A convex polynomial that is not sos-convex

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 see that sos-convexity is a sufficient condition for convexity of $p(x)$. Moreover, the problem of deciding sos-convexity of a polynomial can be cast as the feasibility of a semidefinite program, which can be solved efficiently. Motivated by this computational tractability, it has been recently speculated whether sos-convexity is also a necessary condition for convexity of polynomials. In this paper, we give a negative answer to this question by presenting an explicit example of a trivariate homogeneous polynomial of degree eight that is convex but not sos-convex. Interestingly, our example is found with software using sum of squares programming techniques and the duality theory of semidefinite optimization. As a byproduct of our numerical procedure, we obtain a simple method for searching over a restricted family of nonnegative polynomials that are not sums of squares.Comment: 15 page

Smaller SDP for SOS Decomposition

A popular numerical method to compute SOS (sum of squares of polynomials) decompositions for polynomials is to transform the problem into semi-definite programming (SDP) problems and then solve them by SDP solvers. In this paper, we focus on reducing the sizes of inputs to SDP solvers to improve the efficiency and reliability of those SDP based methods. Two types of polynomials, convex cover polynomials and split polynomials, are defined. A convex cover polynomial or a split polynomial can be decomposed into several smaller sub-polynomials such that the original polynomial is SOS if and only if the sub-polynomials are all SOS. Thus the original SOS problem can be decomposed equivalently into smaller sub-problems. It is proved that convex cover polynomials are split polynomials and it is quite possible that sparse polynomials with many variables are split polynomials, which can be efficiently detected in practice. Some necessary conditions for polynomials to be SOS are also given, which can help refute quickly those polynomials which have no SOS representations so that SDP solvers are not called in this case. All the new results lead to a new SDP based method to compute SOS decompositions, which improves this kind of methods by passing smaller inputs to SDP solvers in some cases. Experiments show that the number of monomials obtained by our program is often smaller than that by other SDP based software, especially for polynomials with many variables and high degrees. Numerical results on various tests are reported to show the performance of our program.Comment: 18 page

Polynomial Norms

In this paper, we study polynomial norms, i.e. norms that are the $d^{\text{th}}$ root of a degree-$d$ homogeneous polynomial $f$. We first show that a necessary and sufficient condition for $f^{1/d}$ to be a norm is for $f$ to be strictly convex, or equivalently, convex and positive definite. Though not all norms come from $d^{\text{th}}$ roots of polynomials, we prove that any norm can be approximated arbitrarily well by a polynomial norm. We then investigate the computational problem of testing whether a form gives a polynomial norm. We show that this problem is strongly NP-hard already when the degree of the form is 4, but can always be answered by testing feasibility of a semidefinite program (of possibly large size). We further study the problem of optimizing over the set of polynomial norms using semidefinite programming. To do this, we introduce the notion of r-sos-convexity and extend a result of Reznick on sum of squares representation of positive definite forms to positive definite biforms. We conclude with some applications of polynomial norms to statistics and dynamical systems

ROBUST STATE FEEDBACK CONTROL OF UNCERTAIN POLYNOMIAL DISCRETE-TIME SYSTEMS: AN INTEGRAL ACTION APPROACH

his paper examines the problem of designing a nonlinear state feedback controller for polynomial discrete-time systems with parametric uncertainty. In general, this is a challenging controller design problem due to the fact that the relation between Lyapunov function and the control input is not jointly convex; hence, this problem cannot be solved by a semidenite programming (SDP). In this paper, a novel approach is proposed, where an integral action is incorporated into the controller design to convexify the controller design problem of polynomial discrete-time systems. Based on the sum of squares (SOS) approach, sufficient conditions for the existence of a nonlinear state feedback controller for polynomial discrete-time systems are given in terms of solvability of polynomial matrix inequalities, which can be solved by the recently developed SOS solver. Numerical examples are provided to demonstrate the validity of this integral action approach

Contributions to the moment-SOS approach in global polynomial optimization

L''Optimisation Polynomiale' s'intéresse aux problèmes d'optimisation P de la forme min {f(x): x dans K} où f est un polynôme et K est un ensemble semi-algébrique de base, c'est-à-dire défini par un nombre fini de contraintes inégalité polynomiales, K={x dans Rn : gj(x) <= 0}. Cette sous discipline de l'optimisation a émergé dans la dernière décennie grâce à la combinaison de deux facteurs: l'existence de certains résultats puissants de géométrie algébrique réelle et la puissance de l'optimisation semidéfinie (qui permet d'exploiter les premiers). Il en a résulté une méthodologie générale (que nous appelons ``moments-SOS') qui permet d'approcher aussi près que l'on veut l'optimum global de P en résolvant une hiérarchie de relaxations convexes. Cependant, chaque relaxation étant un programme semi-défini dont la taille augmente avec le rang dans la hiérarchie, malheureusement, au vu de l'état de l'art actuel des progiciels de programmation semidéfinie, cette méthodologie est pour l'instant limitée à des problèmes P de taille modeste sauf si des symétries ou de la parcimonie sont présentes dans la définition de P. Cette thèse essaie donc de répondre à la question: Peux-t-on quand même utiliser la méthodologie moments-SOS pour aider à résoudre P même si on ne peut résoudre que quelques (voire une seule) relaxations de la hiérarchie? Et si oui, comment? Nous apportons deux contributions: I. Dans une première contribution nous considérons les problèmes non convexes en variables mixtes (MINLP) pour lesquelles dans les contraintes polynomiales {g(x) <=0} où le polynôme g n'est pas concave, g est concerné par peu de variables. Pour résoudre de tels problèmes (de taille est relativement importante) on utilise en général des méthodes de type ``Branch-and-Bound'. En particulier, pour des raisons d'efficacité évidentes, à chaque nœud de l'arbre de recherche on doit calculer rapidement une borne inférieure sur l'optimum global. Pour ce faire on utilise des relaxations convexes du problème obtenues grâce à l'utilisation de sous estimateurs convexes du critère f (et des polynômes g pour les contraintes g(x)<= 0 non convexes). Notre contribution est de fournir une méthodologie générale d'obtention de tels sous estimateurs polynomiaux convexes pour tout polynôme g, sur une boite. La nouveauté de notre contribution (grâce à la méthodologie moment-SOS) est de pouvoir minimiser directement le critère d'erreur naturel qui mesure la norme L_1 de la différence f-f' entre f et son sous estimateur convexe polynomial f'. Les résultats expérimentaux confirment que le sous estimateur convexe polynomial que nous obtenons est nettement meilleur que ceux obtenus par des méthodes classiques de type ``alpha-BB' et leurs variantes, tant du point de vue du critère L_1 que du point de vue de la qualité des bornes inférieures obtenus quand on minimise f' (au lieu de f) sur la boite. II: Dans une deuxième contribution on considère des problèmes P pour lesquels seules quelques relaxations de la hiérarchie moments-SOS peuvent être implantées, par exemple celle de rang k dans la hiérarchie, et on utilise la solution de cette relaxation pour construire une solution admissible de P. Cette idée a déjà été exploitée pour certains problèmes combinatoire en variables 0/1, parfois avec des garanties de performance remarquables (par exemple pour le problème MAXCUT). Nous utilisons des résultats récents de l'approche moment-SOS en programmation polynomiale paramétrique pour définir un algorithme qui calcule une solution admissible pour P à partir d'une modification mineure de la relaxation convexe d'ordre k. L'idée de base est de considérer la variable x_1 comme un paramètre dans un intervalle Y_1 de R et on approxime la fonction ``valeur optimale' J(y) du problème d'optimisation paramétrique P(y)= min {f(x): x dans K; x_1=y} par un polynôme univarié de degré d fixé. Cette étape se ramène à la résolution d'un problème d'optimisation convexe (programme semidéfini). On calcule un minimiseur global y de J sur l'intervalle Y (un problème d'optimisation convexe ``facile') et on fixe la variable x_1=y. On itère ensuite sur les variables restantes x_2,...,x_n en prenant x_2 comme paramètre dans un intervalle Y_2, etc. jusqu'à obtenir une solution complète x de R^n qui est faisable si K est convexe ou dans certains problèmes en variables 0/1 où la faisabilité est facile à vérifier (e.g., MAXCUT, k-CLUSTTER, Knapsack). Sinon on utilise le point obtenu x comme initialisation dans un procédure d'optimisation locale pour obtenir une solution admissible. Les résultats expérimentaux obtenus sur de nombreux exemples sont très encourageants et prometteurs.Polynomial Optimization is concerned with optimization problems of the form (P) : f* = { f(x) with x in set K}, where K is a basic semi-algebraic set in Rn defined by K={x in Rn such as gj(x) less or equal 0}; and f is a real polynomial of n variables x = (x1, x2, ..., xn). In this thesis we are interested in problems (P) where symmetries and/or structured sparsity are not easy to detect or to exploit, and where only a few (or even no) semidefinite relaxations of the moment-SOS approach can be implemented. And the issue we investigate is: How can the moment-SOS methodology be still used to help solve such problem (P)? We provide two applications of the moment-SOS approach to help solve (P) in two different contexts. * In a first contribution we consider MINLP problems on a box B = [xL, xU] of Rn and propose a moment-SOS approach to construct polynomial convex underestimators for the objective function f (if non convex) and for -gj if in the constraint gj(x) less or equal 0, the polynomial gj is not concave. We work in the context where one wishes to find a convex underestimator of a non-convex polynomial f of a few variables on a box B of Rn. The novelty with previous works on this topic is that we want to compute a polynomial convex underestimator p of f that minimizes the important tightness criterion which is the L1 norm of (f-h) on B, over all convex polynomials h of degree d _fixed. Indeed in previous works for computing a convex underestimator L of f, this tightness criterion is not taken into account directly. It turns out that the moment-SOS approach is well suited to compute a polynomial convex underestimator p that minimizes the tightness criterion and numerical experiments on a sample of non-trivial examples show that p outperforms L not only with respect to the tightness score but also in terms of the resulting lower bounds obtained by minimizing respectively p and L on B. Similar improvements also occur when we use the moment-SOS underestimator instead of the aBB-one in refinements of the aBB method. * In a second contribution we propose an algorithm that also uses an optimal solution of a semidefinite relaxation in the moment-SOS hierarchy (in fact a slight modification) to provide a feasible solution for the initial optimization problem but with no rounding procedure. In the present context, we treat the first variable x1 of x = (x1, x2, ...., xn) as a parameter in some bounded interval Y of R. Notice that f*=min { J(y) : y in Y} where J is the function J(y) := inf {f(x) : x in K ; x1=y}. That is one has reduced the original n-dimensional optimization problem (P) to an equivalent one-dimensional optimization problem on an interval. But of course determining the optimal value function J is even more complicated than (P) as one has to determine a function (instead of a point in Rn), an infinite-dimensional problem. But the idea is to approximate J(y) on Y by a univariate polynomial p(y) with the degree d and fortunately, computing such a univariate polynomial is possible via solving a semidefinite relaxation associated with the parameter optimization problem. The degree d of p(y) is related to the size of this semidefinite relaxation. The higher the degree d is, the better is the approximation of J(y) by p(y) and in fact, one may show that p(y) converges to J(y) in a strong sense on Y as d increases. But of course the resulting semidefinite relaxation becomes harder (or impossible) to solve as d increases and so in practice d is fixed to a small value. Once the univariate polynomial p(y) has been determined, one computes x1* in Y that minimizes p(y) on Y, a convex optimization problem that can be solved efficiently. The process is iterated to compute x2 in a similar manner, and so on, until a point x in Rn has been computed. Finally, as x* is not feasible in general, we then use x* as a starting point for a local optimization procedure to find a final feasible point x in K. When K is convex, the following variant is implemented. After having computed x1* as indicated, x2* is computed with x1 fixed at the value x1*, and x3 is computed with x1 and x2 fixed at the values x1* and x2* respectively, etc., so that the resulting point x* is feasible, i.e., x* in K. The same variant applies for 0/1 programs for which feasibility is easy to detect like e.g., for MAXCUT, k-CLUSTER or 0/1-KNAPSACK problems

Filter And Observer Design For Polynomial Discrete-Time Systems: A Sum Of Squares Based Approach

The polynomial discrete-time systems are the type of systems where the dynamics of the systems are described in polynomial forms.This system is classified as an important class of nonlinear systems due to the fact that many nonlinear systems can be modelled as,transformed into,or approximated by polynomial systems.The focus of this thesis is to address the problem of filter and observer design for polynomial discrete-time systems.The main reason for focusing on this area is because the filter and observer design for such polynomial discrete-time systems is categorised as a difficult problem.This is due to the fact that the relation between the Lyapunov matrix and the filter and observer gain is not jointly convex when the parameter-dependent or state-dependent Lyapunov function is under consideration.Therefore the problem cannot possibly be solved via semidefinite programming (SDP).In light of the aforementioned problem, we establish novel methodologies of designing filters for estimating the state of the systems both with and without H∞ performance and also designing an observer for state estimation and also as a controller.We show that through our proposed methodologies,a less conservative design procedure can be rendered for the filter and observer design.In particular,a so-called integrator method is proposed in this research work where an integrator is incorporated into the filter and observer structures.In doing so, the original systems can be transformed into augmented systems.Furthermore,the state-dependent function is selected in a way that its matrix is dependent only upon the original system state.Through this selection,a convex solution to the filter and observer design can be obtained efficiently.The existence of such filter and observer are given in terms of the solvability of polynomial matrix inequalities (PMIs).The problem is then formulated as sum of squares (SOS) constraints,therefore it can be solved by any SOS solvers.In this research work,SOSTOOLS is used as a SOS solver.Finally,to demonstrate the effectiveness and advantages of the proposed design methodologies in this thesis,numerical examples are given in filter design system.The simulation results show that the proposed design methodologies can estimate and stabilise the systems and achieve the prescribed performance requirements

A Complete Characterization of the Gap between Convexity and SOS-Convexity

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 characterization, and its second order characterization, are equivalent. These three equivalent algebraic conditions, henceforth referred to as sos-convexity, can be checked by semidefinite programming whereas deciding convexity is NP-hard. If we denote the set of convex and sos-convex polynomials in $n$ variables of degree $d$ with $\tilde{C}_{n,d}$ and $\tilde{\Sigma C}_{n,d}$ respectively, then our main contribution is to prove that $\tilde{C}_{n,d}=\tilde{\Sigma C}_{n,d}$ if and only if $n=1$ or $d=2$ or $(n,d)=(2,4)$. We also present a complete characterization for forms (homogeneous polynomials) except for the case $(n,d)=(3,4)$ which is joint work with G. Blekherman and is to be published elsewhere. Our result states that the set $C_{n,d}$ of convex forms in $n$ variables of degree $d$ equals the set $\Sigma C_{n,d}$ of sos-convex forms if and only if $n=2$ or $d=2$ or $(n,d)=(3,4)$. To prove these results, we present in particular explicit examples of polynomials in $\tilde{C}_{2,6}\setminus\tilde{\Sigma C}_{2,6}$ and $\tilde{C}_{3,4}\setminus\tilde{\Sigma C}_{3,4}$ and forms in $C_{3,6}\setminus\Sigma C_{3,6}$ and $C_{4,4}\setminus\Sigma C_{4,4}$, and a general procedure for constructing forms in $C_{n,d+2}\setminus\Sigma C_{n,d+2}$ from nonnegative but not sos forms in $n$ variables and degree $d$. Although for disparate reasons, the remarkable outcome is that convex polynomials (resp. forms) are sos-convex exactly in cases where nonnegative polynomials (resp. forms) are sums of squares, as characterized by Hilbert.Comment: 25 pages; minor editorial revisions made; formal certificates for computer assisted proofs of the paper added to arXi