A Novel Unified Approach to Invariance in Control

In this paper, we propose a novel, unified, general approach to investigate sufficient and necessary conditions under which four types of convex sets, polyhedra, polyhedral cones, ellipsoids and Lorenz cones, are invariant sets for a linear continuous or discrete dynamical system. In proving invariance of ellipsoids and Lorenz cones for discrete systems, instead of the … Read more

Strong formulations for convex functions over nonconvex sets

In this paper we derive strong linear inequalities for sets of the form {(x, q) ∈ R^d × R : q ≥ Q(x), x ∈ R^d − int(P ) }, where Q(x) : R^d → R is a quadratic function, P ⊂ R^d and “int” denotes interior. Of particular but not exclusive interest is the … Read more

Immunizing conic quadratic optimization problems against implementation errors

We show that the robust counterpart of a convex quadratic constraint with ellipsoidal implementation error is equivalent to a system of conic quadratic constraints. To prove this result we first derive a sharper result for the S-lemma in case the two matrices involved can be simultaneously diagonalized. This extension of the S-lemma may also be … Read more

A survey of the S-lemma

In this survey we review the many faces of the S-lemma, a result about the correctness of the S-procedure. The basic idea of this widely used method came from control theory but it has important consequences in quadratic and semidefinite optimization, convex geometry and linear algebra as well. These were active research areas, but as … Read more