Let V be an affine algebraic variety over R (or any other real closed field R). We ask when it is true that every positive semidefinite (psd) polynomial function on V is a sum of squares (sos). We show that for dimV ≥ 3 the answer is always negative if V has a real point. Also, if V is a smooth non-rational curve all of whose points at infinity are real, the answer is again negative. The same h...

It has often been claimed in recent papers that one can find a degree d Sum-of-Squares proof if one exists via the Ellipsoid algorithm. In [16], Ryan O’Donnell notes this widely quoted claim is not necessarily true. He presents an example of a polynomial system with bounded coefficients that admits low-degree proofs of non-negativity, but these proofs necessarily involve numbers with an exponen...

These notes survey and explore an emerging method, which we call the low-degree for understanding statistical-versus-computational tradeoffs in high-dimensional inference problems. In short, method posits that a certain quantity—the second moment of likelihood ratio—gives insight into how much computational time is required to solve given hypothesis testing problem, can turn be used predict har...

This paper presents a sum of squares (SOS) approach to guaranteed cost control of polynomial discrete fuzzy systems. First, we present a polynomial discrete fuzzy model that is more general representation of the well-known discrete Takagi-Sugeno (T-S) fuzzy model. Secondly, we derive a design condition based on polynomial Lyapunov functions that contain quadratic Lyapunov functions as a special...

IN THIS chapter we review a series of results obtained in the fi eld of localization that are based on polynomial optimization. First, we provide a review of a set of polynomial function optimization tools, including sum of squares ( SOS ). Then we present several applications of these tools in various sensor network localization tasks. As the fi rst application, we propose a method based on SO...

We introduce a method for proving Sum-of-Squares (SoS)/ Lasserre hierarchy lower bounds when the initial problem formulation exhibits a high degree of symmetry. Our main technical theorem allows us to reduce the study of the positive semidefiniteness to the analysis of “well-behaved” univariate polynomial inequalities. We illustrate the technique on two problems, one unconstrained and the other...

This paper presents a switch methodology together with a sum of squares (SOS) techniques to synthesize a nonlinear controller for a polynomial Takagi-Sugeno (TS) fuzzy model. A polynomial T-S fuzzy model adopts a polynomial representation of the nonlinear dynamics in its consequent part, which make it less susceptible to linearization errors. With respect to polynomial T-S fuzzy models, a fuzzy...

1 Overview In this lecture, we show that a number of sets and functions related to the notion of sum-of-squares (SOS) are SD-representable. We will start with positive polynomials. Then, we introduce a general algebraic framework in which the notion of sum-of-squares can be formulated in very general setting. Recall the cone of nonnegative univariate polynomials: P 2d [t] = p(t) = p 0 + p 1 t +...

We examine linear programming (LP) based relaxations for synthesizing polynomial Lyapunov functions to prove the stability of polynomial ODEs. Our approach starts from a desired parametric polynomial form of the polynomial Lyapunov function. Subsequently, we encode the positive-definiteness of the function, and the negation of its derivative, over the domain of interest. We first compare two cl...

We study a surprising phenomenon related to the representation of a cloud of data points using polynomials. We start with the previously unnoticed empirical observation that, given a collection (a cloud) of data points, the sublevel sets of a certain distinguished polynomial capture the shape of the cloud very accurately. This distinguished polynomial is a sum-of-squares (SOS) derived in a simp...