an extension of sums of squares relaxations to polynomial optimization problems over symmetric cones this paper is based on a recent work by kojima which extended sums of squares relaxations of polynomial optimization problems to polynomial semidefinite programs let varepsilon and varepsilon be a finite dimensional real vector space and a symmetric cone embedded in varepsilon examples of varepsilon and varepsilon include a pair of the n dimensional euclidean space and its nonnegative orthant a pair of the n dimensional euclidean space and n dimensional second order cones and a pair of the space of m m real symmetric or complex hermitian matrices and the cone of their positive semidefinite matrices sums of squares relaxations are further extended to a polynomial optimization problem over varepsilon i e a minimization of a real valued polynomial a x in the n dimensional real variable vector x over a compact feasible region bf x b bf x in varepsilon where b x denotes an varepsilon valued polynomial in x it is shown under a certain moderate assumption on the varepsilon valued polynomial b x that optimal values of a sequence of sums of squares relaxations of the problem which are converted into a sequence of semidefinite programs when they are numerically solved converge to the optimal value of the problem

exploiting sparsity in sdp relaxation for sensor network localization a sensor network localization problem can be formulated as a quadratic optimization problem qop for qops semidefinite programming sdp relaxation by lasserre with relaxation order 1 for general polynomial optimization problems pops is known to be equivalent to the sparse sdp relaxation by waki et al with relaxation order 1 except for the size and sparsity of the resulting sdp relaxation problems we show that the sparse sdp relaxation applied to the qop is at least as strong as the biswas ye sdp relaxation for the sensor network localization problem a sparse variant of the biswas ye sdp relaxation which is equivalent to the original biswas ye sdp relaxation is also derived we compare numerically the sparse sdp relaxation applied to the qop the biswas ye sdp relaxation its sparse variant and the edge based sdp relaxation by wang et al to confirm the effectiveness of the proposed techniques for exploiting the sparsity in sdp relaxation for sensor network localization problems the sparse sdp relaxation applied to the qop is much faster than the biswas ye sdp relaxation and the sparse variant of the biswas ye sdp relaxation outperforms all other sdp relaxations in speed

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

robust global optimization with polynomials we consider the optimization problems maxz minx k p z x and minx k maxz p z x where the criterion p is a polynomial linear in the variables z the set can be described by lmis and k is a basic closed semi algebraic set the first problem is a robust analogue of the generic sdp problem maxz p z whereas the second problem is a robust analogue of the generic problem minx k p x of minimizing a polynomial over a semi algebraic set we show that the optimal values of both robust optimization problems can be approximated as closely as desired by solving a hierarchy of sdp relaxations we also relate and compare the sdp relaxations associated with the max min and the min max robust optimization problems

robust performance analysis of uncertain lti systems dual lmi approach and verifications for exactness this paper addresses robust performance analysis problems of linear time invariant lti systems affected by real parametric uncertainties these problems known also as a special class of structured singular value computation problems are inherently intractable np hard problems as such intensive research effort has been made to obtain computationally tractable and less conservative analysis conditions where linear matrix inequality lmi plays an important nevertheless since lmi based conditions are expected to be conservative in general it is often the case that we cannot conclude anything if the lmi at hand turns out to be infeasible this motivates us to consider the dual of the lmi and examine the structure of the dual solution by pursuing this direction in this paper we provide rank conditions on the dual solution matrix under which we can conclude that the underlying robust performance is never attained in particular a set of uncertain parameters that violates the specified performance can be computed these results come from block moment matrix structure of the dual variable which is consistent with the recent results on polynomial optimization this particular structure enables us to make good use of simultaneous diagonalizability property of commuting diagonalizable matrices so that the sound rank conditions for the exactness verification can be obtained

a sharper bonferroni procedure for multiple tests of significance summary a simple procedure for multiple tests of significance based on individual p values is derived this simple procedure is sharper than holm s 1979 sequentially rejective procedure both procedures contrast the ordered p values with the same set of critical values holm s procedure rejects an hypothesis only if its p value and each of the smaller p values are less than their corresponding critical values the new procedure rejects all hypotheses with smaller or equal p values to that of any one found less than its critical value

finding low rank solutions of sparse linear matrix inequalities using convex optimization this paper is concerned with the problem of finding a low rank solution of an arbitrary sparse linear matrix inequality lmi to this end we map the sparsity of the lmi problem into a graph we develop a mathematical framework to relate the rank of the minimum rank solution of the lmi problem to the sparsity of its underlying graph furthermore we propose three graph theoretic convex programs to obtain a low rank solution two of these convex optimization problems are based on a tree decomposition of the sparsity graph the third one does not rely on any computationally expensive graph analysis and is always polynomial time solvable at the cost of offering a milder theoretical guarantee on the rank of the obtained solution compared to the other two methods the results of this work can be readily applied to three separate problems of minimum rank matrix completion conic relaxation for polynomial optimization and affine rank minimization the results are finally illustrated on two applications of opt

enclosing ellipsoids and elliptic cylinders of semialgebraic sets and their application to error bounds in polynomial optimization this paper is concerned with a class of ellipsoidal sets ellipsoids and elliptic cylinders in mathbb r m which are determined by a freely chosen m m positive semidefinite matrix all ellipsoidal sets in this class are similar to each other through a parallel transformation and a scaling around their centers by a constant factor based on the basic idea of lifting we first present a conceptual min max problem to determine an ellipsoidal set with the smallest size in this class which encloses a given subset of mathbb r m then we derive a numerically tractable enclosing ellipsoidal set of a given semialgebraic subset of mathbb r m as a convex relaxation of the min max problem in the lifting space a main feature of the proposed method is that it is designed to incorporate into existing sdp relaxations with exploiting sparsity for various optimization problems to compute error bounds of their optimal solutions we discuss how we adapt the method to a standard sdp relaxation for quadratic optimization problems and a sparse variant of lasserre s hierarchy sdp relaxation for polynomial optimization problems some numerical results on the sensor network localization problem and polynomial optimization problems are also presented

lower bound for the number of iterations in semidefinite hierarchies for the cut polytope hierarchies of semidefinite relaxations for 0 1 polytopes have been constructed by lasserre 2001a and by lovaisz and schrijver 1991 the cut polytope of a graph onn nodes can be expressed as a projection of such a semidefinite relaxation after at mostn steps we show that n 2 iterations are needed for finding the cut polytope of the complete graphk n

positive definite completions of partial hermitian matrices abstract the question of which partial hermitian matrices some entries specified some free may be completed to positive definite matrices is addressed it is shown that if the diagonal entries are specified and principal minors composed of specified entries are positive then if the undirected graph of the specified entries is chordal a positive definite completion necessarily exists furthermore if this graph is not chordal then examples exist without positive definite completions in case a positive definite completion exists there is a unique matrix in the class of all positive definite completions whose determinant is maximal and this matrix is the unique one whose inverse has zeros in those positions corresponding to unspecified entries in the original partial hermitian matrix additional observations regarding positive definite completions are made

convergent sdp relaxations in polynomial optimization with sparsity we consider a polynomial programming problem p on a compact basic semialgebraic set k subset r n described by m polynomial inequalities g j x geq0 and with criterion f in r x we propose a hierarchy of semidefinite relaxations in the spirit of those of waki e siam j optim 17 2006 pp 218 242 in particular the sdp relaxation of order r has the following two features a the number of variables is o kappa 2r where kappa max kappa 1 kappa 2 with kappa 1 resp kappa 2 being the maximum number of variables appearing in the monomials of f resp appearing in a single constraint g j x geq0 b the largest size of the linear matrix inequalities lmis is o kappa r this is to compare with the respective number of variables o n 2r and lmi size o n r in the original sdp relaxations defined in j b lasserre siam j optim 11 2001 pp 796 817 therefore great computational savings are expected in case of sparsity in the data g j f i e when k is small a frequent case in practical applications of interest the novelty with respect to h waki s kim m kojima and m maramatsu siam j optim 17 2006 pp 218 242 is that we prove convergence to the global optimum of p when the sparsity pattern satisfies a condition often encountered in large size problems of practical applications and known as the running intersection property in graph theory in such cases and as a by product we also obtain a new representation result for polynomials positive on a compact basic semialgebraic set a sparse version of putinar s positivstellensatz m putinar indiana univ math j 42 1993 pp 969 984

log determinant relaxation for approximate inference in discrete markov random fields graphical models are well suited to capture the complex and non gaussian statistical dependencies that arise in many real world signals a fundamental problem common to any signal processing application of a graphical model is that of computing approximate marginal probabilities over subsets of nodes this paper proposes a novel method applicable to discrete valued markov random fields mrfs on arbitrary graphs for approximately solving this marginalization problem the foundation of our method is a reformulation of the marginalization problem as the solution of a low dimensional convex optimization problem over the marginal polytope exactly solving this problem for general graphs is intractable for binary markov random fields we describe how to relax it by using a gaussian bound on the discrete entropy and a semidefinite outer bound on the marginal polytope this combination leads to a log determinant maximization problem that can be solved efficiently by interior point methods thereby providing approximations to the exact marginals we show how a slightly weakened log determinant relaxation can be solved even more efficiently by a dual reformulation when applied to denoising problems in a coupled mixture of gaussian model defined on a binary mrf with cycles we find that the performance of this log determinant relaxation is comparable or superior to the widely used sum product algorithm over a range of experimental conditions

decomposition in conic optimization with partially separable structure decomposition techniques for linear programming are difficult to extend to conic op timization problems with general nonpolyhedral convex cones because the conic inequalities introduce an additional nonlinear coupling between the variables however in many applications the convex cones have a partially separable structure that allows them to be characterized in terms of simpler lower dimensional cones the most important example is sparse semidefinite programming with a chordal sparsity pattern here partial separability derives from the clique decomposition theorems that characterize positive semidefinite and positive semidefinite completable matrices with chordal sparsity patterns the paper describes a decomposition method that exploits partial separability in conic linear optimization the method is based on spingarn s method for equality constrained convex optimization combined with a fast interior point method for evaluating proximal operators

convergent relaxations of polynomial matrix inequalities and static output feedback using a moment interpretation of recent results on sum of squares decompositions of nonnegative polynomial matrices we propose a hierarchy of convex linear matrix inequality lmi relaxations to solve nonconvex polynomial matrix inequality pmi optimization problems including bilinear matrix inequality bmi problems this hierarchy of lmi relaxations generates a monotone sequence of lower bounds that converges to the global optimum results from the theory of moments are used to detect whether the global optimum is reached at a given lmi relaxation and if so to extract global minimizers that satisfy the pmi the approach is successfully applied to pmis arising from static output feedback design problems

towards a coherent philosophy for modelling the environment the predominant philosophy underlying most environmental modelling is a form of pragmatic realism the limitations of this approach in practical applications are discussed in particular in relati

conservation equations governing hillslope responses exploring the physical basis of water balance a unifying approach for the derivation of watershed scale conservation equations governing hydrologic responses has recently been introduced the approach is based on space time averaging of the corresponding point scale conservation equations over an averaging region called the representative elementary watershed rew the conservation equations are supplemented by constitutive relationships needed for the closure of various mass and momentum exchange terms in this paper we present a summary of the resulting equations for a somewhat simpler problem and show how these equations can be employed to model the long term water balance of a single hypothetical rew the governing equations are formulated in terms of an average saturation and a flow velocity of the unsaturated zone soils and the average thickness of the saturated zone soils the equations are coupled ordinary differential equations and are nonlinear their solution is carried out simultaneously for a variety of combinations of watershed geometry soil type and atmospheric forcing we show how a similarity analysis of the governing equations can lead to a general classification of rews in terms of meaningful dimensionless similarity variables in addition we investigate the long term water balance and show that the governing equations are able to provide a realistic picture of the impact of changing climate soil and topographic controls on long term water balance

effects of treatment induced mortality and tumor induced mortality on tests for carcinogenicity in small samples statistical tests of carcinogenicity are shown to have varying degrees of robustness to the effects of mortality mortality induced by two different mechanisms is studied mortality due to the tumor of interest and mortality due to treatment independent of the tumor the two most commonly used tests the life table test and the cochran armitage linear trend test are seen to be highly sensitive to increases in treatment lethality using small sample simulations increases in tumor lethality are seen to affect the performance of commonly used prevalence tests such as logistic regression a simple survival adjusted quantal response test appears to be the most robust of all the procedures considered

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

semidefinite programming for min max problems and games we consider two min max problems 1 minimizing the supremum of finitely many rational functions over a compact basic semi algebraic set and 2 solving a 2 player zero sum polynomial game in randomized strategies with compact basic semi algebraic sets of pure strategies in both problems the optimal value can be approximated by solving a hierarchy of semidefinite relaxations in the spirit of the moment approach developed in lasserre siam j optim 11 796 817 2001 math program b 112 65 92 2008 this provides a unified approach and a class of algorithms to compute nash equilibria and min max strategies of several static and dynamic games each semidefinite relaxation can be solved in time which is polynomial in its input size and practice on a sample of experiments reveals that few relaxations are needed for a good approximation and sometimes even for finite convergence a behavior similar to what was observed in polynomial optimization

a multilevel analysis of the lasserre hierarchy abstract this paper analyzes the relation between different orders of the lasserre hierarchy for polynomial optimization pop although for some cases solving the semidefinite programming relaxation corresponding to the first order of the hierarchy is enough to solve the underlying pop other problems require sequentially solving the second or higher orders until a solution is found for these cases and assuming that the lower order semidefinite programming relaxation has been solved we develop prolongation operators that exploit the solutions already calculated to find initial approximations for the solution of the higher order relaxation we can prove feasibility in the higher order of the hierarchy of the points obtained using the operators as well as convergence to the optimal as the relaxation order increases furthermore the operators are simple and inexpensive for problems where the projection over the feasible set is easy to calculate for example integer 0 1 and 1 1 pops our numerical experiments show that it is possible to extract useful information for real applications using the prolongation operators in particular we illustrate how the operators can be used to increase the efficiency of an infeasible interior point method by using them as an initial point we use this technique to solve quadratic integer 0 1 problems as well as max cut and integer partition problems

coherent quantum lqg control based on a recently developed notion of physical realizability for quantum linear stochastic systems we formulate a quantum lqg optimal control problem for quantum linear stochastic systems where the controller itself may also be a quantum system and the plant output signal can be fully quantum such a control scheme is often referred to in the quantum control literature as coherent feedback control it distinguishes the present work from previous works on the quantum lqg problem where measurement is performed on the plant and the measurement signals are used as the input to a fully classical controller with no quantum degrees of freedom the difference in our formulation is the presence of additional non linear and linear constraints on the coefficients of the sought after controller rendering the problem as a type of constrained controller design problem due to the presence of these constraints our problem is inherently computationally hard and this also distinguishes it in an important way from the standard lqg problem we propose a numerical procedure for solving this problem based on an alternating projections algorithm and as an initial demonstration of the feasibility of this approach we provide fully quantum controller design examples in which numerical solutions to the problem were successfully obtained for comparison we also consider the case of classical linear controllers that use direct or indirect measurements and show that there exists a fully quantum linear controller which offers an improvement in performance over the classical ones

a region dividing technique for constructing the sum of squares approximations to robust semidefinite programs in this technical note we present a novel approach to robust semidefinite programs of which coefficient matrices depend polynomially on uncertain parameters the approach is based on approximation with the sum of squares polynomials but in contrast to the conventional sum of squares approach the quality of approximation is improved by dividing the parameter region into several subregions the optimal value of the approximate problem converges to that of the original problem as the resolution of the division becomes finer an advantage of this approach is that an upper bound on the approximation error can be explicitly obtained in terms of the resolution of the division a numerical example on polynomial optimization is presented to show usefulness of the present approach

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

optimality conditions and finite convergence of lasserre s hierarchy lasserre s hierarchy is a sequence of semidefinite relaxations for solving polynomial optimization problems globally this paper studies the relationship between optimality conditions in nonlinear programming theory and finite convergence of lasserre s hierarchy our main results are i lasserre s hierarchy has finite convergence when the constraint qualification strict complementarity and second order sufficiency conditions hold at every global minimizer under the standard archimedean condition the proof uses a result of marshall on boundary hessian conditions ii these optimality conditions are all satisfied at every local minimizer if a finite set of polynomials which are in the coefficients of input polynomials do not vanish at the input data i e they hold in a zariski open set this implies that under archimedeanness lasserre s hierarchy has finite convergence generically

exploiting sparsity in linear and nonlinear matrix inequalities via positive semidefinite matrix completion a basic framework for exploiting sparsity via positive semidefinite matrix completion is presented for an optimization problem with linear and nonlinear matrix inequalities the sparsity characterized with a chordal graph structure can be detected in the variable matrix or in a linear or nonlinear matrix inequality constraint of the problem we classify the sparsity in two types the domain space sparsity d space sparsity for the symmetric matrix variable in the objective and or constraint functions of the problem which is required to be positive semidefinite and the range space sparsity r space sparsity for a linear or nonlinear matrix inequality constraint of the problem four conversion methods are proposed in this framework two for exploiting the d space sparsity and the other two for exploiting the r space sparsity when applied to a polynomial semidefinite program sdp these conversion methods enhance the structured sparsity of the problem called the correlative sparsity as a result the resulting polynomial sdp can be solved more effectively by applying the sparse sdp relaxation preliminary numerical results on the conversion methods indicate their potential for improving the efficiency of solving various problems

coupled equations for mass and momentum balance in a stream network theoretical derivation and computational experiments in previous work by the authors a rigorous procedure for the derivation of global watershedscale balance laws for mass momentum energy and entropy has been pursued to complement these a set of

necessary global optimality conditions for nonlinear programming problems with polynomial constraints in this paper we develop necessary conditions for global optimality that apply to non linear programming problems with polynomial constraints which cover a broad range of optimization problems that arise in applications of continuous as well as discrete optimization in particular we show that our optimality conditions readily apply to problems where the objective function is the difference of polynomial and convex functions over polynomial constraints and to classes of fractional programming problems our necessary conditions become also sufficient for global optimality for polynomial programming problems our approach makes use of polynomial over estimators and a polynomial version of a theorem of the alternative which is a variant of the positivstellensatz in semi algebraic geometry we discuss numerical examples to illustrate the significance of our optimality conditions

exploiting sparsity in semidefinite programming via matrix completion i general framework a critical disadvantage of primal dual interior point methods compared to dual interior point methods for large scale semidefinite programs sdps has been that the primal positive semidefinite matrix variable becomes fully dense in general even when all data matrices are sparse based on some fundamental results about positive semidefinite matrix completion this article proposes a general method of exploiting the aggregate sparsity pattern over all data matrices to overcome this disadvantage our method is used in two ways one is a conversion of a sparse sdp having a large scale positive semidefinite matrix variable into an sdp having multiple but smaller positive semidefinite matrix variables to which we can effectively apply any interior point method for sdps employing a standard block diagonal matrix data structure the other way is an incorporation of our method into primal dual interior point methods which we can apply directly to a given sdp in part ii of this article we will investigate an implementation of such a primal dual interior point method based on positive definite matrix completion and report some numerical results

tight relaxations for polynomial optimization and lagrange multiplier expressions this paper proposes tight semidefinite relaxations for polynomial optimization the optimality conditions are investigated we show that generally lagrange multipliers can be expressed as polynomial functions in decision variables over the set of critical points the polynomial expressions are determined by linear equations based on these expressions new lasserre type semidefinite relaxations are constructed for solving the polynomial optimization we show that the hierarchy of new relaxations has finite convergence or equivalently the new relaxations are tight for a finite relaxation order

a comparison of tumour incidence analyses applicable in single sacrifice animal experiments monte carlo methods are used to evaluate the operating characteristics of several trend tests for comparing incidence rates of occult tumours the candidate tests are applicable in long term animal experiments that have just one sacrifice time and no information on cause of death when survival decreases with dose and the tumour is lethal the tests that do not adjust for survival or that assume tumours are non lethal behave conservatively whereas the test that assumes tumours are rapidly lethal rejects the hypothesis of equal incidence rates too frequently the only test which consistently operates at the correct level is one that specifies a constant difference between the death rates for animals with and without the tumour surprisingly the test that specifies a constant ratio of these death rates often is conservative finally a test based on a simple modification of the unadjusted test which accounts for differential mortality by scaling down the size of the risk set performs reasonably well in many cases among the tests that operate at the proper level the constant risk difference test consistently exhibits high power across a wide range of situations

regularized nonlinear acceleration we describe a convergence acceleration technique for generic optimization problems our scheme computes estimates of the optimum from a nonlinear average of the iterates produced by any optimization method the weights in this average are computed via a simple linear system whose solution can be updated online this acceleration scheme runs in parallel to the base algorithm providing improved estimates of the solution on the fly while the original optimization method is running numerical experiments are detailed on classical classification problems

maximum block improvement and polynomial optimization in this paper we propose an efficient method for solving the spherically constrained homogeneous polynomial optimization problem the new approach has the following three main ingredients first we establish a block coordinate descent type search method for nonlinear optimization with the novelty being that we accept only a block update that achieves the maximum improvement hence the name of our new search method maximum block improvement mbi convergence of the sequence produced by the mbi method to a stationary point is proved second we establish that maximizing a homogeneous polynomial over a sphere is equivalent to its tensor relaxation problem thus we can maximize a homogeneous polynomial function over a sphere by its tensor relaxation via the mbi approach third we propose a scheme to reach a kkt point of the polynomial optimization provided that a stationary solution for the relaxed tensor problem is available numerical experiments have shown that our new method works very efficiently for a majority of the test instances that we have experimented with the method finds the global optimal solution at a low computational cost

transposition theorems and qualification free optimality conditions new theorems of the alternative for polynomial constraints based on the positivstellensatz from real algebraic geometry and for linear constraints generalizing the transposition theorems of motzkin and tucker are proved based on these two karush john optimality conditions holding without any constraint qualification are proved for single or multiobjective constrained optimization problems the first condition applies to polynomial optimization problems only and gives for the first time necessary and sufficient global optimality conditions for polynomial problems the second condition applies to smooth local optimization problems and strengthens known local conditions if some linear or concave constraints are present the new version reduces the number of constraints for which a constraint qualification is needed to get the kuhn tucker conditions

entanglement criteria for dicke states dicke states are a family of multi qubit quantum states with interesting entanglement properties and they have been observed in many experiments we construct entanglement witnesses for detecting genuine multiparticle entanglement in the vicinity of these states we use the approach of positive partial transpose mixtures to derive the conditions analytically for nearly all cases our criteria are stronger than all conditions previously known

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

brief positive polynomial matrices and improved lmi robustness conditions recently several new lmi conditions for stability of linear systems have been proposed introducing additional slack variables to reduce the gap between conservative convex quadratic stability conditions and intractable non convex robust stability conditions in this paper we show that these improved lmi conditions can be derived with the help of some basic results on positive polynomial matrices the approach allows us to derive in a unifying way results in the state space and polynomial frameworks applications to robust stability analysis and robust stabilization of systems with multi linear parametric uncertainty are fully described

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

further results on cauchy tensors and hankel tensors in this article we present various new results on cauchy tensors and hankel tensors we first introduce the concept of generalized cauchy tensors which extends cauchy tensors in the current literature and provide several conditions characterizing positive semi definiteness of generalized cauchy tensors with nonzero entries furthermore we prove that all even order generalized cauchy tensors with positive entries are completely positive tensors which means every such that generalized cauchy tensor can be decomposed as the sum of nonnegative rank 1 tensors we also establish that all the h eigenvalues of nonnegative cauchy tensors are nonnegative secondly we present new mathematical properties of hankel tensors we prove that an even order hankel tensor is vandermonde positive semi definite if and only if its associated plane tensor is positive semi definite we also show that if the vandermonde rank of a hankel tensor a is less than the dimension of the underlying space then positive semi definiteness of a is equivalent to the fact that a is a complete hankel tensor and so is further equivalent to the sos property of a thirdly we introduce a new class of structured tensors called cauchy hankel tensors which is a special case of cauchy tensors and hankel tensors simultaneously sufficient and necessary conditions are established for an even order cauchy hankel tensor to be positive definite

semidefinite approximations for global unconstrained polynomial optimization we consider the problem of minimizing a polynomial function on mathbb r n known to be hard even for degree 4 polynomials therefore approximation algorithms are of interest lasserre siam j optim 11 2001 pp 796 817 and parrilo math program 96 2003 pp 293 320 have proposed approximating the minimum of the original problem using a hierarchy of lower bounds obtained via semidefinite programming relaxations we propose here a method for computing tight upper bounds based on perturbing the original polynomial and using semidefinite programming the method is applied to several examples

positive semidefinite matrices with a given sparsity pattern abstract this paper concerns the cone of positive semidefinite matrices which have zeros in prescribed entries one of the main purposes is to obtain information and if possible describe the ranks of external matrices in such cones in terms of the pattern of prescribed zeros in particular we study those patterns for which the ranks of extremal matrices are less than a given number

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

establishing convexity of polynomial lyapunov functions and their sublevel sets there has been renewed interest in the use of polynomial and homogeneous polynomial lyapunov functions for addressing stability and performance issues in systems and control theory however no condition is yet available to establish the convexity of these functions and the convexity of their sublevel sets properties that can be useful in many applications in this paper new conditions for establishing these convexity properties are proposed which can be handled through convex linear matrix inequalities optimizations by means of sum of squares relaxations

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

solving moment problems by dimensional extension polynomial inequalities the second part of the paper uses the representation results of positive functionals on certain spaces of rational functions developed in the flrst part for decomposing a polynomial which is positive on such a semi algebraic set into a canonical sum of squares of rational functions times explicit multipliers let n 1 be a flxed integer due to the fact that for n 1 not every nonnegative polynomial in r n can be written as a sum of squares of polynomials see for instance 2 x6 3 the moment problems in n variables are more di cult than the classical one variable problems this very intriguing territory has been investigated by many authors see 2 7 12 and their references although characterizations for measures whose support lies in an arbitrary generally unbounded semi algebraic set do not seem to exist the present paper starts from an idea of the second author see 19 about solving moment problems by a change of basis via an embedding of r n into a submanifold of a higher dimensional euclidean space rougly speaking we prove that certain n 1 dimensional extensions of a moment sequence are naturally characterized by positivity conditions and moreover these extensions parametrize all possible solutions of the moment problem to be more speciflc let fi z r n x fi d x fi 2 z n

an age adjusted bootstrap based poly k test the assumption of an asymptotic normal distribution of some test statistics may be invalid in certain dose response trend tests for instance the survival adjusted cochran armitage test known as the poly k test is asymptotically standard normal under the null hypothesis however the asymptotic normality is not valid if there is a deviation from the tumour onset distribution that is assumed in this test or if the competing risks survival rates differ across groups we develop an age adjusted bootstrap based method to assess the significance of assumed asymptotic normal tests for animal carcinogenicity data the proposed method differs from conventional bootstrap methods in the aspect of preserving the mortality rate in each dose group under the null hypothesis of equal tumour incidence rates among the groups we investigate an empirical distribution of the poly 3 p3 trend test statistic using the proposed age adjusted bootstrap based method and compare it with the p3 test statistic referenced to the assumed standard normal distribution a simulation study is conducted to evaluate the robustness of these tests to various weibull family tumour onset distributions the proposed method is applied to national toxicology program data sets to evaluate a dose related trend of a test substance on the incidence of neoplasms

weighted p value adjustments for animal carcinogenicity trend test summary a typical animal carcinogenicity experiment routinely analyzes approximately 10 30 tumor sites comparisons of tumor responses between dosed and control groups and dose related trend tests are often evaluated for each individual tumor site type separately p value adjustment approaches have been proposed for controlling the overall type i error rate or familywise error rate fwe however these adjustments often result in reducing the power to detect a dose effect this paper proposes using weighted adjustments by assuming that each tumor can be classified as either class a or class b based on prior considerations the tumors in class a which are considered as more critical endpoints are given less adjustment two weighted methods of adjustments are presented the weighted p adjustment and weighted oa adjustment a monte carlo simulation shows that both weighted adjustments control the fwe well furthermore the power increases if a treatment dependent tumor is analyzed as in class a tumors and the power decreases if it is analyzed as in class b tumors a data set from a national toxicology program ntp 2 year animal carcinogenicity experiment with 13 tumor types sites observed in male mice was analyzed using the proposed methods the modified poly 3 test was used to test for increased carcinogenicity since it has been adopted by the ntp as a standard test for a dose related trend the unweighted adjustment analysis concluded that there was no statistically significant dose related trend using the food and drug administration classification scheme for the weighted adjustment analyses two rare tumors with background rates of 1 or less were analyzed as class a tumors and 11 common tumors with background rates higher than 1 as class b both weighted analyses showed a significant dose related trend for one rare tumor

a copositive formulation for the stability number of infinite graphs in the last decade copositive formulations have been proposed for a variety of combinatorial optimization problems for example the stability number independence number in this paper we generalize this approach to infinite graphs and show that the stability number of an infinite graph is the optimal solution of some infinite dimensional copositive program for this we develop a duality theory between the primal convex cone of copositive kernels and the dual convex cone of completely positive measures we determine the extreme rays of the latter cone and we illustrate this theory with the help of the kissing number problem

sparsity in sums of squares of polynomials representation of a given nonnegative multivariate polynomial in terms of a sum of squares of polynomials has become an essential subject in recent developments of sums of squares optimization and semidefinite programming sdp relaxation of polynomial optimization problems we discuss effective methods to obtain a simpler representation of a sparse polynomial as a sum of squares of sparse polynomials by eliminating redundancy

distributed semidefinite programming with application to large scale system analysis distributed algorithms for solving coupled semidefinite programs commonly require many iterations to converge they also put high computational demand on the computational agents in this paper we show that in case the coupled problem has an inherent tree structure it is possible to devise an efficient distributed algorithm for solving such problems the proposed algorithm relies on predictor corrector primal dual interior point methods where we use a message passing algorithm to compute the search directions distributedly message passing here is closely related to dynamic programming over trees this allows us to compute the exact search directions in a finite number of steps this is because computing the search directions requires a recursion over the tree structure and hence terminates after an upward and downward pass through the tree furthermore this number can be computed a priori and only depends on the coupling structure of the problem we use the proposed algorithm for analyzing robustness of large scale uncertain systems distributedly we test the performance of this algorithm using numerical examples

error bounds for convex polynomials the purpose of this paper is to investigate error bounds for convex polynomials we prove that for a convex polynomial f in n variables which is not everywhere positive and which is not constant on any affine subspace either f is a sum of a convex polynomial in fewer variables and a linear form with negative coefficients or the negativity set of f is compact as an application we deduce various types of error bounds for unconstrained and polyhedral constrained convex polynomials

universal rigidity and edge sparsification for sensor network localization owing to their high accuracy and ease of formulation there has been great interest in applying convex optimization techniques particularly that of semidefinite programming sdp relaxation to tackle the sensor network localization problem in recent years however a drawback of such techniques is that the resulting convex program is often expensive to solve in order to speed up computation various edge sparsification heuristics have been proposed whose aim is to reduce the number of edges in the input graph although these heuristics do reduce the size of the convex program and hence make it faster to solve they are often ad hoc in nature and do not preserve the localization properties of the input as such one often has to face a tradeoff between solution accuracy and computational effort in this paper we propose a novel edge sparsification heuristic that can provably preserve the localization properties of the original input at the heart of our heuristic is a graph decomposition procedure which allows us to identify certain sparse generically universally rigid subgraphs of the input graph our computational results show that the proposed approach can significantly reduce the computational and memory complexities of sdp based algorithms for solving the sensor network localization problem moreover it compares favorably with existing speedup approaches both in terms of accuracy and solution time

chordal decomposition in operator splitting methods for sparse semidefinite programs we employ chordal decomposition to reformulate a large and sparse semidefinite program sdp either in primal or dual standard form into an equivalent sdp with smaller positive semidefinite psd constraints in contrast to previous approaches the decomposed sdp is suitable for the application of first order operator splitting methods enabling the development of efficient and scalable algorithms in particular we apply the alternating direction method of multipliers admm to solve decomposed primal and dual standard form sdps each iteration of such admm algorithms requires a projection onto an affine subspace and a set of projections onto small psd cones that can be computed in parallel we also formulate the homogeneous self dual embedding hsde of a primal dual pair of decomposed sdps and extend a recent admm based algorithm to exploit the structure of our hsde the resulting hsde algorithm has the same leading order computational cost as those for the primal or dual problems only with the advantage of being able to identify infeasible problems and produce an infeasibility certificate all algorithms are implemented in the open source matlab solver cdcs numerical experiments on a range of large scale sdps demonstrate the computational advantages of the proposed methods compared to common state of the art solvers

linearization functors on real convex sets we prove that linearizing multilinear optimization problems leads to new functorial operations on real convex sets these operations are convex analogues of hom functors tensor products symmetric powers exterior powers and general schur functors on vector spaces and lead to novel constructions even for polyhedra we discuss their general theory and introduce mechanisms to compute them or approximate them in ways amenable to efficient computation

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 direct proof for the matrix decomposition of chordal structured positive semidefinite matrices agler helton mccullough and rodman proved that a graph is chordal if and only if any positive semidefinite psd symmetric matrix whose nonzero entries are specified by a given graph can be decomposed as a sum of psd matrices corresponding to the maximal cliques this decomposition is recently exploited to solve positive semidefinite programming efficiently their proof is based on a characterization for psd matrix completion of a chordal structured matrix due to grone johnson sa and wolkowicz this note gives a direct and simpler proof for the result of agler et al which leads to an alternative proof of grone et al

convergence analysis for lasserre s measure based hierarchy of upper bounds for polynomial optimization we consider the problem of minimizing a continuous function f over a compact set mathbf k k we analyze a hierarchy of upper bounds proposed by lasserre siam j optim 21 3 864 885 2011 obtained by searching for an optimal probability density function h on mathbf k k which is a sum of squares of polynomials so that the expectation int mathbf k f x h x dx kf x h x dx is minimized we show that the rate of convergence is no worse than o 1 sqrt r o 1 r where 2r is the degree bound on the density function this analysis applies to the case when f is lipschitz continuous and mathbf k k is a full dimensional compact set satisfying some boundary condition which is satisfied e g for convex bodies the rth upper bound in the hierarchy may be computed using semidefinite programming if f is a polynomial of degree d and if all moments of order up to 2r d 2r d of the lebesgue measure on mathbf k k are known which holds for example if mathbf k k is a simplex hypercube or a euclidean ball

ratio estimates the delta method and quantal response tests for increased carcinogenicity this paper demonstrates the use of the delta method for estimating the variance of ratio statistics derived from animal carcinogenicity experiments the cochran armitage test cochran 1954 biometrika 10 417 451 and armitage 1955 biometrics 11 375 386 is routinely applied to carcinogenicity data as a test for linear trend in lifetime tumor incidence rates the computing formula for this test derives from the assumption that the denominators of the quantal response rates are fixed however when time at risk weights are introduced to correct for treatment related differences in survival the denominators of the quantal response rates are subject to random variation the delta method and weighted least squares techniques are applied here to approximate the variance of such ratio statistics and test for a linear dose response relationship among treatments this technique is compared to that of bailer and portier 1988 biometrics 44 417 431 who introduced a survival adjusted quantal response test for trend in lifetime tumor incidence rates their test modifies the usual cochran armitage computing formula by weighting the denominators of the response rates to reflect less than whole animal contributions to risk within the framework of a weighted least squares linear regression model that underlies the cochran armitage test the time at risk weights of bailer and portier are incorporated using the delta method although the delta method approach is slightly more computationally intensive small sample simulations indicate that it has superior operating characteristics over the poly 3 trend test of bailer and portier when background tumor incidence rates are low under 3 and survival patterns differ markedly across treatments abstract truncated at 250 words

the classical moment problem as a self adjoint finite difference operator abstract this is a comprehensive exposition of the classical moment problem using methods from the theory of finite difference operators among the advantages of this approach is that the nevanlinna functions appear as elements of a transfer matrix and convergence of pade approximants appears as the strong resolvent convergence of finite matrix approximations to a jacobi matrix as a bonus of this we obtain new results on the convergence of certain pade approximants for series of hamburger

sums of squares and semidefinite program relaxations for polynomial optimization problems with structured sparsity unconstrained and inequality constrained sparse polynomial optimization problems pops are considered a correlative sparsity pattern graph is defined to find a certain sparse structure in the objective and constraint polynomials of a pop based on this graph sets of the supports for sums of squares sos polynomials that lead to efficient sos and semidefinite program sdp relaxations are obtained numerical results from various test problems are included to show the improved performance of the sos and sdp relaxations

approximate multinormal probabilities applied to correlated multiple endpoints in clinical trials clinical trials with multiple endpoints incur increased familywise type i errors the bonferroni correction is a common method used to modify the p values to account for multiple significance testing for independent endpoints the bonferroni method is slightly conservative whereas with high correlation the conservatism is extreme as demonstrated by pocock et al this paper presents a procedure which allows for the correlation present whilst adjusting the multiple p values the method is based on an approximation derived for multinormal probabilities

brief paper exact determinations of the maximal output admissible set for a class of nonlinear systems this paper is concerned with obtaining necessary and sufficient conditions for fulfilling specified state and control pointwise in time constraints against a certain class of nonlinear dynamics the results are generalizations of the maximal output admissible set theory to the case of nonlinear systems main contribution of the present paper is to propose explicit algorithmic procedures to determine the maximal output admissible set another contribution is that we discuss on a finite determinability of the maximal output admissible set for nonlinear systems some relations between observability of nonlinear systems and finite characterizations of the maximal output admissible set are clarified

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

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

sparse psd approximation of the psd cone while semidefinite programming sdp problems are polynomially solvable in theory it is often difficult to solve large sdp instances in practice one technique to address this issue is to relax the global positive semidefiniteness psd constraint and only enforce psd ness on smaller k times k principal submatrices we call this the sparse sdp relaxation surprisingly it has been observed empirically that in some cases this approach appears to produce bounds that are close to the optimal objective function value of the original sdp in this paper we formally attempt to compare the strength of the sparse sdp relaxation vis a vis the original sdp from a theoretical perspective in order to simplify the question we arrive at a data independent version of it where we compare the sizes of sdp cone and the k psd closure which is the cone of matrices where psd ness is enforced on all k times k principal submatrices in particular we investigate the question of how far a matrix of unit frobenius norm in the k psd closure can be from the sdp cone we provide two incomparable upper bounds on this farthest distance as a function of k and n we also provide matching lower bounds which show that the upper bounds are tight within a constant in different regimes of k and n other than linear algebra techniques we extensively use probabilistic methods to arrive at these bounds one of the lower bounds is obtained by observing a connection between matrices in the k psd closure and matrices satisfying the restricted isometry property

exploiting sparsity in semidefinite programming via matrix completion ii implementation and numerical results in part i of this series of articles we introduced a general framework of exploiting the aggregate sparsity pattern over all data matrices of large scale and sparse semidefinite programs sdps when solving them by primal dual interior point methods this framework is based on some results about positive semidefinite matrix completion and it can be embodied in two different ways one is by a conversion of a given sparse sdp having a large scale positive semidefinite matrix variable into an sdp having multiple but smaller positive semidefinite matrix variables the other is by incorporating a positive definite matrix completion itself in a primal dual interior point method the current article presents the details of their implementations we introduce new techniques to deal with the sparsity through a clique tree in the former method and through new computational formulae in the latter one numerical results over different classes of sdps show that these methods can be very efficient for some problems

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