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It is known that $\underset{x}{\max} \underset{y}{\min} f(x,y) \leq \underset{y}{\min} \underset{x}{\max} f(x,y)$ . When does equality hold in this expression?

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3 Answers 3

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Let $\max_x \min_y f(x, y) = \max_x g(x) = g(a)$ and $\min_y \max_x f(x, y) = \min_y h(y) = h(b)$. We have $$\max_x \min_y f(x, y)= g(a) \le f(a, b) \le h(b) = \min_y \max_x f(x, y)$$

So for equality, we need an element $(a, b)$ to exist s.t. $\min_y f(a, y) = f(a, b) = \max_x f(x, b)$.

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The inequality is the "weak duality inequality" and when it is equality it is called strong duality equality, see this for more detail.

The strong duality holds for convex functions (which their domain has non-empty interior)

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Equality holds if $f(w,z)$ is convex in w and concave in $z$ (or convex in $z$ and concave in $w$)

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