In recent years, optimization theory has been greatly impacted by the advent of sum of squares
(SOS) optimization. The reliance of this technique on large-scale semidefinite programs, …

Motivated by the need for formal guarantees on the stability and safety of controllers for
challenging robot control tasks, we present a control design procedure that explicitly seeks to …

We show that unless P = NP, there exists no polynomial time (or even pseudo-polynomial
time) algorithm that can decide whether a multivariate polynomial of degree four (or higher …

We relax the monotonicity requirement of Lyapunov¿s theorem to enlarge the class of
functions that can provide certificates of stability. To this end, we propose two new sufficient …

Sum of squares (SOS) optimization has been a powerful and influential addition to the
theory of optimization in the past decade. Its reliance on relatively large-scale semidefinite …

We introduce the framework of path-complete graph Lyapunov functions for approximation
of the joint spectral radius. The approach is based on the analysis of the underlying switched …

Historically, scalability has been a major challenge for the successful application of semidefinite
programming in fields such as machine learning, control, and robotics. In this article, we …

Our first contribution in this paper is to prove that three natural sum of squares (sos) based
sufficient conditions for convexity of polynomials, via the definition of convexity, its first order …

A multivariate polynomial p(x) = p(x 1 , . . . , x n ) is sos-convex if its Hessian H(x) can be
factored as H(x) = M T (x) M(x) with a possibly nonsquare polynomial matrix M(x). It is easy to …

In this paper, we consider linear programming (LP) and second order cone programming (SOCP)
based alternatives to sum of squares (SOS) programming and apply this framework to …