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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?
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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.

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  • $\begingroup$ Thank you for answering the question. I investigate more detail properties of $f_k$ to ensure some good convergence results. $\endgroup$
    – Keith
    Oct 17, 2022 at 3:44

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