# How do we call this optimization problem?

Let $$f_k\colon\Bbb R^n\to\Bbb R$$, $$k=1,\dots,K$$, be differentiable (possibly nonconvex) functions and $$X\subset\Bbb R^n$$ be a convex set. Consider the following optimization problem:

$$\min_{x\in X}\max_{k\in\{1,\dots,K\}} f_k(x).$$

This is minimax problem but the inner optimization is a maximization discrete set. Rather than a general minimax problem, I guess this may be easier to deal with. I am interested in the following things:

• Are there any papers which propose algorithms for such structured problem?
• Do the papers show non-asymptotic convergence analysis?

You can rewrite your problem as \begin{align*} \min y\\ \textrm{s.t. }y & \ge f_{k}(x)\quad k=1,\dots,K\\ x & \in X. \end{align*}
What algorithms are applicable (and whether they converge in finitely many steps) will depend on the specifics of the $$f_k()$$ functions.
• Thank you for answering the question. I investigate more detail properties of $f_k$ to ensure some good convergence results. Oct 17, 2022 at 3:44