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Here is the question:

Consider a function of two variables $f(x,y)$ on some compact domain. Let it be convex in $x$ and concave in $y$. Is it true that $f(x,y)$ has a uniqe saddle point (stationary point)? Is $(\hat x,\hat y)=\arg \min_x\max_y f(x,y)$, the argument of saddle point, unique? Is there any role of being strictly concave (or convex) in this case?

My guess is that $f(x,y)$ owns a unique saddle point because of the convexity and concavity of the function. I also guess that if the function is strictly concave-convex, then, the argument of the saddle point is also unique.

Any reference for a proof will also suffice. Thx.

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  • $\begingroup$ It is not true that there exists a stationary point. If it is not strictly convex/concave, it is not true that the stationary point, when it exists, is unique. And depending on how you define "convexity" for non-convex domains, there may be other obstructions to uniqueness in those cases. $\endgroup$
    – Willie Wong
    Mar 31, 2015 at 12:07
  • $\begingroup$ Let the domain be convex and compact and the function of interest be strictly convex and concave in its arguments. Will the saddle value be unique and if yes I will be happy to see a proof. $\endgroup$
    – Seyhmus Güngören
    Mar 31, 2015 at 12:24
  • $\begingroup$ @MichaelGrant yes I mentioned it in the question. I thought I could get a reference there. I dont think that I will get any answer here since the question was asked a few days ago.. $\endgroup$
    – Seyhmus Güngören
    Mar 31, 2015 at 15:58
  • $\begingroup$ Believe me, I sympathize. But since they migrated it over I am not optimistic you will find an answer. It's still a duplicate, though. Perhaps what you should do is edit the original question to give it this title: it is, after all, a much better title than the generic one you gave it. $\endgroup$
    – Michael Grant
    Mar 31, 2015 at 15:59

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