1

In the context of (non-)convex optimisation, what is the reason behind requiring that the objective function be lower semicontinuous? From what I understand, 1) a function is continuous iff it is both lower and upper semicontinuous, and 2) a convex function is continuous. So what is the link between l.s.c functions and convex optimisation?

John Lee
  • 97
  • 1
  • 6
  • 1
    If a function on a compact space is lower semicontinuous, then it attains its infimum. Convexity is usually required to state that a local minimum is a global one, as well. – Budenn Jul 22 '15 at 13:31
  • Thanks @Budenn, when a function is l.s.c, its epigraph is only closed, but not convex, so is the definition of convexity weakened for l.s.c functions? – John Lee Jul 22 '15 at 13:38
  • No, these concepts are independent: l.s.c. is for existence of a global minimum, convexity is to guarantee that the optimization method converges to a global minimum, not just a local one. – Budenn Jul 22 '15 at 13:43

1 Answers1

0

A convex function is not necessarily continuous: discontinuities may occur on the boundary of the domain. For instance consider the function $f:[0,1]\rightarrow\mathbb{R}$ defined as $$ f(x)= \left\{ \begin{array}{ll} 1, & \text{for } x=0,\\ x^{2}, & \text{for } 0< x\leqslant 1. \end{array} \right. $$ The function $f$ is convex on the compact $[0,1]$, but $f$ is not lower-semicontinuous and we can easily remark that $f$ does not have a minimum on $[0,1]$.

More generally, lower-semicontinuity is a sufficient condition for the existence of local or global minima on compact domains.

Dali D.
  • 21