1

Let $f \colon \mathbb{R}^n \to \mathbb{R}$ be a function that is convex in each coordinate:

$$ \forall i \in \{1, \dots, n\}. \forall x_1. \dots \forall x_{i-1}, \forall x_{i+1}\dots \forall x_n. f(x_1,\dots,x_{i-1},\cdot,x_{i+1},\dots, x_n) \text{ is convex} $$

Can $f$ have a local optimum that is not also a global optimum?

I already know that $f$ is not necessarily convex (consider, e.g., $f(x,y)=x \cdot y$). Therefore, I suspect that in general, $f$ may have local optimas. However, I could not find a counterexample so far ($x \cdot y$ has no optimum).

Peter
  • 760
  • 4
  • 20

1 Answers1

2

The function $$ f(x,y) = x^2 \, y^2 + 20 \, x \, y + x^2 + y^2 + x $$ has two local minimizers (near $(-3,3)$ and $(3,-3)$) with different function values. Only one of these minimizers is the global minimizer.

gerw
  • 31,359
  • Is it possible to have 3 local minima? In particular, can we say that a coordinate wise convex function on $n$ coordinates can only have $n$ distinct minima? (Unless it is constant.) – Thomas Ahle Feb 25 '20 at 20:15