Can You Certify That a Solution Is Global?

No Guarantees

How can you tell if you have located the global minimum of your objective function? The short answer is that you cannot; you have no guarantee that the result of a Global Optimization Toolbox solver is a global optimum. While all Global Optimization Toolbox solvers repeatedly attempt to locate a global solution, no solver employs an algorithm that can certify a solution as global.

However, you can use the strategies in this section for investigating solutions.

Check if a Solution Is a Local Solution with patternsearch

Before you can determine if a purported solution is a global minimum, first check that it is a local minimum. To do so, run patternsearch on the problem.

To convert the problem to use patternsearch instead of fmincon or fminunc, enter

problem.solver = 'patternsearch';

Also, change the start point to the solution you just found, and clear the options:

problem.x0 = x;
problem.options = [];

For example, Check Nearby Points shows the following:

options = optimoptions(@fmincon,'Algorithm','active-set');
ffun = @(x)(x(1)-(x(1)-x(2))^2);
problem = createOptimProblem('fmincon', ...
'objective',ffun,'x0',[1/2 1/3], ...
'lb',[0 -1],'ub',[1 1],'options',options);
[x,fval,exitflag] = fmincon(problem)

x =
1.0e-007 *
0    0.1614

fval =
-2.6059e-016

exitflag =
1

However, checking this purported solution with patternsearch shows that there is a better solution. Start patternsearch from the reported solution x:

% set the candidate solution x as the start point
problem.x0 = x;
problem.solver = 'patternsearch';
problem.options = [];
[xp,fvalp,exitflagp] = patternsearch(problem)

Optimization terminated: mesh size less than options.MeshTolerance.

xp =

1.0000   -1.0000

fvalp =

-3.0000

exitflagp =

1

Identify a Bounded Region That Contains a Global Solution

Suppose you have a smooth objective function in a bounded region. Given enough time and start points, MultiStart eventually locates a global solution.

Therefore, if you can bound the region where a global solution can exist, you can obtain some degree of assurance that MultiStart locates the global solution.

For example, consider the function

$f={x}^{6}+{y}^{6}+\mathrm{sin}\left(x+y\right)\left({x}^{2}+{y}^{2}\right)-\mathrm{cos}\left(\frac{{x}^{2}}{1+{y}^{2}}\right)\left(2+{x}^{4}+{x}^{2}{y}^{2}+{y}^{4}\right).$

The initial summands x6 + y6 force the function to become large and positive for large values of |x| or |y|. The components of the global minimum of the function must be within the bounds

–10 ≤ x,y ≤ 10,

since 106 is much larger than all the multiples of 104 that occur in the other summands of the function.

You can identify smaller bounds for this problem; for example, the global minimum is between –2 and 2. It is more important to identify reasonable bounds than it is to identify the best bounds.

To check whether there is a better solution to your problem, run MultiStart with additional start points. Use MultiStart instead of GlobalSearch for this task because GlobalSearch does not run the local solver from all start points.

For example, see Example: Searching for a Better Solution.

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