Suppose $X\subseteq\mathbb{R}^n$ is a convex and compact set, and $Y\subseteq\mathbb{R}^m$ is a nonconvex bounded set. Consider
$$
\min_{x\in X}\max_{y\in Y}x^TAy.
$$
Is this equivalent to
$$
\min_{x\in X}\max_{y\in conv(Y)}x^TAy,
$$
where $conv$ is the convex hull? It seems to make sense, since for any fixed $x$, the objective is linear in $y$.
Then, suppose $X$ is a standard simplex, and $Y$ is a set of all $m$ dimensional elementary vectors (zero vector except for one being 1). Does it mean that optimal $y^*$ is also an elementary vector, i.e., pure strategy?
Best Answer
Yes, the maximum of a linear map over any set is equal to the maximum over the convex hull of the set.