1
$\begingroup$

Can someone give an example of a convex function $f$ on a path-connected compact nonconvex set where some point $c$ is a local minimum with $\nabla f(c)=0$ but not a global minimum. Thus showing that the set being not convex makes finding global minimums harder.

  • 0
    Well presumably the focus would be on set with interior having some meat to it.2012-03-11

2 Answers 2

2

Consider the set $A:=[-1,3]\times[-3,3]\ \setminus\ \{(x,y)\ |\ 0<|y| (a rectangle minus an isosceles triangle) and define f(x,y):=\cases{(x-2)^2 & $(x\leq 0 \ \vee\ y<0)$ \cr 2(x-1)^2+2 & $(x>0\ \wedge \ y>0)$\cr}\ . This $f$ is $C^1$ on $A$, convex, and has local minima at $c:=(2,-{5\over2})$, c':=(1,2) with $f(c)=0$, f(c')=2.

  • 0
    @Rahul Narain: There was a typo in the definition of the isosceles triangle. Thank you.2012-03-11
2

For example the function $f(x,y) = x^2+y^2$ restricted to the square with vertices $(-1,-2)$, $(5,-2)$, $(5,4)$ and $(-1,4)$ has four local minima but only one global minimum.

  • 0
    @user782220 As Rahul Narain commented: The boundary of the square. (Otherwise $f$ would clearly only have a single local minimum.)2012-03-11