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Assume $f : \mathbb{R}^n \to \mathbb{R}$ is a function that depends on $x\in \mathbb{R}^m$ and $y\in \mathbb{R}^{n-m}$. If it is known that for any $x_0 \in \mathbb{R}^m$ function $f(x_0,y)$ is convex and also for any $y_0 \in \mathbb{R}^m$ function $f(x,y_0)$ is convex, can we say that $f$ is convex?

If $(x^*,y^*)$ is a known local minimum of $f$, is there any thing that could be said about its global minimum?

Rodrigo de Azevedo
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Leili Esmaeilani
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1 Answers1

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No. If $n=2$ then $f(x,y)=xy$ is an example of such a function. Or $f(x,y)=x^2+y^2-10xy$. More generally, for $n>2$, bilinear functions such as $f(x,y)=x' A y$ are convex in each variable separately but not (typically) jointly.

kimchi lover
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