A Semidefinite Approach for Truncated K-Moment Problems

A truncated moment sequence (tms) of degree d is a vector indexed by monomials whose degree is at most d. Let K be a semialgebraic set.The truncated K-moment problem (TKMP) is: when does a tms y admit a positive Borel measure supported? This paper proposes a semidefinite programming (SDP) approach for solving TKMP. When K is compact, we get the following results: whether a tms y of degree d admits a K-measure or notcan be checked via solving a sequence of SDP problems; when y admits no K-measure, a certificate will be given; when y admits a K-measure, a representing measure for y would be obtained from solving the SDP under some necessary and some sufficient conditions. Moreover, we also propose a practical SDP method for finding flat extensions, which in our numerical experiments always finds a finitely atomic representing measure for a tms when it admits one

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

Linear Optimization with Cones of Moments and Nonnegative Polynomials

Let A be a finite subset of N^n and R[x]_A be the space of real polynomials whose monomial powers are from A. Let K be a compact basic semialgebraic set of R^n such that R[x]_A contains a polynomial that is positive on K. Denote by P_A(K) the cone of polynomials in R[x]_A that are nonnegative on K. The dual cone of P_A(K) is R_A(K), the set of all A-truncated moment sequences in R^A that admit representing measures supported in K. Our main results are: i) We study the properties of P_A(K) and R_A(K) (like interiors, closeness, duality, memberships), and construct a convergent hierarchy of semidefinite relaxations for each of them. ii) We propose a semidefinite algorithm for solving linear optimization problems with the cones P_A(K) and R_A(K), and prove its asymptotic and finite convergence; a stopping criterion is also given. iii) We show how to check whether P_A(K) and R_A(K) intersect affine subspaces; if they do, we show to get get a point in the intersections; if they do not, we prove certificates for the non-intersecting

Convergence of the Lasserre Hierarchy of SDP Relaxations for Convex Polynomial Programs without Compactness

The Lasserre hierarchy of semidefinite programming (SDP) relaxations is an effective scheme for finding computationally feasible SDP approximations of polynomial optimization over compact semi-algebraic sets. In this paper, we show that, for convex polynomial optimization, the Lasserre hierarchy with a slightly extended quadratic module always converges asymptotically even in the face of non-compact semi-algebraic feasible sets. We do this by exploiting a coercivity property of convex polynomials that are bounded below. We further establish that the positive definiteness of the Hessian of the associated Lagrangian at a saddle-point (rather than the objective function at each minimizer) guarantees finite convergence of the hierarchy. We obtain finite convergence by first establishing a new sum-of-squares polynomial representation of convex polynomials over convex semi-algebraic sets under a saddle-point condition. We finally prove that the existence of a saddle-point of the Lagrangian for a convex polynomial program is also necessary for the hierarchy to have finite convergence.Comment: 17 page

Copositive certificates of non-negativity for polynomials on semialgebraic sets

A certificate of non-negativity is a way to write a given function so that its non-negativity becomes evident. Certificates of non-negativity are fundamental tools in optimization, and they underlie powerful algorithmic techniques for various types of optimization problems. We propose certificates of non-negativity of polynomials based on copositive polynomials. The certificates we obtain are valid for generic semialgebraic sets and have a fixed small degree, while commonly used sums-of-squares (SOS) certificates are guaranteed to be valid only for compact semialgebraic sets and could have large degree. Optimization over the cone of copositive polynomials is not tractable, but this cone has been well studied. The main benefit of our copositive certificates of non-negativity is their ability to translate results known exclusively for copositive polynomials to more general semialgebraic sets. In particular, we show how to use copositive polynomials to construct structured (e.g., sparse) certificates of non-negativity, even for unstructured semialgebraic sets. Last but not least, copositive certificates can be used to obtain not only hierarchies of tractable lower bounds, but also hierarchies of tractable upper bounds for polynomial optimization problems.Comment: 27 pages, 1 figur

Exact relaxation for polynomial optimization on semi-algebraic sets

In this paper, we study the problem of computing by relaxation hierarchies the infimum of a real polynomial function f on a closed basic semialgebraic set and the points where this infimum is reached, if they exist. We show that when the infimum is reached, a relaxation hierarchy constructed from the Karush-Kuhn-Tucker ideal is always exact and that the vanishing ideal of the KKT minimizer points is generated by the kernel of the associated moment matrix in that degree, even if this ideal is not zero-dimensional. We also show that this relaxation allows to detect when there is no KKT minimizer. We prove that the exactness of the relaxation depends only on the real points which satisfy these constraints.This exploits representations of positive polynomials as elementsof the preordering modulo the KKT ideal, which only involves polynomials in the initial set of variables. Applications to global optimization, optimization on semialgebraic sets defined by regular sets of constraints, optimization on finite semialgebraic sets, real radical computation are given