Linear Matrix Inequalities in Control

Why is convexity interesting ???
Reason 1: absence of local minima
Definition
f : S → R. Then x0 ∈ S is a
local optimum if ∃ε > 0 such that
f(x0) ≤ f(x) for all x ∈ S with x − x0 ≤ ε
global optimum if f(x0) ≤ f(x) for all x ∈ S
Theorem
If f : S → R is convex then every local optimum x0 is a global optimum
of f. If f is strictly convex, then the global optimum x0 is unique.
Siep Weiland and Carsten Scherer (DISC) Linear Matrix Inequalities in Control Class 1 25 / 59