I want to solve a problem of the form $$ \min_{x \in X} \max_{y \in Y} f(x,y) $$ where $X \subset \mathbb{R}^d$ and $Y \subset \mathbb{R}^k$. Typically $d$ is large ($>100$) and $k$ is small (1-3). Furthermore, in my case, $Y$ is also compact. However, the function $f$ is such that, for fixed $x_0$, i can not compute $\min_{y\in Y}f(x_0,y)$ directly (analyticaly). Thus far i have used a iterated optimization approach, that is, using of the shelf optimization algorithms to compute the outer minimum as well as the inner maximum.
The problem with this is twofold: 1) In every step of the outer algorithm, $\min_{y\in Y}f(x,y)$ has to be computed by the inner algorithm. This means that the whole procedure is computationally expensive.
2) The inner algorithm is of coures not exact and thus yields an aproximation of $\min_{y\in Y}f(x,y)$. This might interfere with the convergence of the outer algorithm.
Since I'm not really fluent in the optimization nomenclature & literature searching online has not been very fruitfull. Any tips towards
- The correct(or common) name for this type of problem
- Any applicable theoretical results for this type of problem
- Algorithms specifically developed to solve these problems
would be greatly appreciated.