Linear Matrix Inequalities in Control

Ellipsoidal algorithm
Aim: Minimize convex function f : R n → R
Step 0 Let x0 ∈ R n and P0 > 0 such that all minimizers of f are
located in the ellipsoid
E0 := {x ∈ R n | (x − x0) ⊤ P −1
0 (x − x0) ≤ 1}.
Set k = 0.
Step 1 Compute a subgradient gk of f at xk. If gk = 0 then stop,
otherwise proceed to Step 2.
Step 2 All minimizers are contained in
Hk := Ek ∩ {x | 〈gk, x − xk〉 ≤ 0}.
Step 3 Compute xk+1 ∈ R n and Pk+1 > 0 with minimal
determinant det Pk+1 such that
Ek+1 := {x ∈ R n | (x − xk+1) ⊤ P −1
k+1 (x − xk+1) ≤ 1}
contains Hk.
Step 4 Set k to k + 1 and return to Step 1.
Siep Weiland and Carsten Scherer (DISC) Linear Matrix Inequalities in Control Class 1 29 / 59