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According to Von Neuman's minimax theorem we have

$$\max_{x\in X} \min_{y\in Y}f(x,y)=\min_{y\in Y} \max_{x\in X} f(x,y)$$

for some compact sets $X$ and $Y$ and a convex (in $y$), concave (in $x$) function $f(x,y)$.

Is there any use of compactness constraint, IF both maximum and minimum are known to exist for any case?

In other words does compactness play another role in addition to guaranteeing the existence of the solutions?

Thanks in advance.

Seyhmus Güngören
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1 Answers1

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I think existence of bounded optima is the only reason to require compactness. You can replace general compactness with lower/upper semi-continuity and compactness of the set of optimal points.

Benjamin Lindqvist
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