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