On the Complexity of Detecting Convexity over a Box

It has recently been shown that the problem of testing global convexity of polynomials of degree four is {strongly} NP-hard, answering an open question of N.Z. Shor. This result is minimal in the degree of the polynomial when global convexity is of concern. In a number of applications however, one is interested in testing convexity … Read more

Robust-to-Dynamics Optimization

A robust-to-dynamics optimization (RDO) problem} is an optimization problem specified by two pieces of input: (i) a mathematical program (an objective function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ and a feasible set $\Omega\subseteq\mathbb{R}^n$), and (ii) a dynamical system (a map $g:\mathbb{R}^n\rightarrow\mathbb{R}^n$). Its goal is to minimize $f$ over the set $\mathcal{S}\subseteq\Omega$ of initial conditions that forever remain in $\Omega$ under … Read more

On the Complexity of Testing Attainment of the Optimal Value in Nonlinear Optimization

We prove that unless P=NP, there exists no polynomial time (or even pseudo-polynomial time) algorithm that can test whether the optimal value of a nonlinear optimization problem where the objective and constraints are given by low-degree polynomials is attained. If the degrees of these polynomials are fixed, our results along with previously-known “Frank-Wolfe type” theorems … Read more

A Globally Asymptotically Stable Polynomial Vector Field with Rational Coefficients and no Local Polynomial Lyapunov Function

We give an explicit example of a two-dimensional polynomial vector field of degree seven that has rational coefficients, is globally asymptotically stable, but does not admit an analytic Lyapunov function even locally. Citation Submitted for publication Article Download View A Globally Asymptotically Stable Polynomial Vector Field with Rational Coefficients and no Local Polynomial Lyapunov Function

SOS-Convex Lyapunov Functions and Stability of Difference Inclusions

We introduce the concept of sos-convex Lyapunov functions for stability analysis of both linear and nonlinear difference inclusions (also known as discrete-time switched systems). These are polynomial Lyapunov functions that have an algebraic certificate of convexity and that can be efficiently found via semidefinite programming. We prove that sos-convex Lyapunov functions are universal (i.e., necessary … Read more

On Algebraic Proofs of Stability for Homogeneous Vector Fields

We prove that if a homogeneous, continuously differentiable vector field is asymptotically stable, then it admits a Lyapunov function which is the ratio of two polynomials (i.e., a rational function). We further show that when the vector field is polynomial, the Lyapunov inequalities on both the rational function and its derivative have sum of squares … Read more

Sum of squares certificates for stability of planar, homogeneous, and switched systems

We show that existence of a global polynomial Lyapunov function for a homogeneous polynomial vector field or a planar polynomial vector field (under a mild condition) implies existence of a polynomial Lyapunov function that is a sum of squares (sos) and that the negative of its derivative is also a sum of squares. This result … Read more

Response to “Counterexample to global convergence of DSOS and SDSOS hierarchies”

In a recent note [8], the author provides a counterexample to the global convergence of what his work refers to as “the DSOS and SDSOS hierarchies” for polynomial optimization problems (POPs) and purports that this refutes claims in our extended abstract [4] and slides in [3]. The goal of this paper is to clarify that … Read more

Improving Efficiency and Scalability of Sum of Squares Optimization: Recent Advances and Limitations

It is well-known that any sum of squares (SOS) program can be cast as a semidefinite program (SDP) of a particular structure and that therein lies the computational bottleneck for SOS programs, as the SDPs generated by this procedure are large and costly to solve when the polynomials involved in the SOS programs have a … Read more

On the Construction of Converging Hierarchies for Polynomial Optimization Based on Certificates of Global Positivity

In recent years, techniques based on convex optimization and real algebra that produce converging hierarchies of lower bounds for polynomial minimization problems have gained much popularity. At their heart, these hierarchies rely crucially on Positivstellens\”atze from the late 20th century (e.g., due to Stengle, Putinar, or Schm\”udgen) that certify positivity of a polynomial on an … Read more