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When is \begin{equation} \min_X \max_Y f(X,Y) \end{equation}

globally solvable? (i.e. we can find global solution for the optimization problem?) I am not looking for reformulations. Is it only when $f$ is concave in $Y$ and convex in $X$?

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There are primarily two things -

  1. convexity/concavity of domain
  2. convexity/concavity of objective function

A convex domain enables us to make strong comments regarding the global maxima and minima.

The objective function will have a maximum iff it is concave in the domain and min iff it is convex. This statement can be made if we have been given that the domain in convex.

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Considering that we have min over max how does this apply? – user25004 Nov 24 '12 at 6:51
I am not very clear about the problem statement(what does min over max mean) but if X is the domain for maximization part of the problem and Y for the minimization then yes the conditions you mention are necessary. – Aseem Dua Nov 24 '12 at 7:22
Also these arguments are made for an open set. – Aseem Dua Nov 24 '12 at 7:28
I did not receive any response after setting the bounty, so I think no one gets bounty from my side. – user25004 Dec 7 '12 at 7:47

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