Syntax for optimization with integer and continuous variables, nonlinear equalities and inequalities?

Happy holidays!

I am trying to write an optimization problem that has (i) integer and continuous variables and (ii) nonlinear equalities and inequalities. Without going into details, the problem is something along the lines of:

x = optimvar('x', 5, 'Type', 'integer', 'LowerBound', 0, 'UpperBound', 1);
y = optimvar('u', 5, 'LowerBound',[-Inf(4,1); 1], ...
    'UpperBound', [Inf(4,1); 1]);

A = optimconstr(10);
B = optimconstr(10);

for i=1:10
    A(i) = f(x, y) == 0;
    B(i) = g(x, y) >= 0;
end

obj = h(x, y);

prob.Objective = obj;
prob.Constraints.A = A;
prob.Constraints.B = B;

[sol, val] = solve(prob)

When I run this on its own, I receive the error:

Error using optim.problemdef.OptimizationProblem/solve
GA does not solve problems with integer and equality
constraints.
For more help see No Equality Constraints in the
documentation.

As an aside, when I try to click on No Equality Constraints, I also get an error:

Error using helpview
Topic ID (no_eqcon_intga) not found in gads.

When I try to switch to use fmincon by adding:

options = optimoptions('fmincon', 'Algorithm', 'interior-point', 'Display', 'iter');
x0 = repelem([1], 5);
u0 = repelem([1], 5);

xu0.x = x0;
xu0.u = u0;

[sol, val] = solve(prob, xu0, options);

I receive the error:

Error using optim.internal.problemdef.ProblemImpl/solveImpl
The value of 'GlobalSolver' is invalid. GlobalSolver
must be a MultiStart or GlobalSearch object.

Error in optim.problemdef.OptimizationProblem/solve

Some guidance on the proper syntax, or at least a reference I can look at, would be greatly appreciated!

Answer 1

[Matt J]

You only have 2^5=32 possible combinations of values for the integer variables x. The best thing to do would be to loop over these combinations and, for each one, solve the sub-problem in which y is the only unknown.

Answer 2

[Torsten]

Concerning your attempt with "ga":

See the Note below from the documentation of "ga":

x = ga(fun,nvars,A,b,Aeq,beq,lb,ub,nonlcon,intcon) or

x = ga(fun,nvars,A,b,Aeq,beq,lb,ub,nonlcon,intcon,options) requires that the variables listed in intcon take integer values.

Note

When there are integer constraints, ga does not accept nonlinear equality constraints, only nonlinear inequality constraints.

Concerning your attempt with "fmincon":

fmincon only accepts continuous solution variables, no binaries as you define for x.

Thus summarizing:

Your problem as stated is not solvable within the MATLAB optimization environment.

Answer 3

[Walter Roberson]

You just cannot do that using ga().

You can potentially do that using surrogateopt()

... But there is a lot to be said for @Matt J's suggestion of running once for each of the possible x values and find the best overall soltuion.