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

on classes of non gaussian asymptotic minimizers in entropic uncertainty principles abstract in this paper we revisit the bialynicki birula and mycielski uncertainty principle and its cases of equality this shannon entropic version of the well known heisenberg uncertainty principle can be used when dealing with variables that admit no variance in this paper we extend this uncertainty principle to renyi entropies we recall that in both shannon and renyi cases and for a given dimension n the only case of equality occurs for gaussian random vectors we show that as n grows however the bound is also asymptotically attained in the cases of n dimensional student t and student r distributions a complete analytical study is performed in a special case of a student t distribution we also show numerically that this effect exists for the particular case of a n dimensional cauchy variable whatever the renyi entropy considered extending the results of abe and illustrating the analytical asymptotic study of the student t case in the student r case we show numerically that the same behavior occurs for uniformly distributed vectors these particular cases and other ones investigated in this paper are interesting since they show that this asymptotic behavior cannot be considered as a gaussianization of the vector when the dimension increases

the hausdorff young inequality on lie groups we prove several results about the best constants in the hausdorff young inequality for noncommutative groups in particular we establish a sharp local central version for compact lie groups and extend known results for the heisenberg group in addition we prove a universal lower bound to the best constant for general lie groups

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

on the lp lq norm of the hankel transform and related operators abstract we investigate the l p 0 l q 0 mapping properties of the operators l f y y 0 x y f x j x y d x f c 0 0 for suitable values of the parameters and we evaluate the operator norm of l in some special and significant cases

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

open problems from the 12th international conference on fuzzy set theory and its applications eighteen open problems posed during fsta 2014 liptovsk jan slovakia are presented these problems concern fuzzy logics fuzzy partitions copulas triangular norms and related aggregation functions some open problems concerning effect and mv algebras are also included

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

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

the smallest semicopula based universal integrals remarks and improvements abstract we provide several improvements and corrections of results dealing with the smallest universal integral i s based on a semicopula s we deal with a transformation of the integral into another semicopula based universal integral for continuous and commutative semicopulas monotone convergence theorems with new proofs and frechet type mappings properties corrected during this way we discover several fine properties of the integral functional i s we also discuss further convergences of measurable functions and the corresponding integrals in the light of the newest results from the literature answering several open questions stated in our previous articles of the series

a central limit theorem for operators abstract we prove an analogue of the central limit theorem for operators for every operator k defined on c x we construct a sequence of operators k n defined on c x 1 x n given by the n fold tensor product of k with itself modulo a proper scaling and we demonstrate that under certain orthogonality conditions this sequence converges in a weak sense to an unique operator c we show that this operator c is a member of a family of operators c that we call centered gaussian operators and which coincides with the family of operators given by a centered gaussian kernel inspired in the approximation method used by beckner in 1 to prove the sharp form of the hausdorff young inequality the present article shows that beckner s method is a special case of a general approximation method for operators in particular we characterize the hermite semi group as the family of centered gaussian operators associated with any semi group of operators

on the nonlinear brascamp lieb inequality we prove a nonlinear variant of the general brascamp lieb inequality our proof consists of running an efficient or tight induction on scales argument which uses the existence of gaussian near extremizers to the underlying linear brascamp lieb inequality lieb s theorem in a fundamental way a key ingredient is an effective version of lieb s theorem which we establish via a careful analysis of near minimizers of weighted sums of exponential functions instances of this inequality are quite prevalent in mathematics and we illustrate this with some applications in harmonic analysis

the smallest semicopula based universal integrals i properties and characterizations abstract in this article we provide a detailed study of basic properties of the smallest universal integral i s with s being the underlying semicopula and review some of the recent developments in this direction the class of integrals under study is also known under the name seminormed integrals and includes the well known sugeno and shilkret integrals as special cases we present some representations and equivalent formulations for i s further we discuss basic properties such as translatability linearity homogeneity and maxitivity in particular we provide the characterizations of the two prominent integrals of sugeno and shilkret we also raise and discuss some open questions

an entropy based uncertainty principle for a locally compact abelian group abstract we classify all functions on a locally compact abelian group giving equality in an entropy inequality generalizing the heisenberg uncertainty principle in particular for functions on a real line we proof a conjecture of hirschman published in 1957

a reverse sidorenko inequality let h be a graph allowing loops as well as vertex and edge weights we prove that for every triangle free graph g without isolated vertices the weighted number of graph homomorphisms hom g h satisfies the inequality begin aligned hom g h le prod uv in e g hom k d u d v h 1 d ud v end aligned where d u denotes the degree of vertex u in g in particular one has begin aligned hom g h 1 e g le hom k d d h 1 d 2 end aligned for every d regular triangle free g the triangle free hypothesis on g is best possible more generally we prove a graphical brascamp lieb type inequality where every edge of g is assigned some two variable function these inequalities imply tight upper bounds on the partition function of various statistical models such as the ising and potts models which includes independent sets and graph colorings for graph colorings corresponding to h k q we show that the triangle free hypothesis on g may be dropped this is also valid if some of the vertices of k q are looped a corollary is that among d regular graphs g k d d maximizes the quantity c q g 1 v g for every q and d where c q g counts proper q colorings of g finally we show that if the edge weight matrix of h is positive semidefinite then begin aligned hom g h le prod v in v g hom k d v 1 h 1 d v 1 end aligned this implies that among d regular graphs g k d 1 maximizes hom g h 1 v g for 2 spin ising models our results give a complete characterization of extremal graphs complete bipartite graphs maximize the partition function of 2 spin antiferromagnetic models and cliques maximize the partition function of ferromagnetic models these results settle a number of conjectures by galvin tetali galvin and cohen csikvari perkins tetali and provide an alternate proof to a conjecture by kahn

translation invariant semicopula based integrals in this note we give a solution to problem 9 2 which was presented by mesiar and stup anova 2015 6 we show that the class of semicopulas solving problem 9 2 contains only the ukasiewicz t norm

integral bounds for radar ambiguity functions and wigner distributions an upper bound is proved for the lp norm of woodward s ambiguity function in radar signal analysis and of the wigner distribution in quantum mechanics when p 2 a lower bound is proved for 1 p 2 in addition a lower bound is proved for the entropy these bounds set limits to the sharpness of the peaking of the ambiguity function or wigner distribution the bounds are best possible and equality is achieved in the lp bounds if and only if the functions f and g that enter the definition are both gaussians

on the multilinear restriction and kakeya conjectures we prove d linear analogues of the classical restriction and kakeya conjectures in rd our approach involves obtaining monotonicity formulae pertaining to a certain evolution of families of gaussians closely related to heat flow we conclude by giving some applications to the corresponding variable coefficient problems and the so called joints problem as well as presenting some n linear analogues for n d

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

on an integral criterion for hypercontractivity of diffusion semigroups and extremal functions abstract in the line of investigation of the works by d bakry and m emery lecture notes in math vol 1123 pp 175 206 springer verlag new york berlin 1985 and o s rothaus j funct anal 42 1981 102 109 j funct anal 65 1986 358 367 we study an integral inequality behind the 2 criterion of d bakry and m emery see previous reference and its applications to hypercontractivity of diffusion semigroups with in particular a short proof of the hypercontractivity property of the ornstein uhlenbeck semigroup our exposition unifies in a simple way several previous results interpolating smoothly from the spectral gap inequalities to logarithmic sobolev inequalities and even true sobolev inequalities we examine simultaneously the extremal functions for hypercontractivity and logarithmic sobolev inequalities of the ornstein uhlenbeck semigroup and heat semigroup on spheres

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

a strong entropy power inequality when one of the random summands is gaussian we sharpen the entropy power inequality epi in terms of the strong data processing function for gaussian channels among other consequences this strong epi generalizes the vector extension of costa s epi to non gaussian channels in a precise sense this leads to a new reverse epi and as a corollary sharpens stam s uncertainty principle relating entropy power and fisher information or equivalently gross logarithmic sobolev inequality applications to network information theory are also given including a short self contained proof of the rate region for the two encoder quadratic gaussian source coding problem and a new outer bound for the one sided gaussian interference channel

phase space localization of antisymmetric functions upper and lower bounds are written down for the minimum information entropy in phase space of an antisymmetric wave function in any number of dimensions similar bounds are given when the wave function is restricted to belong to any of the proper subspaces of the fourier transform operator

on lieb s conjecture for the wehrl entropy of bloch coherent states lieb s conjecture for the wehrl entropy of bloch coherent states is proved for spin 1 and spin 3 2 using a geometric representation we solve the entropy integrals for states of arbitrary spin and evaluate them explicitly in the cases of spin 1 3 2 and 2 we also give a group theoretic proof for all spin of a related inequality

extremals of functionals with competing symmetries abstract we present a new method of producing optimizing sequences for highly symmetric functionals the sequences have good convergence properties built in we apply the method in different settings to give elementary proofs of some classical inequalities such as the hardy littlewood sobolev and the logarithmic sobolev inequality in their sharp form

entropic uncertainty bound for open pointer based simultaneous measurements of conjugate observables we discuss the information entropy for a general open pointer based simultaneous measurement and show how it is bound from below this entropic uncertainty bound is a direct consequence of the structure of the entropy and can be obtained from the formal solution of the measurement dynamics furthermore the structural properties of the entropy allow us to give an intuitive interpretation of the noisy influence of the pointers and the environmental heat bath on the measurement results

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

sum rate capacity for symmetric gaussian multiple access channels with feedback the feedback sum rate capacity is established for the symmetric j user gaussian multiple access channel gmac the main contribution is a converse bound that combines the dependence balance argument of hekstra and willems 1989 with a variant of the factorization of a convex envelope of geng and nair 2014 the converse bound matches the achievable sum rate of the fourier modulated estimate correction strategy of kramer 2002

gaussian kernels have only gaussian maximizers a gaussian integral kernel g x y on rn x rn is the exponential of a quadratic form in x and y the fourier transform kernel is an example the problem addressed here is to find the sharp bound of g as an operator from l p rp to l q rn and to prove that the l p rn functions that saturate the bound are necessarily gaussians this is accomplished generally for 1 q in some special cases besides greatly extending previous results in this area the proof technique is also essentially different from earlier ones a corollary of these results is a fully multidimensional multilinear generalization of young s inequality

the smallest semicopula based universal integrals ii in this paper we continue studying the smallest universal integral i s having s as the underlying semicopula we present convergence theorems for i s integral sequences including monotone almost everywhere almost uniform in measure and in mean converging sequences of measurable functions respectively it emerges that these convergences characterize the underlying measure properties such as null additivity monotone autocontinuity and autocontinuity we provide many examples and counter examples as well as a few interesting open problems

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

the weakly nonlinear large box limit of the 2d cubic nonlinear schrodinger equation we consider the cubic nonlinear schrodinger nls equation set on a two dimensional box of size l with periodic boundary conditions by taking the large box limit l to infty in the weakly nonlinear regime characterized by smallness in the critical space we derive a new equation set on r 2 that approximates the dynamics of the frequency modes this nonlinear equation turns out to be hamiltonian and enjoys interesting symmetries such as its invariance under fourier transform as well as several families of explicit solutions a large part of this work is devoted to a rigorous approximation result that allows to project the long time dynamics of the limit equation into that of the cubic nls equation on a box of finite size

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

dimension dependent hypercontractivity for gaussian kernels we derive sharp local and dimension dependent hypercontractive bounds on the markov kernel of a large class of diffusion semigroups unlike the dimension free ones they capture refined properties of markov kernels such as trace estimates they imply classical bounds on the ornstein uhlenbeck semigroup and a dimensional and refined transportation talagrand inequality when applied to the hamilton jacobi equation hypercontractive bounds on the ornstein uhlenbeck semigroup driven by a non diffusive levy semigroup are also investigated curvature dimension criteria are the main tool in the analysis

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

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

the hypercontractive approach to exactly bounding an operator with complex gaussian kernel abstract we study operators g l q r n l p r n given by a nondegenerate complex gaussian kernel in many cases such an operator attains its bound at a maximizing gaussian function g l q r n to show this we prove a complexified version of edward nelson s hypercontractive theorem

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

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

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