

For example, suppose you want to use the active-set algorithm in fmincon to solve the problem in Example of Run with MultiStart. Further suppose that you want to have tolerances of 0.01 for both TolX and TolFun. The run method groups local solutions whose objective function values are within TolFun of each other, and which are also less than TolX apart from each other. To obtain the solution:

% % Set the random stream to get exactly the same output
% rng(14,'twister')
ms = MultiStart('TolFun',0.01,'TolX',0.01);
opts = optimoptions(@fmincon,'Algorithm','active-set');
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,someminsm] = run(ms,problem,50);

MultiStart completed the runs from all start points.

All 50 local solver runs converged with a
positive local solver exit flag.

someminsm

someminsm = 

  1x5 GlobalOptimSolution

  Properties:
    X
    Fval
    Exitflag
    Output
    X0

In this case, MultiStart generated five distinct solutions. Here "distinct" means that the solutions are more than 0.01 apart in either objective function value or location.

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