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?
