# Question about implementation method for optimization problem

Suppose we wanted to solve the following optimization problem: $$\inf_{x \geq 0}\sup_{y \in [0, 1];\ z > 0} f(x, y, z),$$ where $$f(x, y, z)$$ is some objective function with a closed form that can be specified in terms of parameters $$x$$, $$y$$, $$z$$. How would we implement this optimization, say using the Python programming language?

I am only vaguely familiar with implementing optimization in the Bayesian setting (eg using variational inference) or by a grid search. But I do not think Bayesian optimization works here as there are any priors here (I could be wrong though) and I'm not quite sure how to discretize these parameters in a reasonable way.

(The motivation for the optimization problem above comes from my personal study of robust optimization models from the paper https://arxiv.org/pdf/1908.05659.pdf.)

• If you want to implement a solver for that, you could think about using a metaheuristics-based approach. There are several packages that allow you to build your own metaheuristics, like Mealpy. Feb 9 at 20:43
• Also, if you want to code your own solver, you could take a look to the Julia language. Feb 9 at 20:44
• Thank you very much for the suggestions. I'm quite unfamiliar with using metaheuristics - can you please direct me to some resources for how I might set up a solver for $\inf$-$\sup$ optimization problems? Feb 9 at 21:51
• Do you know anything about the function f , convex in x,z etc?
– skr
Feb 10 at 1:59
• For simplicity, let us assume that it is convex. I haven't evaluated its closed form, but I know for sure that it has a closed form. Feb 10 at 3:52