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?

回答(2 个)

John D'Errico
John D'Errico 2023-1-10
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.

Michael Hesse
Michael Hesse 2023-1-11
编辑:Michael Hesse 2023-1-11
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?
  3 个评论
Michael Hesse
Michael Hesse 2023-1-11
编辑:Michael Hesse 2023-1-11
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
Torsten 2023-1-11
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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