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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.

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

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 (Optimization Toolbox) 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

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

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

The initial summands *x*^{6} + *y*^{6} 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 10^{6} is much larger than all
the multiples of 10^{4} 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.

If you use `GlobalSearch`

on an unconstrained
problem, change your `problem`

structure before using `MultiStart`

.
You have two choices in updating a `problem`

structure
for an unconstrained problem using `MultiStart`

:

Change the

`solver`

field to`'fminunc'`

:problem.solver = 'fminunc';

To avoid a warning if your objective function does not compute a gradient, change the local

`options`

to have`Algorithm`

set to`'quasi-newton'`

:problem.options.Algorithm = 'quasi-newton';

Add an artificial constraint, retaining

`fmincon`

as the local solver:problem.lb = -Inf(size(x0));

To search a larger region than the default, see Refine Start Points.

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