How to optimize when the objective can be negatively influenced

Hello, I want to solve an optimization problem:

x* = arg max_x f(x; c) for any c (where x and c are real numbers)

f is some kind of utility function and c can be set in such a way that it describes a worst case scenario

How can I solve such a problem with matlab?

Answer 1

[John D'Errico]

This is just an optimization problem. Use any appropriate optimization solver. Note that the optimizers are typically minimizers, but that just means you will minimize -f(x). As far as the parameter c is concerned, are you looking to find a solution that is parametric as a function of c? Do you want to see a formula, as a function of c? For example,

syms x c
f = -x^2 + c*x;

The maximum is a simple to solve problem of course. It lies at the point x==c/2. We can find that by differentiating and setting the result to zero.

solve(diff(f,x) == 0,x)

ans = 

Of course, many far more complex problems will not have a simple solution like this. So you might decide to formulate the problem in terms of a solver.

fun = @(x,c) -x.^2 + c*x;
xmax = @(c) fminsearch(@(x) -fun(x,c),1);

Now you can solve for the max for any given numerical value of c.

xmax(3)

ans = 1.5000

If these cases are not what you are thinking about, then you need to be far less vague in terms of your real problem.

Answer 2

[Michael Hesse]

Hello John, thank you for your response. I guess I have to reformulate my problem.

x* = arg max_x min_c f(x, c), where x and c are vectors.

I guess the symbolic solution to this problem is to solve the inner optimization problem by setting it's gradient with respect to c to zero and solve for c and then solve the outer optimization problem, right? Let's say this approach is not applicable, how can I solve such a problem numerically within matlab?

[Torsten]

Maybe fminimax by taking in mind the identity

max_x min_i F_i(x) = −min_x max_i (−F_i(x))

[Michael Hesse]

Probably, but not feasible if c is high dimensional and we have to discretize f(x, c_i) = F_i with a fine grid.

[Torsten]

That's the numerical method ready-made for your problem.

If you think it's not feasible for your problem, you will have to program a better solution or search elsewhere in numerical libraries.