symmetric sums of squares over k subset hypercubes we consider the problem of finding sum of squares sos expressions to establish the non negativity of a symmetric polynomial over a discrete hypercube whose coordinates are indexed by the k element subsets of n for simplicity we focus on the case k 2 but our results extend naturally to all values of k ge 2 we develop a variant of the gatermann parrilo symmetry reduction method tailored to our setting that allows for several simplifications and a connection to flag algebras we show that every symmetric polynomial that has a sos expression of a fixed degree also has a succinct sos expression whose size depends only on the degree and not on the number of variables our method bypasses much of the technical difficulties needed to apply the gatermann parrilo method and offers flexibility in obtaining succinct sos expressions that are combinatorially meaningful as a byproduct of our results we arrive at a natural representation theoretic justification for the concept of flags as introduced by razborov in his flag algebra calculus furthermore this connection exposes a family of non negative polynomials that cannot be certified with any fixed set of flags answering a question of razborov in the context of our finite setting

an adaptive linear approximation algorithm for copositive programs we study linear optimization problems over the cone of copositive matrices these problems appear in nonconvex quadratic and binary optimization for instance the maximum clique problem and other combinatorial problems can be reformulated as such problems we present new polyhedral inner and outer approximations of the copositive cone which we show to be exact in the limit in contrast to previous approximation schemes our approximation is not necessarily uniform for the whole cone but can be guided adaptively through the objective function yielding a good approximation in those parts of the cone that are relevant for the optimization and only a coarse approximation in those parts that are not using these approximations we derive an adaptive linear approximation algorithm for copositive programs numerical experiments show that our algorithm gives very good results for certain nonconvex quadratic problems

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

symmetric tensor approximation hierarchies for the completely positive cone in this paper we construct two approximation hierarchies for the completely positive cone based on symmetric tensors we show that one hierarchy corresponds to dual cones of a known polyhedral approximation hierarchy for the copositive cone and the other hierarchy corresponds to dual cones of a known semidefinite approximation hierarchy for the copositive cone as an application we consider a class of bounds on the stability number of a graph obtained from the polyhedral approximation hierarchy and we construct a primal optimal solution with its tensor lifting for each of such linear programs

copositive programming motivated bounds on the stability and the chromatic numbers the lovasz theta number of a graph g can be viewed as a semidefinite programming relaxation of the stability number of g it has recently been shown that a copositive strengthening of this semidefinite program in fact equals the stability number of g we introduce a related strengthening of the lovasz theta number toward the chromatic number of g which is shown to be equal to the fractional chromatic number of g solving copositive programs is np hard this motivates the study of tractable approximations of the copositive cone we investigate the parrilo hierarchy to approximate this cone and provide computational simplifications for the approximation of the chromatic number of vertex transitive graphs we provide some computational results indicating that the lovasz theta number can be strengthened significantly toward the fractional chromatic number of g on some hamming graphs

quantization based filtering method using first order approximation the quantization based filtering method see g pages and h pham bernoulli 11 2005 pp 893 932 g pages h pham and j printems optimal quantization methods and applications to numerical problems in finance in handbook of computational and numerical methods in finance s t rachev ed birkhauser boston 2004 pp 253 297 is a grid based approximation method to solve nonlinear filtering problems with discrete time observations it relies on off line preprocessing of some signal grids in order to construct fast recursive schemes for filter approximation we give here an improvement of this method by taking advantage of the stationary quantizer property the key ingredient is the use of vanishing correction terms to describe schemes based on piecewise linear approximations convergence results are given and numerical results are presented for the particular cases of linear gaussian model and stochastic volatility models

copositivity and constrained fractional quadratic problems we provide completely positive and copositive optimization formulations for the constrained fractional quadratic problem cfqp and standard fractional quadratic problem stfqp based on these formulations semidefinite programming relaxations are derived for finding good lower bounds to these fractional programs which can be used in a global optimization branch and bound approach applications of the cfqp and stfqp related with the correction of infeasible linear systems and eigenvalue complementarity problems are also discussed

non gaussian state space modeling of nonstationary time series abstract a non gaussian state space approach to the modeling of nonstationary time series is shown the model is expressed in state space form where the system noise and the observational noise are not necessarily gaussian recursive formulas of prediction filtering and smoothing for the state estimation and identification of the non gaussian state space model are given also given is a numerical method based on piecewise linear approximation to the density functions for realizing these formulas significant merits of non gaussian modeling and the wide range of applicability of the method are illustrated by some numerical examples a typical application of this non gaussian modeling is the smoothing of a time series that has mean value function with both abrupt and gradual changes simple gaussian state space modeling is not adequate for this situation here the model with small system noise variance cannot detect jump whereas the one with large system noise variance yields unfavorable wiggle to work

reduction of symmetric semidefinite programs using the regular ast representation we consider semidefinite programming problems on which a permutation group is acting we describe a general technique to reduce the size of such problems exploiting the symmetry the technique is based on a low order matrix representation of the commutant centralizer ring of the matrix algebra generated by the permutation matrices we apply it to extending a method of de klerk et al that gives a semidefinite programming lower bound to the crossing number of complete bipartite graphs it implies that cr k 8 n 2 9299n 2 6n cr k 9 n 3 8676n 2 8n and for any m 9 lim n to infty frac rm cr k m n z m n geq 0 8594 frac m m 1 where z m n is the zarankiewicz number lfloor frac 1 4 m 1 2 rfloor lfloor frac 1 4 n 1 2 rfloor which is the conjectured value of cr k m n here the best factor previously known was 0 8303 instead of 0 8594

cones of matrices and set functions and 0 1 optimization it has been recognized recently that to represent a polyhedron as the projection of a higher dimensional but simpler polyhedron is a powerful tool in polyhedral combinatorics a general method is developed to construct higher dimensional polyhedra or in some cases convex sets whose projection approximates the convex hull of 0 1 valued solutions of a system of linear inequalities an important feature of these approximations is that one can optimize any linear objective function over them in polynomial time in the special case of the vertex packing polytope a sequence of systems of inequalities is obtained such that the first system already includes clique odd hole odd antihole wheel and orthogonality constraints in particular for perfect and many other graphs this first system gives the vertex packing polytope for various classes of graphs including t perfect graphs it follows that the stable set polytope is the projection of a polytope with a polynomial number of facets an extension o

approximation of the stability number of a graph via copositive programming lovasz and schrijver siam j optim 1 1991 pp 166 190 showed how to formulate increasingly tight approximations of the stable set polytope of a graph by solving semidefinite programs sdps of increasing size lift and project method in this paper we present a similar idea we show how the stability number can be computed as the solution of a conic linear program lp over the cone of copositive matrices subsequently we show how to approximate the copositive cone ever more closely via a hierarchy of linear or semidefinite programs of increasing size liftings the latter idea is based on recent work by parrilo structured semidefinite programs and semi algebraic geometry methods in robustness and optimization ph d thesis california institute of technology pasadena ca 2000 in this way we can compute the stability number alpha g of any graph g v e after at most alpha g 2 successive liftings for the lp based approximations one can compare this to the n alpha g 1 bound for the lp based lift and project scheme of lovasz and schrijver our approach therefore requires fewer liftings for families of graphs where alpha g o sqrt n we show that the first sdp based approximation for alpha g in our series of increasingly tight approximations coincides with the vartheta function of schrijver ieee trans inform theory 25 1979 pp 425 429 we further show that the second approximation is tight for complements of triangle free graphs and for odd cycles

strengthening the lovasz bound for graph coloring the lovasz number is a way to approximate the independence number of a graph but also its chromatic number we express the lovasz bound as the continuous relaxation of a discrete lovasz number which we derive from karger et al s formulation and which is equal to the chromatic number we also give another relaxation a la schrijver mceliece which is better than the lovasz number

a variational approach to copositive matrices this work surveys essential properties of the so called copositive matrices the study of which has been spread over more than fifty five years special emphasis is given to variational aspects related to the concept of copositivity in addition some new results on the geometry of the cone of copositive matrices are presented here for the first time

an axiomatic duality framework for the theta body and related convex corners lovasz theta function and the related theta body of graphs have been in the center of the intersection of four research areas combinatorial optimization graph theory information theory and semidefinite optimization in this paper utilizing a modern convex optimization viewpoint we provide a set of minimal conditions axioms under which certain key desired properties are generalized including the main equivalent characterizations of the theta function the theta body of graphs and the corresponding antiblocking duality relations our framework describes several semidefinite and polyhedral relaxations of the stable set polytope of a graph as generalized theta bodies as a by product of our approach we introduce the notion of schur lifting of cones which is dual to psd lifting more commonly used in sdp relaxations of combinatorial optimization problems in our axiomatic generalization we also generalize the notion of complements of graphs to diagonally scaling invariant polyhedral cones finally we provide a weighted generalization of the copositive formulation of the fractional chromatic number by dukanovic and rendl from 2010

the operator psi for the chromatic number of a graph we investigate hierarchies of semidefinite approximations for the chromatic number chi g of a graph g we introduce an operator psi mapping any graph parameter beta g nested between the stability number alpha g and chi overline g to a new graph parameter psi beta g nested between alpha overline g and chi g psi beta g is polynomial time computable if beta g is as an application there is no polynomial time computable graph parameter nested between the fractional chromatic number chi cdot and chi cdot unless text p text np moreover based on the motzkin straus formulation for alpha g we give quadratically constrained quadratic and copositive programming formulations for chi g under some mild assumptions n beta g le psi beta g but while n beta g remains below chi g psi beta g can reach chi g e g for beta cdot alpha cdot we also define new polynomial time computable lower bounds for chi g improving the classic lovasz theta number and its strengthenings obtained by adding nonnegativity and triangle inequalities experimental results on hamming graphs kneser graphs and dimacs benchmark graphs will be given in the follow up paper n gvozdenovic and m laurent siam j optim 19 2008 pp 592 615

on the copositive representation of binary and continuous nonconvex quadratic programs in this paper we model any nonconvex quadratic program having a mix of binary and continuous variables as a linear program over the dual of the cone of copositive matrices this result can be viewed as an extension of earlier separate results which have established the copositive representation of a small collection of np hard problems a simplification which reduces the dimension of the linear conic program and an extension to complementarity constraints are established and computational issues are discussed

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

algorithmic enumeration of ideal classes for quaternion orders we provide algorithms to count and enumerate representatives of the right ideal classes of an eichler order in a quaternion algebra defined over a number field we analyze the run time of these algorithms and consider several related problems including the computation of two sided ideal classes isomorphism classes of orders connecting ideals for orders and ideal principalization we conclude by giving the complete list of definite eichler orders with class number at most 2

some np complete problems in quadratic and nonlinear programming in continuous variable smooth nonconvex nonlinear programming we analyze the complexity of checking whether a a given feasible solution is not a local minimum and b the objective function is not bounded below on the set of feasible solutions we construct a special class of indefinite quadratic programs with simple constraints and integer data and show that checking a or b on this class is np complete as a corollary we show that checking whether a given integer square matrix is not copositive is np complete

semidefinite programming relaxations for graph coloring and maximal clique problems the semidefinite programming formulation of the lovasz theta number does not only give one of the best polynomial simultaneous bounds on the chromatic number g or the clique number g of a graph but also leads to heuristics for graph coloring and extracting large cliques this semidefinite programming formulation can be tightened toward either g or g by adding several types of cutting planes we explore several such strengthenings and show that some of them can be computed with the same effort as the theta number we also investigate computational simplifications for graphs with rich automorphism groups

the np completeness of edge coloring we show that it is np complete to determine the chromatic index of an arbitrary graph the problem remains np complete even for cubic graphs

bounds on entanglement dimensions and quantum graph parameters via noncommutative polynomial optimization in this paper we study optimization problems related to bipartite quantum correlations using techniques from tracial noncommutative polynomial optimization first we consider the problem of finding the minimal entanglement dimension of such correlations we construct a hierarchy of semidefinite programming lower bounds and show convergence to a new parameter the minimal average entanglement dimension which measures the amount of entanglement needed to reproduce a quantum correlation when access to shared randomness is free then we study optimization problems over synchronous quantum correlations arising from quantum graph parameters we introduce semidefinite programming hierarchies and unify existing bounds on quantum chromatic and quantum stability numbers by placing them in the framework of tracial polynomial optimization

a tutorial on particle filters for online nonlinear non gaussian bayesian tracking increasingly for many application areas it is becoming important to include elements of nonlinearity and non gaussianity in order to model accurately the underlying dynamics of a physical system moreover it is typically crucial to process data on line as it arrives both from the point of view of storage costs as well as for rapid adaptation to changing signal characteristics in this paper we review both optimal and suboptimal bayesian algorithms for nonlinear non gaussian tracking problems with a focus on particle filters particle filters are sequential monte carlo methods based on point mass or particle representations of probability densities which can be applied to any state space model and which generalize the traditional kalman filtering methods several variants of the particle filter such as sir asir and rpf are introduced within a generic framework of the sequential importance sampling sis algorithm these are discussed and compared with the standard ekf through an illustrative example

exploiting symmetry in copositive programs via semidefinite hierarchies copositive programming is a relative young field which has evolved into a highly active research area in mathematical optimization an important line of research is to use semidefinite programming to approximate conic programming over the copositive cone two major drawbacks of this approach are the rapid growth in size of the resulting semidefinite programs and the lack of information about the quality of the semidefinite programming approximations these drawbacks are an inevitable consequence of the intractability of the generic problems that such approaches attempt to solve to address such drawbacks we develop customized solution approaches for highly symmetric copositive programs which arise naturally in several contexts for instance symmetry properties of combinatorial problems are typically inherited when they are addressed via copositive programming as a result we are able to compute new bounds for crossing number instances in complete bipartite graphs

computing semidefinite programming lower bounds for the fractional chromatic number via block diagonalization recently we investigated in siam j optim 19 2008 pp 572 591 hierarchies of semidefinite approximations for the chromatic number chi g of a graph g in particular we introduced two hierarchies of lower bounds the psi hierarchy converging to the fractional chromatic number and the psi hierarchy converging to the chromatic number of a graph in both hierarchies the first order bounds are related to the lovasz theta number while the second order bounds would already be too costly to compute for large graphs as an alternative relaxations of the second order bounds are proposed in siam j optim 19 2008 pp 572 591 we present here our experimental results with these relaxed bounds for hamming graphs kneser graphs and dimacs benchmark graphs symmetry reduction plays a crucial role as it permits us to compute the bounds by using more compact semidefinite programs in particular for hamming and kneser graphs we use the explicit block diagonalization of the terwilliger algebra given by schrijver ieee trans inform theory 51 2005 pp 2859 2866 our numerical results indicate that the new bounds can be much stronger than the lovasz theta number for some of the dimacs instances we improve the best known lower bounds significantly

approximate graph coloring by semidefinite programming we consider the problem of coloring k colorable graphs with the fewest possible colors we present a randomized polynomial time algorithm that colors a 3 colorable graph on n vertices with min o 1 3 log 1 2 log n o n 1 4 log 1 2 n colors where is the maximum degree of any vertex besides giving the best known approximation ratio in terms of n this marks the first nontrivial approximation result as a function of the maximum degree this result can be generalized to k colorable graphs to obtain a coloring using min o 1 2 k log 1 2 log n o n 1 3 k 1 log 1 2 n colors our results are inspired by the recent work of goemans and williamson who used an algorithm for semidefinite optimization problems which generalize linear programs to obtain improved approximations for the max cut and max 2 sat problems an intriguing outcome of our work is a duality relationship established between the value of the optimum solution to our semidefinite program and the lovasz t function we show lower bounds on the gap between the optimum solution of our semidefinite program and the actual chromatic number by duality this also demonstrates interesting new facts about the t function

intrinsic stationarity for vector quantization foundation of dual quantization we develop a new approach to vector quantization which guarantees an intrinsic stationarity property that also holds in contrast to regular quantization for nonoptimal quantization grids this goal is achieved by replacing the usual nearest neighbor projection operator for voronoi quantization by a random splitting operator which maps the random source to the vertices of a triangle of d simplex in the quadratic euclidean case it is shown that these triangles or d simplices make up a delaunay triangulation of the underlying grid furthermore we prove the existence of an optimal grid for this delaunay or dual quantization procedure we also provide a stochastic optimization method to compute such optimal grids here for higher dimensional uniform and normal distributions a crucial feature of this new approach is the fact that it automatically leads to a second order quadrature formula for computing expectations regardless of the optimality of the underlying grid

observations preprocessing and quantization for nonlinear filters methods of preprocessing the observations for nonlinear filters are investigated the aim being to reduce the computational labor involved in their implementation an example of preprocessing is quantization which involves the replacement of real valued observation samples by discrete values because the resulting likelihood functions have finite support they can be held in look up tables kept in some type of rapid access memory which renders their real time evaluation trivial preprocessed observations of this sort carry less information than raw observations this loss of information is characterized here for filters with substantially noisy observation samples by means of a functional central limit theorem among other things this supplies an asymptotic effective signal to noise ratio for preprocessed filters methods of optimizing preprocessing operations with respect to this quantified information loss are developed in particular optimal quantization thresholds are found for observations that are contaminated by gaussian noise and it is shown that the loss of information for quite coarse quantization schemes is small for example the asymptotic effective signal to noise ratio for a filter with one bit quantized observations is 2 pi times that for the same filter with raw observations simulations on two examples demonstrate the validity of the asymptotic characterization even when the observation samples are only modestly contaminated by noise

chromatic gallai identities operating on lovasz number if g is a triangle free graph then two gallai identities can be written as alpha g overline chi l g v g alpha l g overline chi g where alpha and overline chi denote the stability number and the clique partition number and l g is the line graph of g we show that surprisingly both equalities can be preserved for any graph g by deleting the edges of the line graph corresponding to simplicial pairs of adjacent arcs according to any acyclic orientation of g as a consequence one obtains an operator phi which associates to any graph parameter beta such that alpha g le beta g le overline chi g for all graph g a graph parameter phi beta such that alpha g le phi beta g le overline chi g for all graph g we prove that vartheta g le phi vartheta g and that phi overline chi f g le overline chi f g for all graph g where vartheta is lovasz theta function and overline chi f is the fractional clique partition number moreover overline chi f g le phi vartheta g for triangle free g comparing to the previous strengthenings psi vartheta and vartheta triangle of vartheta numerical experiments show that phi vartheta is a significant better lower bound for overline chi than vartheta

computing the stability number of a graph via linear and semidefinite programming we study certain linear and semidefinite programming lifting approximation schemes for computing the stability number of a graph our work is based on and refines de klerk and pasechnik s approach to approximating the stability number via copositive programming siam j optim 12 2002 pp 875 892 we provide a closed form expression for the values computed by the linear programming approximations we also show that the exact value of the stability number alpha g is attained by the semidefinite approximation of order alpha g 1 as long as alpha g leq 6 our results reveal some sharp differences between the linear and the semidefinite approximations for instance the value of the linear programming approximation of any order is strictly larger than alpha g whenever alpha g 1

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