How to do "not equal to" constraints in fmincon/Global search?

Is there (in the meantime) a better way than:

https://de.mathworks.com/matlabcentral/answers/409510-set-not-equal-to-constraint-for-a-linear-system-of-equations-solver

existing to "add" a not equal constraint to the optimization Problem?

I want to say for example "x not equal to [0,1,0,1]".

[Torsten]

Is x supposed to be a binary integer solution ?

Or are the components of x continuous variables ?

[Daniela Würmseer]

Continuous variables

[Matt J]

@Daniela Würmseer not-equal constraints are not a valid concept in continuous-domain optimization. For example, what would you would consider to be the minimum of f(x)=x^2, subject to the constraint x~=0? It is not well-defined.

Keep in mind as well that iterative optimization algorithms rarely reach the exact minimizer. Rather, they merely converge toward it asymptotically. So, forbidding a specific point with a not-equal constraint, when that point will never be reached exactly anyway has questionable benefits.

[Daniela Würmseer]

@Matt J thank you, I need to test somehow if the solution i get from Global Search is unique. So i wanted to solve my Problem twice, once normal and after the same problem again but with the constraint that x notequal to the solution i just got.

[Walter Roberson]

If you implemented an anti-equality constraint, fmincon would happily return the next adjacent representable number(s).

GlobalSearch is often configured to use random starting points; a trivial change in starting point is likely to return locations that are not bit-for-bit identical.

[Walter Roberson]

Matt's suggestion of requiring a minimum distance from the distinguished point might be useful for this situation.

And possibly paretosearch() instead of global search.

Answer 1

[Matt J]

I need to test somehow if the solution i get from Global Search is unique.

The run() command, with 5 output arguments, can return all candidate solutions located by the search

[x,fval,exitflag,output,solutions] = run(___)

If you see that multiple solutions have the same objective value, it would indicate the solution is not unique.

[Walter Roberson]

... after doing uniquetol by rows to filter points that are essentially the same.

Answer 2

[Walter Roberson]

You could try a nonlinear equality constraint

ceq = double(isequal(x, [0 1 0 1]));

This would return 1 if the equality holds, but non-zero is disfavored. Favoured is 0 which corresponds to false which would be the case when the x is anything else.

I am not convinced that this will work well.

[John D'Errico]

Ugh, no. This form of constraint would be discontinuous, and fmincon presumes not only continuity, but differentiability. And since fmincon will be passing in real values for the parameters, NOT integer values, exact equality will essentially never happen anyway. So I would doubt that constraint will be of any value at all.

[Walter Roberson]

I was thinking about the continuity / differentiability when I posted that. I was thinking that it would seem plausible that you could use the nonlinear constraints to exclude (for example) a circle from consideration, and a point is a circle shrunk down to no radius. The nonlinear constraints have never been documented as being required to establish a concave area of solution.

Answer 3

[Matt J]

You could impose a constraint like as a smooth approximation to .

function [c,ceq,gradc,gradceq]=nonlcon(x,r)

  ceq=[]; gradceq=[];
  
  gradc=[0;1;0;1]-x(:);
  
  c=(r^2-norm(gradc)^2)/2;
  
end