Linear Matrix Inequalities in Control

Optimization problems
1 What is least possible cost? Compute optimal value
fopt := inf f(x) = inf{f(x) | x ∈ S} ≥ −∞
x∈S
Convention: S = ∅ then fopt = +∞
Convention: If fopt = −∞ then problem is said to be unbounded
2 How to determine almost optimal solutions? For arbitrary ε > 0 find
xε ∈ S with fopt ≤ f(xε) ≤ fopt + ε.
3 Is there an optimal solution (or minimizer)? Does there exist
xopt ∈ S with fopt = f(xopt)
4 Can we calculate all optimal solutions? (Non)-uniqueness
arg min f(x) := {x ∈ S | fopt = f(x)}
Siep Weiland and Carsten Scherer (DISC) Linear Matrix Inequalities in Control Class 1 12 / 59