Recap and context

The notion of convexity will allow us to:

• Efficiently verify whether a given point is a global minimum
• Efficiently find (approximate) global optimal solutions

Convexity is the “great watershed of optimization”: in a very general sense, convex optimization problems are “easy”, while non-convex optimization problems are very often ”hard” (we have seen examples in the previous lecture).

Before we can define what it means for a function to be convex, we first have to introduce the concept of a convex set.

Convex sets

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Definition: Convex set

A set $S \subseteq \R^d$ is convex iff

$$ \lambda x + (1-\lambda)y \in S ~,

\qquad \forall x,y \in S ~, 0 \leq \lambda \leq 1. $$

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• In words, with any pair $x,y \in S$, the set $S$ also includes any point on the segment $[x,y]$:

Simple examples of convex sets