I would like to minimize a set of similar objective functions
$$f_\boldsymbol{s}(\boldsymbol{x}),$$
where $\boldsymbol{x} \in A \subseteq \mathbb{R}^M$ and the parameterization $\boldsymbol{s} \in S \subseteq \mathbb{R}^N$ distinguishes the objective functions that I want to minimize.
edit1: I assume $f_\boldsymbol{s}(\boldsymbol{x})$ to be continuous in $\boldsymbol{s}$.
The solution that I am looking for is a model $\hat{g}$ of $g: S \rightarrow A$ so that $$\forall \boldsymbol{s} \in S, \forall \boldsymbol{x} \in A: f_\boldsymbol{s}(g(\boldsymbol{s})) \leq f_\boldsymbol{s}(\boldsymbol{x}).$$
In other words, I try to solve the problem
$$\hat{g} = \arg \min_{g} \int_{\boldsymbol{s} \in S} f_{\boldsymbol{s}}(g(\boldsymbol{s}))d\boldsymbol{s}$$
- Is this kind of problem known in the optimization community?
- Are there any algorithms that are able to solve these kind of problems?
I am not looking for multi-objective optimizers!