Let $(\Omega,\mathcal E,\mu)$ be a probability space, $k\in\mathbb N$, $$W:=\left\{w:E\to[0,\infty)^k:\sum_{i=1}^kw_i=1\;\mu\text{-almost surely}\right\},$$ $G$ be a finite nonempty set and $a^{(g)}:E\to[0,\infty)^k$ be $\mathcal E$-measurable for $g\in G$.
How can we solve the saddle-point problem $$\inf_{w\in W}\sup_{g\in G}\sum_{i\in I}\int a^{(g)}_iw_i\:{\rm d}\mu\tag1$$ (at least numerically)?
Note that $W$ is a closed, convex subset of $\mathcal L^r(\mu;\mathbb R^k)$, with empty interior, contained in the unit closed ball, for all $r\ge1$.
Note that for fixed $g\in G$, a minimizer $w\in W$ of $\sum_{i\in I}\int a^{(g)}_iw_i\:{\rm d}\mu$ is given by $$w^{(g)}_i(x):=\delta_{ij^{(g)}(x)}\;\;\;\text{for }x\in E\text{ and }i\in\{1,\ldots,k\},$$ where $\delta$ denotes the Kronecker delta and $$j^{(g)}(x):=\min\underset{i\in I}{\operatorname{arg\:min}}\:a^{(g)}_i(x)\;\;\;\text{for }x\in E$$ ($\operatorname{arg\:min}$ is treated as being set-valued and we break ties by selecting the smallest index).
Remark: I'm actually not interested in the $G$-component of a saddle-point, i.e. I'm only interested in a minimizer $w\in W$ of $\sup_{g\in G}\sum_{i\in I}\int a^{(g)}_iw_i\:{\rm d}\mu$.
