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Is a convex combination of two quasiconvex functions necessarily quasiconvex? If not, what can be said about the convex combination?

Somabha Mukherjee
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  • Okay, I think I've got it. The function f(x) = (1/2)(log(x) + log(1-x)) (x in (0,1) is a convex combination of two quasiconvex functions, but is itself not quasiconvex. – Somabha Mukherjee Jun 23 '15 at 00:58

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You are right: quasiconvexity is not preserved under convex combinations. Another example is $\sqrt{|x|}+\sqrt{|x-1|}$ where the summands are quasiconvex but their sum is not (it has two local minima at $0$ and $1$).

You can't really say much about such combinations. For any $L$-Lipschitz function $f$ the sum $g(x) = f(x)+Lx$ is quasiconvex, and so is $h(x) = f(x)-Lx$. So the mean $\frac12(g(x)+h(x)) = f(x)$ can be any Lipschitz function. In particular, it can be something like $\sin x$, with infinitely many local minima.