Are convexity and concavity sufficient or necessary condition for $\min \max = \max \min$

Question

If $f$ is convex in $x$ and concave in $y$, then $\min_{x} \max_{y} f(x,y) = \max_{y} \min_{x} f(x,y)$. Is that a sufficient or necessary condition? My intuition tells that it is a necessary but from my understanding from reading literature is sufficient since the equality (i.e., strong duality) can also happen if Slater's condition is met. If is it necessary, is there a connection between Slater's condition and a saddle point?

Answer 1

The condition is neither necessary nor sufficient.

For equality to hold, just having a convex-concave function is not enough. You need something more, e.g., that the domain of either $X$ or $Y$ is compact, or that the equality comes from an optimization problem for which Slater's condition holds. A simple counterexample when no regularity condition is met, is $f(x,y) = xy$, for which $\min_x \max_y f(x,y) = \infty$ while $\max_y \min_x f(x,y) = -\infty$.

On the other hand, you can have equality even when $f$ is not convex-concave. A trivial example is a constant function defined on just two points, e.g., $f : \{0,1\} \to \mathbb{R}$ with $f(0)=f(1)=0$.