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How to optimize when the objective can be negatively influenced

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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?

Answers (2)

John D'Errico
John D'Errico on 10 Jan 2023
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.
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.

Michael Hesse
Michael Hesse on 11 Jan 2023
Edited: Michael Hesse on 11 Jan 2023
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?
Michael Hesse
Michael Hesse on 11 Jan 2023
Edited: Michael Hesse on 11 Jan 2023
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 on 11 Jan 2023
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.

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