You can set the `StartPointsToRun`

option so
that `MultiStart`

and `GlobalSearch`

use
only start points that satisfy inequality constraints. This option
can speed your optimization, since the local solver does not have
to search for a feasible region. However, the option can cause the
solvers to miss some basins of attraction.

There are three settings for the `StartPointsToRun`

option:

`all`

— Accepts all start points`bounds`

— Rejects start points that do not satisfy bounds`bounds-ineqs`

— Rejects start points that do not satisfy bounds or inequality constraints

For example, suppose your objective function is

function y = tiltcircle(x) vx = x(:)-[4;4]; % ensure vx is in column form y = vx'*[1;1] + sqrt(16 - vx'*vx); % complex if norm(x-[4;4])>4

`tiltcircle`

returns complex values for `norm(x - [4 4]) > 4`

.

Code for Generating the Figure

Write a constraint function that is positive on the set where `norm(x - [4 4]) > 4`

function [c ceq] = myconstraint(x) ceq = []; cx = x(:) - [4;4]; % ensure x is a column vector c = cx'*cx - 16; % negative where tiltcircle(x) is real

Set `GlobalSearch`

to use only start points satisfying
inequality constraints:

gs = GlobalSearch('StartPointsToRun','bounds-ineqs');

To complete the example, create a problem structure and run the solver:

opts = optimoptions(@fmincon,'Algorithm','interior-point'); problem = createOptimProblem('fmincon',... 'x0',[4 4],'objective',@tiltcircle,... 'nonlcon',@myconstraint,'lb',[-10 -10],... 'ub',[10 10],'options',opts); rng(7,'twister'); % for reproducibility [x,fval,exitflag,output,solutionset] = run(gs,problem) GlobalSearch stopped because it analyzed all the trial points. All 5 local solver runs converged with a positive local solver exit flag. x = 1.1716 1.1716 fval = -5.6530 exitflag = 1 output = struct with fields: funcCount: 3242 localSolverTotal: 5 localSolverSuccess: 5 localSolverIncomplete: 0 localSolverNoSolution: 0 message: 'GlobalSearch stopped because it analyzed all the trial po...' solutionset = 1x4 GlobalOptimSolution array with properties: X Fval Exitflag Output X0

**tiltcircle With Local Minima**

Code for Generating the Figure

The `tiltcircle`

function has just one local
minimum. Yet `GlobalSearch`

(`fmincon`

)
stops at several points. Does this mean `fmincon`

makes
an error?

The reason that `fmincon`

stops at several
boundary points is subtle. The `tiltcircle`

function
has an infinite gradient on the boundary, as you can see from a one-dimensional
calculation:

$$\frac{d}{dx}\sqrt{16-{x}^{2}}=\frac{-x}{\sqrt{16-{x}^{2}}}=\pm \infty \text{at}\left|x\right|=4.$$

So there is a huge gradient normal to the boundary. This gradient
overwhelms the small additional tilt from the linear term. As far
as `fmincon`

can tell, boundary points are stationary
points for the constrained problem.

This behavior can arise whenever you have a function that has a square root.