Degree bounds for Putinar's Positivstellensatz on the hypercube

The Positivstellens\"atze of Putinar and Schm\"udgen show that any polynomial $f$ positive on a compact semialgebraic set can be represented using sums of squares. Recently, there has been large interest in proving effective versions of these results, namely to show bounds on the required degree of the sums of squares in such representations. These effective Positivstellens\"atze have direct implications for the convergence rate of the celebrated moment-SOS hierarchy in polynomial optimization. In this paper, we restrict to the fundamental case of the hypercube $\mathrm{B}^{n} = [-1, 1]^n$. We show an upper degree bound for Putinar-type representations on $\mathrm{B}^{n}$ of the order $O(f_{\max}/f_{\min})$, where $f_{\max}$, $f_{\min}$ are the maximum and minimum of $f$ on $\mathrm{B}^{n}$, respectively. Previously, specialized results of this kind were available only for Schm\"udgen-type representations and not for Putinar-type ones. Complementing this upper degree bound, we show a lower degree bound in $\Omega(\sqrt[8]{f_{\max}/f_{\min}})$. This is the first lower bound for Putinar-type representations on a semialgebraic set with nonempty interior described by a standard set of inequalities.Comment: Final version. Improved presentation and included a more detailed comparison with Bach and Rudi, Exponential Convergence of Sum-of-Squares Hierarchies for Trigonometric Polynomials, SIAM Opt., 202

Asymptotic analysis of semidefinite bounds for polynomial optimization and independent sets in geometric hypergraphs

The goal of a mathematical optimization problem is to maximize an objective (or minimize a cost) under a given set of rules, called constraints. Optimization has many applications, both in other areas of mathematics and in the real world. Unfortunately, some of the most interesting problems are also very hard to solve numerically. To work around this issue, one often considers relaxations: approximations of the original problem that are much easier to solve. Naturally, it is then important to understand how (in)accurate these relaxations are. This thesis consists of three parts, each covering a different method that uses semidefinite programming to approximate hard optimization problems. In Part 1 and Part 2, we consider two hierarchies of relaxations for polynomial optimization problems based on sums of squares. We show improved guarantees on the quality of Lasserre's measure-based hierarchy in a wide variety of settings (Part 1). We establish error bounds for the moment-SOS hierarchy in certain fundamental special cases. These bounds are much stronger than the ones obtained from existing, general results (Part 2). In Part 3, we generalize the celebrated Lovász theta number to (geometric) hypergraphs. We apply our generalization to formulate relaxations for a type of independent set problem in the hypersphere. These relaxations allow us to improve some results in Euclidean Ramsey theory

Polynomial Optimization Methods

This thesis is an exposition of ideas and methods that help un- derstanding the problem of minimizing a polynomial over a basic closed semi-algebraic set. After the introduction of some the- ory on mathematical tools such as sums of squares, nonnegative polynomials and moment matrices, several Positivstellensa ̈tze are considered. Positivstellens ̈atze provide sums of squares represen- tations of polynomials, positive on basic closed semi-algebraic sets. Subsequently, semi-definite programming methods, in par- ticular based on Putinar’s Postivstellensatz, are considered. In order to use semi-definite programming, certain degree bounds are set. These bounds give rise to a hierarchy of approximations of the minimum of a polynomial, which will also be discussed. Finally, some new results are given that are obtained by looking at sums of squares representations of a positive polynomial when minimizing over the unit hypercube

Approximability and proof complexity

This work is concerned with the proof-complexity of certifying that optimization problems do \emph{not} have good solutions. Specifically we consider bounded-degree "Sum of Squares" (SOS) proofs, a powerful algebraic proof system introduced in 1999 by Grigoriev and Vorobjov. Work of Shor, Lasserre, and Parrilo shows that this proof system is automatizable using semidefinite programming (SDP), meaning that any $n$-variable degree-$d$ proof can be found in time $n^{O(d)}$. Furthermore, the SDP is dual to the well-known Lasserre SDP hierarchy, meaning that the "$d/2$-round Lasserre value" of an optimization problem is equal to the best bound provable using a degree-$d$ SOS proof. These ideas were exploited in a recent paper by Barak et al.\ (STOC 2012) which shows that the known "hard instances" for the Unique-Games problem are in fact solved close to optimally by a constant level of the Lasserre SDP hierarchy. We continue the study of the power of SOS proofs in the context of difficult optimization problems. In particular, we show that the Balanced-Separator integrality gap instances proposed by Devanur et al.\ can have their optimal value certified by a degree-4 SOS proof. The key ingredient is an SOS proof of the KKL Theorem. We also investigate the extent to which the Khot--Vishnoi Max-Cut integrality gap instances can have their optimum value certified by an SOS proof. We show they can be certified to within a factor .952 ($> .878$) using a constant-degree proof. These investigations also raise an interesting mathematical question: is there a constant-degree SOS proof of the Central Limit Theorem?Comment: 34 page

A version of Putinar's Positivstellensatz for cylinders

We prove that, under some additional assumption, Putinar's Positivstellensatz holds on cylinders of type S×R with S={x¯∈Rn|g1(x¯)≥0,…,gs(x¯)≥0} such that the quadratic module generated by g1,…,gs in R[X1,…,Xn] is archimedean, and we provide a degree bound for the representation of a polynomial f∈R[X1,…,Xn,Y] which is positive on S×R as an explicit element of the quadratic module generated by g1,…,gs in R[X1,…,Xn,Y]. We also include an example to show that an additional assumption is necessary for Putinar's Positivstellensatz to hold on cylinders of this type.Fil: Escorcielo, Paula Micaela. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Perrucci, Daniel Roberto. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentin

An Efficient Framework for Global Non-Convex Polynomial Optimization over the Hypercube

We present a novel efficient theoretical and numerical framework for solving global non-convex polynomial optimization problems. We analytically demonstrate that such problems can be efficiently reformulated using a non-linear objective over a convex set; further, these reformulated problems possess no spurious local minima (i.e., every local minimum is a global minimum). We introduce an algorithm for solving these resulting problems using the augmented Lagrangian and the method of Burer and Monteiro. We show through numerical experiments that polynomial scaling in dimension and degree is achievable for computing the optimal value and location of previously intractable global polynomial optimization problems in high dimension

Practical polynomial optimization through positivity certificates with and without denominators

Les certificats de positivité ou Positivstellens"atze fournissent des représentations de polynômes positifs sur des ensembles semialgébriques de basiques, c'est-à-dire des ensembles définis par un nombre fini d'inégalités polynomiales. Le célèbre Positivstellensatz de Putinar stipule que tout polynôme positif sur un ensemble semialgébrique basique fermé $S$ peut être écrit comme une combinaison pondérée linéaire des polynômes décrivant $S$, sous une certaine condition sur $S$ légèrement plus forte que la compacité. Lorsqu'il est écrit comme ceci, il devient évident que le polynôme est positif sur $S$, et donc cette description alternative fournit un certificat de positivité sur $S$. De plus, comme les poids polynomiaux impliqués dans le Positivstellensatz de Putinar sont des sommes de carrés (SOS), de tels certificats de positivité permettent de concevoir des relaxations convexes basées sur la programmation semidéfinie pour résoudre des problèmes d'optimisation polynomiale (POP) qui surviennent dans diverses applications réelles, par exemple dans la gestion des réseaux d'énergie et l'apprentissage automatique pour n'en citer que quelques unes. Développée à l'origine par Lasserre, la hiérarchie des relaxations semidéfinies basée sur le Positivstellensatz de Putinar est appelée la emph{hiérarchie Moment-SOS}. Dans cette thèse, nous proposons des méthodes d'optimisation polynomiale basées sur des certificats de positivité impliquant des poids SOS spécifiques, sans ou avec dénominateurs.Positivity certificates or Positivstellens"atze provide representations of polynomials positive on basic semialgebraic sets, i.e., sets defined by finitely many polynomial inequalities. The famous Putinar's Positivstellensatz states that every positive polynomial on a basic closed semialgebraic set $S$ can be written as a linear weighted combination of the polynomials describing $S$, under a certain condition on $S$ slightly stronger than compactness. When written in this it becomes obvious that the polynomial is positive on $S$, and therefore this alternative description provides a certificate of positivity on $S$. Moreover, as the polynomial weights involved in Putinar's Positivstellensatz are sums of squares (SOS), such Positivity certificates enable to design convex relaxations based on semidefinite programming to solve polynomial optimization problems (POPs) that arise in various real-life applications, e.g., in management of energy networks and machine learning to cite a few. Originally developed by Lasserre, the hierarchy of semidefinite relaxations based on Putinar's Positivstellensatz is called the emph{Moment-SOS hierarchy}. In this thesis, we provide polynomial optimization methods based on positivity certificates involving specific SOS weights, without or with denominators

Improved convergence rates for Lasserre-type hierarchies of upper bounds for box-constrained polynomial optimization

We consider the problem of minimizing a given multivariate polynomial f over the hypercube [-1,1]^n. An idea, introduced by Lasserre, is to find a probability distribution on the hypercube with polynomial density function h (of given degree r) that minimizes the expectation of f over the hypercube with respect to this probability distribution. It is known that, for the Lebesgue measure one may show an error bound in 1/sqrt{r} if h is a sum-of-squares density, and an error bound in 1/r if h is the density of a beta distribution. In this paper, we show another probability distribution that permits to show an error bound in 1/r^2 when selecting a density function h with a Schmuedgen-type sum-of-squares decomposition. The convergence rate analysis relies on the theory of polynomial kernels, and in particular on Jackson kernels. We also show that the resulting upper bounds may be computed as generalized eigenvalue problems, as is also the case for sum-of-squares densitie

Handelman 's hierarchy for the maximum stable set problem.

The maximum stable set problem is a well-known NP-hard problem in combinatorial optimization, which can be formulated as the maximization of a quadratic square-free polynomial over the (Boolean) hypercube. We investigate a hierarchy of linear programming relaxations for this problem, based on a result of Handelman showing that a positive polynomial over a polytope with non-empty interior can be represented as conic combination of products of the linear constraints defining the polytope. We relate the rank of Handelman's hierarchy with structural properties of graphs. In particular we show a relation to fractional clique covers which we use to upper bound the Handelman rank for perfect graphs and determine its exact value in the vertex-transitive case. Moreover we show two upper bounds on the Handelman rank in terms of the (fractional) stability number of the graph and compute the Handelman rank for several classes of graphs including odd cycles and wheels and their complements. We also point out links to several other linear and semidefinite programming hierarchies