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What, if any, is the relationship between the convexity of a function $f:\mathbb{R}^N\rightarrow \mathbb{R}$ and the convexity of the same function with all parameters held constant except for a single parameter?

In other words, if $g_n:\mathbb{R}\rightarrow \mathbb{R}$ where $g_n(x_n) = f(a_1, a_2, .. , x_n , ..)$ for a particular set of values of $a_i$ , what is the relationship between the convexity of $f$ and the convexity of $g_n$ ?

user7693
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1 Answers1

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Of course, if $f$ is convex, then it is convex w.r.t. each single parameter. The converse, however, fails blatantly. To see this, just consider a quadratic function $f(x) = \frac12 \, x^\top M \, x$ for some symmetric matrix $M$. The convexity of $f$ is equivalent to $M$ being positive semi-definite, whereas the separate convexity requires that the diagonal of $M$ is non-negative.

One famous example is $f(x) = x_1 \, x_2$.

gerw
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