calculation of acoustic radiation using equivalent sphere methods among the methods generally used to solve a problem in the domain of acoustic radiation the equivalent sources method offers an interesting alternative it consists in replacing the vibrating surface with a distribution of acoustic sources placed inside the structure the contribution of each source is determined in such a way that the acoustic field radiated by these sources verifies the same boundary conditions on the structure the number of unknowns in the problem is no longer directly linked to the number of mesh points on the structure as with boundary elements methods but to the number of equivalent sources employed in the model the equivalent source method is therefore of major interest if the acoustic radiation of the structure can be approximated with a sufficiently low number of sources this paper proposes its application when the equivalent source is a sphere in this case the number of unknowns is equal to the number of modes in contrast to the one point multipole the sphere has a surface surrounding a closed volume to express the boundary conditions although sphere multipole equivalence has been demonstrated the surface of the sphere allows normalization of the functions used leading to stabilization of the system to be resolved first the main acoustic radiation characteristics of a sphere and of the linear system verified by the modal coefficients of an equivalent sphere are presented the different parameters of the model are then studied position and radius of the equivalent sphere truncation of the series and influence of the spatial sampling mesh in the same vein a second approach is presented it consists of making each point of the structure correspond to a point of the sphere and the vibrating field at the surface of the sphere is deduced from that of the structure by simple geometric projection results can be obtained very quickly as no matrix inversion is required the accuracy of the results depends on the distance between the sphere and the structure finally an experimental validation that uses both methods is presented and shows interesting results when the structure is closed and when its shape is not too far removed from a sphere

algebraic theory for robust stability of interconnected systems necessary and sufficient conditions we consider an interconnected system s o made of linear mulrivariable subsystems which are specified by matrix fractions with elements in a ring of stable scalar transfer functions h given that the k th subsystem is perturbed from g k n rk d k 1 to tilde g k n rk delta n rk d k delta d k 1 and that the system s o is h stable we derive a computationally efficient necessary and sufficient condition for the h stability of the perturbed system these fractional perturbations are more general than the conventional additive and multiplicative perturbations the result is generalized to handle simultaneous perturbations of two or more subsystems

the truncated complex moment problem let 2n denote a sequence of complex numbers 00 01 10 0 2n 2n 0 00 0 ij ji and let k denote a closed subset of the complex plane c the truncated complex k moment problem for entails determining whether there exists a positive borel measure on c such that ij zizj d 0 i j 2n and supp k for k kp a semi algebraic set determined by a collection of complex polynomials p pi z z mi 1 we characterize the existence of a finitely atomic representing measure with the fewest possible atoms in terms of positivity and extension properties of the moment matrix m n and the localizing matrices mpi we prove that there exists a rankm n atomic representing measure for 2n supported in kp if and only if m n 0 and there is some rankpreserving extension m n 1 for which mpi n ki 0 where deg pi 2ki or 2ki 1 1 i m

completely positive reformulations for polynomial optimization polynomial optimization encompasses a very rich class of problems in which both the objective and constraints can be written in terms of polynomials on the decision variables there is a well established body of research on quadratic polynomial optimization problems based on reformulations of the original problem as a conic program over the cone of completely positive matrices or its conic dual the cone of copositive matrices as a result of this reformulation approach novel solution schemes for quadratic polynomial optimization problems have been designed by drawing on conic programming tools and the extensively studied cones of completely positive and of copositive matrices in particular this approach has been applied to solve key combinatorial optimization problems along this line of research we consider polynomial optimization problems that are not necessarily quadratic for this purpose we use a natural extension of the cone of completely positive matrices namely the cone of completely positive tensors we provide a general characterization of the class of polynomial optimization problems that can be formulated as a conic program over the cone of completely positive tensors as a consequence of this characterization it follows that recent related results for quadratic problems can be further strengthened and generalized to higher order polynomial optimization problems also we show that the conditions underlying the characterization are conceptually the same regardless of the degree of the polynomials defining the problem to illustrate our results we discuss in further detail special and relevant instances of polynomial optimization problems

cubic column relations in truncated moment problems for the truncated moment problem associated to a complex sequence 2n ij i j z i j 2n to have a representing measure it is necessary for the moment matrix m n to be positive semidefinite and for the algebraic variety v to satisfy rankm n cardv as well as a consistency condition the riesz functional vanishes on every polynomial of degree at most 2n that vanishes on v in previous work with l fialkow and h m moller the first named author proved that for the extremal case rankm n cardv positivity and consistency are sufficient for the existence of a representing measure in this paper we solve the truncated moment problem for cubic column relations in m 3 of the form z3 itz uz u t r we do this by checking consistency for u t in the open cone determined by 0 u t 2 u we first prove that the algebraic variety has exactly 7 points and rankm 3 7 we then apply the above mentioned result to obtain a concrete computable necessary and sufficient condition for the existence of a representing measure

iterated rings of bounded elements and generalizations of schmudgen s positivstellensatz let a be a commutative r algebra of finite transcendence degree d 2 n we investigate the relationship between the subring of geometrically bounded elements

global optimization of rational functions a semidefinite programming approach we consider the problem of global minimization of rational functions on unconstrained case and on an open connected semi algebraic subset of or the partial closure of such a set constrained case we show that in the univariate case n 1 these problems have exact reformulations as semidefinite programming sdp problems by using reformulations introduced in the phd thesis of jibetean 16 this extends the analogous results by nesterov 13 for global minimization of univariate polynomials for the bivariate case n 2 we obtain a fully polynomial time approximation scheme fptas for the unconstrained problem if an a priori lower bound on the infimum is known by using results by de klerk and pasechnik 1 for the np hard multivariate case we discuss semidefinite programming based relaxations for obtaining lower bounds on the infimum by using results by parrilo 15 and lasserre 12

non extremal sextic moment problems abstract positive semidefiniteness recursiveness and the variety condition of a moment matrix are necessary and sufficient conditions to solve the quadratic and quartic moment problems also positive semidefiniteness combined with another necessary condition consistency is a sufficient condition in the case of extremal moment problems i e when the rank of the moment matrix denoted by r and the cardinality of the associated algebraic variety denoted by v are equal however these conditions are not sufficient for non extremal sextic or higher order truncated moment problems in this paper we settle three key instances of the non extremal i e r v sextic moment problem as follows when r 7 positive semidefiniteness consistency and the variety condition guarantee the existence of a 7 atomic representing measure when r 8 we construct two determining algorithms corresponding to the cases v 9 and v to accomplish this we generalize the rank reduction technique developed in previous work where we solved the nonsingular quartic moment problem and found an explicit way to build a representing measure

strict positivstellensatze for matrix polynomials with scalar constraints abstract we extend krivine s strict positivstellensatz for usual real multivariate polynomials to symmetric matrix polynomials with scalar constraints the proof is an elementary computation with schur complements analogous extensions of schmudgen s and putinar s strict positivstellensatz were recently proved by hol and scherer using methods from optimization theory

global optimization of polynomials using gradient tentacles and sums of squares we consider the problem of computing the global infimum of a real polynomial f on mathbb r n every global minimizer of f lies on its gradient variety i e the algebraic subset of mathbb r n where the gradient of f vanishes if f attains a minimum on mathbb r n it is therefore equivalent to look for the greatest lower bound of f on its gradient variety nie demmel and sturmfels proved recently a theorem about the existence of sums of squares certificates for such lower bounds based on these certificates they find arbitrarily tight relaxations of the original problem that can be formulated as semidefinite programs and thus be solved efficiently we deal here with the more general case when f is bounded from below but does not necessarily attain a minimum in this case the method of nie demmel and sturmfels might yield completely wrong results in order to overcome this problem we replace the gradient variety by larger semialgebraic subsets of mathbb r n which we call gradient tentacles it now gets substantially harder to prove the existence of the necessary sums of squares certificates

symmetric sign pattern matrices that require unique inertia abstract a sign pattern matrix is a matrix whose entries are from the set 0 the purpose of this paper is to characterize symmetric sign patterns that require unique inertia that is all the real symmetric matrices with the given sign pattern must have the same inertia further some constructions to obtain sign patterns that require unique inertia are provided sign patterns corresponding to some special graphs are also considered finally extensions to complex sign patterns are mentioned

an analogue of the riesz haviland theorem for the truncated moment problem abstract let 2 n i i 2 n denote a d dimensional real multisequence let k denote a closed subset of r d and let p 2 n p r x 1 x d deg p 2 n corresponding to the riesz functional l l p 2 n r is defined by l a i x i a i i we say that l is k positive if whenever p p 2 n and p k 0 then l p 0 we prove that admits a k representing measure if and only if l admits a k positive linear extension l p 2 n 2 r this provides a generalization from the full moment problem to the truncated moment problem of the riesz haviland theorem we also show that a semialgebraic set solves the truncated moment problem in terms of natural degree bounded positivity conditions if and only if each polynomial strictly positive on that set admits a degree bounded weighted sum of squares representation

first order conditions for semidefinite representations of convex sets defined by rational or singular polynomials a set is called semidefinite representable or semidefinite programming sdp representable if it equals the projection of a higher dimensional set which is defined by some linear matrix inequality lmi this paper discusses the semidefinite representability conditions for convex sets of the form s mathcal d f x in mathcal d f x geq 0 here mathcal d x in mathbb r n g 1 x geq 0 ldots g m x geq 0 is a convex domain defined by some nice concave polynomials g i x they satisfy certain concavity certificates and f x is a polynomial or rational function when f x is concave over mathcal d we prove that s mathcal d f has some explicit semidefinite representations under certain conditions called preordering concavity or q module concavity which are based on the positivstellensatz certificates for the first order concavity criteria f u nabla f u t x u f x geq 0 quad forall x u in mathcal d when f x is a polynomial or rational function having singularities on the boundary of s mathcal d f a perspective transformation is introduced to find some explicit semidefinite representations for s mathcal d f under certain conditions in the special case n 2 if the laurent expansion of f x around one singular point has only two consecutive homogeneous parts we show that s mathcal d f always admits an explicitly constructible semidefinite representation

non negative hereditary polynomials in a free algebra we prove a non negative stellensatz and a null stellensatz for a class of polynomials called hereditary polynomials in a free algebra

the l problem of moments in two dimensions abstract the classical solution to the l problem of moments due to n i akhiezer and m g krein is partially generalized to the two dimensional case a parallel between the apparently quite different one dimensional and two dimensional l problems of moments is established by means of the theories of the phase shift and of the principal function this point of view brings into the field the geometry of hilbert spaces and the analysis of the pairs of not necessarily commuting self adjoint operators acting on them

probleme des moments sur un compact de rn et decomposition de polynomes a plusieurs variables on donne la solution du probleme des moments pour un compact d interieur non vide quelconque de r n on utilise cette resolution pour obtenir un resultat de decomposition des polynomes strictement positifs sur un compact d interieur non vide

the full infinite dimensional moment problem on semi algebraic sets of generalized functions we consider a generic basic semi algebraic subset s of the space of generalized functions that is a set given by not necessarily countably many polynomial constraints we derive necessary and sufficient conditions for an infinite sequence of generalized functions to be realizable on s namely to be the moment sequence of a finite measure concentrated on s our approach combines the classical results about the moment problem on nuclear spaces with the techniques recently developed to treat the moment problem on basic semi algebraic sets of rd in this way we determine realizability conditions that can be more easily verified than the well known haviland type conditions our result completely characterizes the support of the realizing measure in terms of its moments as concrete examples of semi algebraic sets of generalized functions we consider the set of all radon measures and the set of all the measures having bounded radon nikodym density w r t the lebesgue measure

a comparison of the sherali adams lov 341sz schrijver and lasserre relaxations for 0 1 programming sherali and adams sa90 lov asz and schrijver ls91 and recently lasserre las01b have proposed lift and project methods for constructing hierarchies of successive linear or semidefinite relaxations of a 0 1 polytope p subseteq or n converging to p in n steps lasserre s approach uses results about representations of positive polynomials as sums of squares and the dual theory of moments we present the three methods in a common elementary framework and show that the lasserre construction provides the tightest relaxations of p as an application this gives a direct simple proof for the convergence of the lasserre s hierarchy we describe applications to the stable set polytope and to the cut polytope

nonlinear optimal control via occupation measures and lmi relaxations we consider the class of nonlinear optimal control problems ocps with polynomial data i e the differential equation state and control constraints and cost are all described by polynomials and more generally for ocps with smooth data in addition state constraints as well as state and or action constraints are allowed we provide a simple hierarchy of lmi linear matrix inequality relaxations whose optimal values form a nondecreasing sequence of lower bounds on the optimal value under some convexity assumptions the sequence converges to the optimal value of the ocp preliminary results show that good approximations are obtained with few moments

semidefinite characterization of sum of squares cones in algebras we extend nesterov s semidefinite programming characterization of squared functional systems and faybusovich s abstraction to bilinear symmetric maps to cones of sum of squares elements in general abstract algebras using algebraic techniques such as isomorphism linear isomorphism tensor products sums and direct sums we show that many concrete cones are in fact sum of squares cones with respect to some algebra and thus are representable by the cone of positive semidefinite matrices we also consider nonnegativity with respect to a proper cone mathcal k and show that in some cases cones of mathcal k nonnegative functions are either sum of squares or at least semidefinite representable for example we show that some well known chebyshev systems when extended to euclidean jordan algebras induce cones that are semidefinite representable finally we will discuss some concrete examples and applications including minimum ellipsoid enclosing given space curves minimization of eigenvalues of po

sums of squares on real algebraic curves given an affine algebraic variety v over with real points v compact and a non negative polynomial function f v with finitely many real zeros we establish a local global criterion for f to be a sum of squares in v we then specialize to the case where v is a curve the notion of virtual compactness is introduced and it is shown that in the local global principle compactness of v can be relaxed to virtual compactness the irreducible curves on which every non negative polynomial is a sum of squares are classified all results are extended to the more general framework of preorders moreover applications to the k moment problem from analysis are given in particular schmudgen s solution of the k moment problem for compact k is extended for dim k 1 to the case when k is virtually compact

near optimal analysis of lasserre s univariate measure based bounds for multivariate polynomial optimization we consider a recent hierarchy of upper approximations proposed by lasserre arxiv 1907 097784 2019 for the minimization of a polynomial f over a compact set k n this hierarchy relies on using the push forward measure of the lebesgue measure on k by the polynomial f and involves univariate sums of squares of polynomials with growing degrees 2r hence it is weaker but cheaper to compute than an earlier hierarchy by lasserre siam journal on optimization 21 3 864 885 2011 which uses multivariate sums of squares we show that this new hierarchy converges to the global minimum of f at a rate in o log2 r r2 whenever k satisfies a mild geometric condition which holds e g for convex bodies as an application this rate of convergence also applies to the stronger hierarchy based on multivariate sums of squares which extends earlier convergence results to a wider class of compact sets furthermore we show that our analysis is near optimal by proving a lower bound on the convergence rate in 1 r2 for a class of polynomials on k 1 1 obtained by exploiting a connection to orthogonal polynomials

lmi approximations for cones of positive semidefinite forms an interesting recent trend in optimization is the application of semidefinite programming techniques to new classes of optimization problems in particular this trend has been successful in showing that under suitable circumstances polynomial optimization problems can be approximated via a sequence of semidefinite programs similar ideas apply to conic optimization over the cone of copositive matrices and to certain optimization problems involving random variables with some known moment information we bring together several of these approximation results by studying the approximability of cones of positive semidefinite forms homogeneous polynomials our approach enables us to extend the existing methodology to new approximation schemes in particular we derive a novel approximation to the cone of copositive forms that is the cone of forms that are positive semidefinite over the nonnegative orthant the format of our construction can be extended to forms that are positive semidefinite over more general conic domains we also construct polyhedral approximations to cones of positive semidefinite forms over a polyhedral domain this opens the possibility of using linear programming technology in optimization problems over these cones

optimization of polynomials on compact semialgebraic sets a basic closed semialgebraic subset s of r n is defined by simultaneous polynomial inequalities g 1 ge 0 dotsc g m ge 0 we give a short introduction to lasserre s method for minimizing a polynomial f on a compact set s of this kind it consists of successively solving tighter and tighter convex relaxations of this problem which can be formulated as semidefinite programs we give a new short proof for the convergence of the optimal values of these relaxations to the infimum f ast of f on s which is constructive and elementary in the case where f possesses a unique minimizer x ast we prove that every sequence of nearly optimal solutions of the successive relaxations gives rise to a sequence of points in r n converging to x ast

error bounds for some semidefinite programming approaches to polynomial minimization on the hypercube we consider the problem of minimizing a polynomial on the hypercube 0 1 n and derive new error bounds for the hierarchy of semidefinite programming approximations to this problem corresponding to the positivstellensatz of schmudgen math ann 289 1991 pp 203 206 the main tool we employ is bernstein approximations of polynomials which also gives constructive proofs and degree bounds for positivity certificates on the hypercube

sum of squares approximation of polynomials nonnegative on a real algebraic set wih every real polynomial f we associate a family f epsilon r epsilon r of real polynomials in explicit form in terms of f and the parameters epsilon 0 r in n and such that vert f f epsilon r vert 1 to 0 as epsilon to 0 let v subset r n be a real algebraic set described by finitely many polynomials equations g j x 0 j in j and let f be a real polynomial nonnegative on v we show that for every epsilon 0 there exist nonnegative scalars lambda j epsilon j in j such that for all r sufficiently large f epsilon r sum j in j lambda j epsilon g j 2 quad mbox is a sum of squares this representation is an obvious certificate of nonnegativity of f epsilon r on v and very specific in terms of the g j that define the set v in particular it is valid with no assumption on v in addition this representation is also useful from a computation point of view as we can define semidefinite programming relaxations to approximate the global minimum of f on a real algebraic set v or a semialgebraic set k and again with no assumption on v or k

a sum of squares approximation of nonnegative polynomials we show that every real nonnegative polynomial f can be approximated as closely as desired in the l 1 norm of its coefficient vector by a sequence of polynomials f epsilon that are sums of squares the novelty is that each f epsilon has a simple and explicit form in terms of f and epsilon

moment problems for multi sequences of operators abstract we give an explicit solution to the scalar moment problem on semi algebraic compact subsets of r n and apply this result to the study of some operator multi sequences

a strict positivstellensatz for the weyl algebra let c be an element of the weyl algebra open image in new window which is given by a strictly positive operator in the schrodinger representation it is shown that under some conditions there exist certain elements b1 bd from open image in new window such that dj 1bjcb j is a finite sum of squares

convexity in semialgebraic geometry and polynomial optimization we review several and provide new results on the theory of moments sums of squares and basic semialgebraic sets when convexity is present in particular we show that under convexity the hierarchy of semidefinite relaxations for polynomial optimization simplifies and has finite convergence a highly desirable feature as convex problems are in principle easier to solve in addition if a basic semialgebraic set mathbf k is convex but its defining polynomials are not we provide two algebraic certificates of convexity which can be checked numerically the second is simpler and holds if a sufficient and almost necessary condition is satisfied it also provides a new condition for mathbf k to have semidefinite representation for this we use and extend some of the recent results from the author and helton and nie math program to appear finally we show that when restricting to a certain class of convex polynomials the celebrated jensen s inequality in convex analysis can be extended to linear functionals that are not necessarily probability measures

moment problems for operator polynomials abstract haviland s theorem states that given a closed subset k in r n each functional l r x r positive on pos k p r x p k 0 admits an integral representation by a positive borel measure schmudgen proved that in the case of compact semialgebraic set k it suffices to check positivity of l on a preordering t having k as the non negativity set further he showed that the compactness of k is equivalent to the archimedianity of t the aim of this paper is to extend these results from functionals on the usual real polynomials to operators mapping from the real matrix or operator polynomials into r m n r or b k

improved convergence rates for lasserre type hierarchies of upper bounds for box constrained polynomial optimization we consider the problem of minimizing a given n variate polynomial f over the hypercube 1 1 n an idea introduced by lasserre is to find a probability distribution on 1 1 n with polynomial density function h of given degree r that minimizes the expectation int 1 1 n f x h x d mu x where d mu x is a fixed finite borel measure supported on 1 1 n it is known that for the lebesgue measure d mu x dx one may show an error bound o 1 sqrt r if h is a sum of squares density and an o 1 r error bound if h is the density of a beta distribution in this paper we show an error bound of o 1 r 2 if d mu x left prod i 1 n sqrt 1 x i 2 right 1 the well known measure in the study of orthogonal polynomials and h has a schmudgen type representation with respect to 1 1 n which is a more general condition than a sum of squares the convergence rate analysis relies on the theory of polynomial kernels and in particular on jackson kerne

approximate volume and integration for basic semialgebraic sets given a basic compact semialgebraic set mathbf k subset mathbb r n we introduce a methodology that generates a sequence converging to the volume of mathbf k this sequence is obtained from optimal values of a hierarchy of either semidefinite or linear programs not only the volume but also every finite vector of moments of the probability measure that is uniformly distributed on mathbf k can be approximated as closely as desired which permits the approximation of the integral on mathbf k of any given polynomial the extension to integration against some weight functions is also provided finally some numerical issues associated with the algorithms involved are briefly discussed

polynomials nonnegative on a grid and discrete optimization we characterize the real valued polynomials on r n that are nonnegative not necessarily strictly positive on a grid k of points of r n in terms of a weighted sum of squares whose degree is bounded and known in advance we also show that the mimimization of an arbitrary polynomial on k a discrete optimization problem reduces to a convex continuous optimization problem of fixed size the case of concave polynomials is also investigated the proof is based on a recent result of curto and fialkow on the k moment problem

conic approach to quantum graph parameters using linear optimization over the completely positive semidefinite cone we investigate the completely positive semidefinite cone mathcal cs n a new matrix cone consisting of all n times n matrices that admit a gram representation by positive semidefinite matrices of any size in particular we study relationships between this cone and the completely positive and the doubly nonnegative cone and between its dual cone and trace positive noncommutative polynomials we use this new cone to model quantum analogues of the classical independence and chromatic graph parameters alpha g and chi g which are roughly obtained by allowing variables to be positive semidefinite matrices instead of 0 1 scalars in the programs defining the classical parameters we can formulate these quantum parameters as conic linear programs over the cone mathcal cs n using this conic approach we can recover the bounds in terms of the theta number and define further approximations by exploiting the link to trace positive polynomials

the strict positivstellensatz for global analytic functions and the moment problem for semianalytic sets analytic functions strictly positive on a global semianalytic set x f1 0 fk 0 in rn are characterized as functions expressible as g a0 a1f1 akfk for strictly positive global analytic functions a0 ak the proof is elementary using the fact that the analytic functions are dense in c rn r in the whitney topology the same proof works for nash functions this is an improvement of the standard analytic version of stengle s positivstellensatz in two directions the hypothesis is weaker there is no requirement that x be compact and the conclusion is stronger several applications are given including i a new proof of the weak positivstellensatz for semianalytic sets and ii the solution of thek moment problem for basic closed semianalytic

backstepping pde design a convex optimization approach backstepping design for boundary linear partial differential equation pde is formulated as a convex optimization problem some classes of parabolic pdes and a first order hyperbolic pde are studied with particular attention to nonstrict feedback structures based on the compactness of the volterra and fredholm type operators involved their kernels are approximated via polynomial functions the resulting kernel pdes are optimized using sum of squares decomposition and solved via semidefinite programming with sufficient precision to guarantee the stability of the system in the mathcal l 2 norm this formulation allows optimizing extra degrees of freedom where the kernel pdes are included as constraints uniqueness and invertibility of the fredholm type transformation are proved for polynomial kernels in the space of continuous functions the effectiveness and limitations of the approach proposed are illustrated by numerical solutions of some kernel pdes

the moment problem on compact semi algebraic sets in this chapter we begin the study of the multidimensional moment problem the passage to dimensions d 2 brings new difficulties and unexpected phenomena in sect 3 2 we derived solvability criteria of the moment problem on intervals in terms of positivity conditions it seems to be natural to look for similar characterizations in higher dimensions as well this leads us immediately into the realm of real algebraic geometry and to descriptions of positive polynomials on semi algebraic sets in this chapter we treat this approach for basic closed compact semi algebraic subsets of mathbb r d it turns out that for such sets there is a close interaction between the moment problem and positivstellensatze for strictly positive polynomials

polyhedra related to integer convex polynomial systems this paper deals with the reformulation of a polynomial integer program for deducing a linear integer relaxation of such a program a class of polyhedra that are associated with nonlinear functions is introduced a characterization of the family of polynomials for which our approach leads to an equivalent linear integer program is given finally the family of so called integer convex polynomials is defined and polyhedra related to such a polynomial are investigated

positivstellensatze for real function algebras we look for algebraic certificates of positivity for functions which are not necessarily polynomial functions similar questions were examined earlier by lasserre and putinar positivity and optimization for semi algebraic functions to appear proposition 1 and by putinar a striktpositivestellensatz for measurable functions corrected version to appear theorem 2 1 we explain how these results can be understood as results on hidden positivity the required positivity of the functions implies their positivity when considered as polynomials on the real variety of the respective algebra of functions this variety is however not directly visible in general we show how algebras and quadratic modules with this hidden positivity property can be constructed we can then use known results for example jacobi s representation theorem jacobi in math z 237 259 273 2001 theorem 4 or the krivine stengle positivstellensatz marshall in positive polynomials and sums of squares mathematical surveys and monographs 146 2008 page 25 to obtain certificates of positivity relative to a quadratic module of an algebra of real valued functions our results go beyond the results of lasserre and putinar for example when dealing with non continuous functions the conditions are also easier to check we explain the application of our result to various sorts of real finitely generated algebras of semialgebraic functions the emphasis is on the case where the quadratic module is also finitely generated our results also have application to optimization of real valued functions using the semidefinite programming relaxation methods pioneered by lasserre siam j optim 11 3 796 817 2001 lasserre in moments positive polynomials and their applications imperial college press london 2009 lasserre and putinar in positivity and optimization for semi algebraic functions to appear marshall in positive polynomials and sums of squares mathematical surveys and monographs 146 2008 page 25

closure of the cone of sums of 2d powers in commutative real topological algebras abstract let r be a unitary commutative r algebra and k x r hom r r closed with respect to the product topology we consider r endowed with the topology t k induced by the family of seminorms a a for k and a r in case k is compact we also consider the topology induced by a k sup k a for a r if k is zariski dense then those topologies are hausdorff in this paper we prove that the closure of the cone of sums of 2d powers r 2 d with respect to those two topologies is equal to psd k a r a 0 for all k in particular any continuous linear functional l on the polynomial ring r r x r x 1 x n with l h 2 d 0 for each h r x is integration with respect to a positive borel measure supported on k finally we give necessary and sufficient conditions to ensure the continuity of a linear functional with respect to those two topologies

sums of squares on reducible real curves we ask whether every polynomial function that is non negative on a real algebraic curve can be expressed as a sum of squares in the coordinate ring scheiderer has classified all irreducible curves for which this is the case for reducible curves we show how the answer depends on the configuration of the irreducible components and give complete necessary and sufficient conditions we also prove partial results in the more general case of finitely generated preorderings and discuss applications to the moment problem for semialgebraic sets

the k moment problem with densities given a compact basic semi algebraic set mathbf k subset mathbb r n a rational fraction f mathbb r n to mathbb r and a sequence of scalars y y we investigate when y alpha int mathbf k x alpha f d mu holds for all alpha in mathbb n n i e when y is the moment sequence of some measure fd supported on k this yields a set of convex linear matrix inequalities lmi we also use semidefinite programming to develop a converging approximation scheme to evaluate the integral fd when the moments of are known and f is a given rational fraction numerical expreriments are also provided we finally provide again lmi conditions on the moments of two measures nu mu with support contained in k to have d nu f d mu for some rational fraction f

positivity sums of squares and the multi dimensional moment problem let k be the basic closed semi algebraic set in r n defined by some finite set of polynomials s and t the preordering generated by s for k compact f a polynomial in n variables nonnegative on k and real e 0 we have that f e t in particular the k moment problem has a positive solution in the present paper we study the problem when k is not compact for n 1 we show that the k moment problem has a positive solution if and only if s is the natural description of k see section 1 for n 2 we show that the k moment problem fails if k contains a cone of dimension 2 on the other hand we show that if k is a cylinder with compact base then the following property holds f r x f 0 on k q t such that real 0 f eq t this property is strictly weaker than the one given in schmudgen 1991 but in turn it implies a positive solution to the k moment problem using results of marshall 2001 we provide many noncompact examples in hypersurfaces for which holds finally we provide a list of 8 open problems

archimedean operator theoretic positivstellensatze abstract we prove a general archimedean positivstellensatz for hermitian operator valued polynomials and show that it implies the multivariate fejer riesz theorem of dritschel rovnyak and positivstellensatze of ambrozie vasilescu and scherer hol we also obtain several generalizations of these and related results the proof of the main result depends on an extension of the abstract archimedean positivstellensatz for algebras that is interesting in its own right

limits of positive flat bivariate moment matrices the bivariate moment problem for a sequence 6 of degree 6 remains unsolved but we prove that if the associated 10 10 moment matrix m3 satisfies m3 0 and rank m3 6 then admits a sequence of approximate representing measures and 5 has a representing measure more generally let fd denote the closure of the positive flat moment matrices of degree 2d in n variables each matrix in fd admits computable approximate representing measures and in 2013 jiawang nie and the author began to study concrete conditions for membership in this class let 2d i i zn i 2d 0 0 denote a real n dimensional sequence of degree 2d if the corresponding moment matrix md md is the limit of a sequence of positive flat moment matrices m k d i e m k d 0 and rank m k d rank m k d 1 then i md 0 ii rank md d 1 dim r x1 xn d 1 and iii 2d 1 admits a representing measure we extend our earlier results by proving conversely that for n 2 if md satisfies certain positivity and rank conditions related to i iii then md is the limit of positive flat moment matrices

global optimization with polynomials and the problem of moments we consider the problem of finding the unconstrained global minimum of a real valued polynomial p x mathbb r n to mathbb r as well as the global minimum of p x in a compact set k defined by polynomial inequalities it is shown that this problem reduces to solving an often finite sequence of convex linear matrix inequality lmi problems a notion of karush kuhn tucker polynomials is introduced in a global optimality condition some illustrative examples are provided

extending positive definiteness the main result of this paper gives criteria for extendibility of mappings defined on symmetric subsets of semigroups to positive definite ones by specifying the mappings in question we obtain new solutions of relevant issues in harmonic analysis concerning truncations of some important multivariate moment problems like complex two sided complex and multidimensional trigonometric moment problems in addition unbounded subnormality and existence of unitary power dilation of several contractions is treated as an application of our general scheme

certifying convergence of lasserre s hierarchy via flat truncation consider the optimization problem of minimizing a polynomial function subject to polynomial constraints a typical approach for solving it globally is applying lasserre s hierarchy of semidefinite relaxations based on either putinar s or schmudgen s positivstellensatz a practical question in applications is how to certify its convergence and get minimizers in this paper we propose flat truncation as a certificate for this purpose assume the set of global minimizers is nonempty and finite our main results are 1 putinar type lasserre s hierarchy has finite convergence if and only if flat truncation holds under some generic assumptions the same conclusion holds for the schmudgen type one under weaker assumptions 2 flat truncation is asymptotically satisfied for putinar type lasserre s hierarchy if the archimedean condition holds the same conclusion holds for the schmudgen type one if the feasible set is compact 3 we show that flat truncation can be used as a certificate to check exactness of standard sos relaxations and jacobian sdp relaxations

how fast do polynomials grow on semialgebraic sets abstract we study the growth of polynomials on semialgebraic sets for this purpose we associate a graded algebra with the set and address all kinds of questions about finite generation we show that for a certain class of sets the algebra is finitely generated this implies that the total degree of a polynomial determines its growth on the set at least modulo bounded polynomials we however also provide several counterexamples where there is no connection between total degree and growth in the plane we give a complete answer to our questions for certain simple sets and we provide a systematic construction for examples and counterexamples some of our counterexamples are of particular interest for the study of moment problems since none of the existing methods seems to be able to decide the problem there we finally also provide new three dimensional sets for which the algebra of bounded polynomials is not finitely generated

distinguished representations of non negative polynomials abstract let g 1 g r r x 1 x n such that the set k g 1 0 g r 0 in r n is compact we study the problem of representing polynomials f with f k 0 in the form f s 0 s 1 g 1 s r g r with sums of squares s i with particular emphasis on the case where f has zeros in k assuming that the quadratic module of all such sums is archimedean we establish a local global condition for f to have such a representation vis a vis the zero set of f in k this criterion is most useful when f has only finitely many zeros in k we present a number of concrete situations where this result can be applied as another application we solve an open problem from s kuhlmann et al positivity sums of squares and the multi dimensional moment problem ii adv geometry in press on one dimensional quadratic modules

solution of the truncated moment problem with variety y x 3 we show that positivity consistency and the variety condition which are sufficient to solve the truncated moment problem on planar curves of degree 2 are not sufficient for curves of higher degree using new partly algorithmic conditions based on positive moment matrix extensions we present a concrete solution to the truncated moment problem on the curve y x 3 we also use moment matrix extensions to solve in a less concrete sense truncated moment problems on curves of the form y g x and yg x 1 g x leading to degree bounded weighted sum of squares representations for polynomials which are strictly positive on such curves

pade approximation for a multivariate markov transform methods of pade approximation are used to analyse a multivariate markov transform which has been recently introduced by the authors the first main result is a characterization of the rationality of the markov transform via hankel determinants the second main result is a cubature formula for a special class of measures

on the moment problem of closed semi algebraic sets the moment problem for compact semi algebraic sets has been solved in s1 see pd section 6 4 for a refinement of the result in the terminology explained below this means that each defining sequence f of a compact semialgebraic set kf has property smp on the other hand for many noncompact semi algebraic sets for instance sets containing a cone of dimension two km ps the moment problem is not solvable only very few noncompact semi algebraic sets classes of real algebraic curves so km ps and cylinder sets mc are known to have a positive solution of the moment problem in this paper we study semi algebraic setskf such that there exist polynomials h1 hn which are bounded on the set kf our main result theorem 1 reduces the moment problem for the set kf to the moment problem for the fiber sets kf c where c is real algebraic variety x r h1 x 1 hn x n from this theorem new classes of non compact closed semi algebraic sets are obtained for which the moment problem has an affirmative solution combined with a result of v powers and c scheiderer ps it follows that tube sets around certain real algebraic curves have property smp see theorem 9 let f f1 fk be a finite set of polynomials fj r x r x1 xd and let k kf be the associated closed semi algebraic subset defined by

on the real multidimensional rational k moment problem we present a solution to the real multidimensional rational k moment problem where k is defined by finitely many polynomial inequalities more precisely let s be a finite set of real polynomials in x x1 xn such that the corresponding basic closed semialgebraic set ks is nonempty let e d 1r x be a localization of the real polynomial algebra and t s the preordering on e generated by s we show that every linear functional l on e such that l t s 0 is represented by a positive measure on a certain subset of ks provided d contains an element that grows fast enough on ks

copositive optimization recent developments and applications due to its versatility copositive optimization receives increasing interest in the operational research community and is a rapidly expanding and fertile field of research it is a special case of conic optimization which consists of minimizing a linear function over a cone subject to linear constraints the diversity of copositive formulations in different domains of optimization is impressive since problem classes both in the continuous and discrete world as well as both deterministic and stochastic models are covered copositivity appears in local and global optimality conditions for quadratic optimization but can also yield tighter bounds for np hard combinatorial optimization problems here some of the recent success stories are told along with principles algorithms and applications

a conic programming approach to generalized tchebycheff inequalities consider the problem of finding optimal bounds on the expected value of piecewise polynomials over all measures with a given set of moments this is a special class ofgeneralized tchebycheff inequalities in probability theory we study this problem within the framework of conic programming relying on a general approximation scheme for conic programming we show that these bounds can be numerically computed or approximated via semidefinite programming we also describe some applications of this class of generalized tchebycheff inequalities in probability finance and inventory theory

a representation theorem for certain partially ordered commutative rings let a be a commutative ring with 1 let p subset a be a preordering of higher level i e 0 1 in p 1 not in p p p subset p p cdot p subset p and a 2n subset p for some n in n and let m subset a be an archimedean p module i e 1 in m 1 not in m m m subset m p cdot m subset m and forall a in a exists n in n n a in m we endow x m varphi in rm hom a r mid varphi m subset r with the weak topology with respect to all mappings widehat a x m rightarrow r widehat a varphi varphi a and consider the representation phi m a rightarrow cal c x m r a mapsto widehat a we find that x m is a non empty compact hausdorff space further we prove that

real radical initial ideals abstract we explore the consequences of an ideal i r x 1 x n having a real radical initial ideal both for the geometry of the real variety of i and as an application to sums of squares representations of polynomials we show that if in w i is real radical for a vector w in the tropical variety then w is in the logarithmic set of the real variety we also give algebraic sufficient conditions for w to be in the logarithmic limit set of a more general semialgebraic set if in addition w r 0 n then the corresponding quadratic module is stable in particular if in w i is real radical for some w r 0 n then r x 1 x n 2 i is stable this provides a method for checking the conditions for stability given by powers and scheiderer 2001 ps

semidefinite programming vs lp relaxations for polynomial programming we consider the global minimization of a multivariate polynomial on a semi algebraic set o defined with polynomial inequalities we then compare two hierarchies of relaxations namely lp relaxations based on products of the original constraints in the spirit of the rlt procedure of sherali and adams 1990 and recent semidefinite programming sdp relaxations introduced by the author the comparison is analyzed in light of recent results in real algebraic geometry on various representations of polynomials positive on a compact semi algebraic set

exponentially stable nonlinear systems have polynomial lyapunov functions on bounded regions this paper presents a proof that existence of a polynomial lyapunov function is necessary and sufficient for exponential stability of a sufficiently smooth nonlinear vector field on a bounded set the main result states that if there exists an n times continuously differentiable lyapunov function which proves exponential stability on a bounded subset of rn then there exists a polynomial lyapunov function which proves exponential stability on the same region such a continuous lyapunov function will exist if for example the vector field is at least n times continuously differentiable the proof is based on a generalization of the weierstrass approximation theorem to differentiable functions in several variables specifically polynomials can be used to approximate a differentiable function using the sobolev norm w 1 infin to any desired accuracy this approximation result is combined with the second order taylor series expansion to show that polynomial lyapunov functions can approximate continuous lyapunov functions arbitrarily well on bounded sets the investigation is motivated by the use of polynomial optimization algorithms to construct polynomial lyapunov functions

positivity and optimization for semi algebraic functions we describe algebraic certificates of positivity for functions belonging to a finitely generated algebra of borel measurable functions with particular emphasis on algebras generated by semi algebraic functions in this case the standard global optimization problem with constraints given by elements of the same algebra is reduced via a natural change of variables to the better understood case of polynomial optimization a collection of simple examples and numerical experiments complement the theoretical parts of the article

positive polynomials and tame preorderings from the geometric point of view it is one of the essential merits of theartinschreier theory of ordered fields to have shed new light on certain problems concerning the behaviour of regular functions on an irreducible variety v over r this eventually led to a purely algebraic characterization of those f r v which are positive or nonnegative on v r reg a first example is provided by e artin s positive solution of hilbert s 17 th problem in the seventies classical artin schreier theory was extended to commutative rings and generalizations of artin s result were proven for the purpose of this introduction we consider only a special situation let us fix polynomials f1 fk r x r x1 xn

the complex moment problem and subnormality a polar decomposition approach abstract it has been known that positive definiteness does not guarantee a bisequence to be a complex moment however it turns out that positive definite extendibility does theorems 1 and 22 and this is the main theme of this paper the main tool is generally understood polar decomposition to strengthen applicability of our approach we work out a criterion for positive definite extendibility in a fairly wide context theorems 9 and 29 all this enables us to prove characterizations of subnormality of unbounded operators having invariant domain theorems 37 and 39 and their further applications theorems 41 and 43 and a description of the complex moment problem on real algebraic curves theorems 52 and 56 the latter question is completed in the appendix in which we relate the complex moment problem to the two dimensional real one with emphasis on real algebraic sets

polynomial programming lp relaxations also converge we consider the global minimization of a multivariate polynomial on a compact semialgebraic set using a result of krivine j analyse math 12 1964 pp 307 326 c r acad sci paris 258 1964 pp 3417 3418 it is shown that lp relaxations based on products of the original constraints in the spirit of the reformulation linearization technique procedure of sherali and adams converge to the global optimum this extends to compact semialgebraic sets previous results known for polytopes and 0 1 programs the asymptotic behavior of the lp relaxations is also analyzed and it is shown that in important cases finite convergence cannot occur