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

a new a posteriori error estimate for the morley element in this paper we present a posteriori error analysis for the nonconforming morley element of the fourth order elliptic equation we propose a new residual based a posteriori error estimator and prove its reliability and efficiency these results refine those of beirao da veiga et al numer math 106 165 179 2007 by dropping two edge jump terms in both the energy norm of the error and the estimator and those of wang and zhang local a priori and a posteriori error estimates of finite elements for biharmonic equation research report 13 2006 by showing the efficiency in the sense of verfurth a review of a posteriori error estimation and adaptive mesh refinement techniques wiley teubner new york 1996 moreover the normal component in the estimators of beirao da veiga et al numer math 106 165 179 2007 and wang and zhang local a priori and a posteriori error estimates of finite elements for biharmonic equation research report 13 2006 is dropped and therefore only the tangential component of the stress on each edge comes into the estimator in addition we generalize these results to three dimensional case

approximation schemes for infinite linear programs this paper presents approximation schemes for an infinite linear program in particular it is shown that under suitable assumptions the program s optimum value can be approximated by the values of finite dimensional linear programs and that in addition every accumulation point of a sequence of optimal solutions for the approximating programs is an optimal solution for the original problem

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

a mixed method for the biharmonic problem based on a system of first order equations we introduce a new mixed method for the biharmonic problem the method is based on a formulation where the biharmonic problem is rewritten as a system of four first order equations a hybrid form of the method is introduced which allows us to reduce the globally coupled degrees of freedom to only those associated with lagrange multipliers which approximate the solution and its derivative at the faces of the triangulation for k ge 1 a projection of the primal variable error superconverges with order k 3 while the error itself converges with order k 1 only this fact is exploited by using local postprocessing techniques that produce new approximations to the primal variable converging with order k 3 we provide numerical experiments that validate our theoretical results

a posteriori error analysis of the linked interpolation technique for plate bending problems we develop a posteriori error estimates for the so called linked interpolation technique to approximate the solution of plate bending problems we show that the proposed residual based estimator is both reliable and efficient

an explicit equivalent positive semidefinite program for nonlinear 0 1 programs we consider the general nonlinear optimization problem in 0 1 variables and provide an explicit equivalent positive semidefinite program in 2n 1 variables the optimal values of both problems are identical from every optimal solution of the former one easily finds an optimal solution of the latter and conversely from every solution of the latter one may construct an optimal solution of the former for illustration the equivalent positive semidefinite program is explicated for quadratic 0 1 programs and max cut in mathbb r 3 for unconstrained 0 1 programs a special representation in terms of a weighted sum of squares is provided

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

a family of c0 finite elements for kirchhoff plates ii numerical results abstract a new family of c0 kirchhoff plate elements has been introduced by the authors in the theoretical counterpart of the present paper a family of c0 finite elements for kirchhoff plates i error analysis the method presented is a displacement formulation with the deflection and the rotation as unknowns in the theoretical part an a priori and an a posteriori error analysis has been accomplished for the family in the present contribution the authors first give a supplementary derivation of the method and recall the main theoretical results then focus on the computational aspects of the method and finally present a set of numerical results on various benchmark computations these tests verify the optimal convergence rate of the method and illustrate the robustness of the reliable and efficient residual based a posteriori error estimator for adaptive mesh refinements

a family of c 0 finite elements for kirchhoff plates i error analysis a new finite element formulation for the kirchhoff plate model is presented the method is a displacement formulation with the deflection and the rotation vector as unknowns and it is based on ideas stemming from a stabilized method for the reissner mindlin model r stenberg in asymptotic methods for elastic structures p ciarlet l trabucho and j m viano eds de gruyter berlin 1995 and a method to treat a free boundary p destuynder and t nevers rairo model math anal numer 22 1988 pp 217 242 optimal a priori and a posteriori error estimates are derived

distributionally robust optimization with polynomial densities theory models and algorithms in distributionally robust optimization the probability distribution of the uncertain problem parameters is itself uncertain and a fictitious adversary e g nature chooses the worst distribution from within a known ambiguity set a common shortcoming of most existing distributionally robust optimization models is that their ambiguity sets contain pathological discrete distributions that give nature too much freedom to inflict damage we thus introduce a new class of ambiguity sets that contain only distributions with sum of squares sos polynomial density functions of known degrees we show that these ambiguity sets are highly expressive as they conveniently accommodate distributional information about higher order moments conditional probabilities conditional moments or marginal distributions exploiting the theoretical properties of a measure based hierarchy for polynomial optimization due to lasserre siam j optim 21 3 864 885 2011 we prove that certain worst case expectation constraints are polynomial time solvable under these new ambiguity sets we also show how sos densities can be used to approximately solve the general problem of moments we showcase the applicability of the proposed approach in the context of a stylized portfolio optimization problem and a risk aggregation problem of an insurance company

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

on the capacity of two dimensional run length constrained channels two dimensional binary patterns that satisfy one dimensional d k run length constraints both horizontally and vertically are considered for a given d and k the capacity c sub d k is defined as c sub d k lim sub m n spl rarr spl infin log sub 2 n sub m n sup d k mn where n sub m n sup d k denotes the number of m spl times n rectangular patterns that satisfy the two dimensional d k run length constraint bounds on c sub d k are given and it is proven for every d spl ges 1 and every k d that c sub d k 0 if and only if k d 1 encoding algorithms are also discussed

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

series expansions from the corner transfer matrix renormalization group method ii asymmetry and high density hard squares the corner transfer matrix renormalization group method is a method claimed to be subexponential for numerically calculating physical quantities of statistical mechanical models in a previous paper we extended this method to generate series expansions here we show how to extend both the original and series methods to deal with asymmetry in any model we discuss the cases of rotational translational and reflection asymmetry and give some improvements to the method this is demonstrated by an application of the method to generate series for the hard square model in the high density regime producing 51 terms of the partition function and 48 terms of the order parameter these series are analysed producing estimates of the critical point and exponents and showing the likely presence of a confluent singularity with exponent 17 8 in the order parameter

a posteriori error analysis for the morley plate element with general boundary conditions this paper introduces and analyses a local residual based a posteriori error indicator for the morley finite element method of the biharmonic kirchhoff plate bending problem in the theoretical part of the paper a recent approach presented by the authors for clamped boundaries is extended to general boundary conditions the error indicator is proven to be both reliable and efficient the numerical part of the paper presents a set of results on various benchmark computations with different kinds of domains and boundary conditions these tests verify the reliability and efficiency of the error estimator and illustrate the robustness of the method for adaptive mesh refinements copyright 2009 john wiley sons ltd

isogeometric analysis for sixth order boundary value problems of gradient elastic kirchhoff plates abstract sixth order boundary value problems of a one parameter gradient elastic kirchhoff plate model are formulated in a weak form within an h 3 sobolev space setting with the corresponding equilibrium equations and general boundary conditions the corresponding conforming galerkin method is proposed with error estimates for discretizations satisfying c 2 continuity requirements continuity coercivity and consistency of the corresponding bilinear form are utilized for proving the theoretical results numerical computations with conforming isogeometric discretizations of c p 1 continuous nurbs basis functions of order p 3 confirm the theoretical results and illustrate the features of the problem for both statics and free vibrations in particular the effects of the additional boundary conditions and parameter dependent boundary layers corresponding to the gradient elasticity theory are addressed by the numerical examples

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

finite element methods for a modified reissner mindlin free plate model the solution of the reissner mindlin plate problem with free boundary conditions presents a strong layer effect near the free edges as a consequence the solution is not even uniformly bounded even in h3 2 which implies that at most an o h1 2 uniform convergence rate can be reached by finite element methods in the h1 norm following instead the modified free boundary model presented by beirao da veiga and brezzi which gives more regular solutions better error estimates can be obtained in principle in this paper we present and analyze the extension of different families of well known optimal plate methods to this new model all the modified methods presented are proved to be optimal and free of locking

bounds on the capacity of constrained two dimensional codes bounds on the capacity of constrained two dimensional 2 d codes are presented the bounds of calkin and wilf see siam j discr math vol 11 no 1 p 54 60 1998 apply to first order symmetric constraints the bounds are generalized in a weaker form to higher order and nonsymmetric constraints results are given for constraints specified by run length limits or a minimum distance between pixels of a given value

optimal inequalities in probability theory a convex optimization approach we propose a semidefinite optimization approach to the problem of deriving tight moment inequalities for p x in s for a set s defined by polynomial inequalities and a random vector x defined on omega subseteq cal r n that has a given collection of up to kth order moments in the univariate case we provide optimal bounds on p x in s when the first k moments of x are given as the solution of a semidefinite optimization problem in k 1 dimensions in the multivariate case if the sets s and omega are given by polynomial inequalities we obtain an improving sequence of bounds by solving semidefinite optimization problems of polynomial size in n for fixed k we characterize the complexity of the problem of deriving tight moment inequalities we show that it is np hard to find tight bounds for k geq 4 and omega cal r n and for k geq 2 and omega cal r n when the data in the problem is rational for k 1 and omega cal r n we show that we can find tight upper bounds by solving n convex optimization problems when the set s is convex and we provide a polynomial time algorithm when s and omega are unions of convex sets over which linear functions can be optimized efficiently for the case k 2 and omega cal r n we present an efficient algorithm for finding tight bounds when s is a union of convex sets over which convex quadratic functions can be optimized efficiently

upper bounds on the growth rates of independent sets in two dimensions via corner transfer matrices abstract we devise an algorithm to calculate upper bounds on the growth rates of the number of independent sets on a variety of regular two dimensional lattices using an amalgamation of techniques from linear algebra combinatorics and statistical mechanics our method uses calkin and wilf s transfer matrix eigenvalue upper bound together with the collatz wielandt formula from linear algebra to obtain a good bound we need an approximate eigenvector which we find using baxter s corner transfer matrix ansatz and nishino and okunishi s corner transfer matrix renormalisation group method this results in an algorithm for computing upper bounds which is far faster in practice than all other known methods it is also the first algorithm for this problem with a polynomial rather than exponential memory requirement and it is extremely parallelisable this allows us to make dramatic improvements to the previous best known upper bounds we apply our algorithm to five models including independent sets on the square lattice also known as the hard squares lattice gas from statistical mechanics in all cases we extend the number of rigorously known digits of the growth rate by 4 6 digits

discontinuous petrov galerkin method with optimal test functions for thin body problems in solid mechanics we study the applicability of the discontinuous petrov galerkin dpg variational framework for thin body problems in structural mechanics our numerical approach is based on discontinuous piecewise polynomial finite element spaces for the trial functions and approximate local computation of the corresponding optimal test functions in the timoshenko beam problem the proposed method is shown to provide the best approximation in an energy type norm which is equivalent to the l2 norm for all the unknowns uniformly with respect to the thickness parameter the same formulation remains valid also for the asymptotic euler bernoulli solution as another one dimensional model problem we consider the modelling of the so called basic edge effect in shell deformations in particular we derive a special norm for the test space which leads to a robust method in terms of the shell thickness finally we demonstrate how a posteriori error estimator arising directly from the discontinuous variational framework can be utilized to generate an optimal hp mesh for resolving the boundary layer

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

a unified approach for embedded boundary conditions for fourth order elliptic problems summary an efficient procedure for embedding kinematic boundary conditions in the biharmonic equation for problems such as the pure streamfunction formulation of the navier stokes equations and thin plate bending is based on a stabilized variational formulation obtained by nitsche s approach for enforcing boundary constraints the absence of kinematic admissibility constraints allows the use of non conforming meshes with non interpolatory approximations thereby providing added flexibility in addressing the higher continuity requirements typical of these problems variationally conjugate pairs weakly enforce kinematic boundary conditions the use of a scaling factor leads to a formulation with a single stabilization parameter for plates the enforcement of tangential derivatives of deflections obviates the need for pointwise enforcement of corner values in the presence of corners the single stabilization parameter is determined from a local generalized eigenvalue problem guaranteeing coercivity of the discrete bilinear form the accuracy of the approach is verified by representative computations with bicubic b splines providing guidance to the determination of the scaling and exhibiting optimal rates of convergence and robust performance with respect to values of the stabilization parameter copyright 2014 john wiley sons ltd

analysis of some low order finite element schemes for mindlin reissner and kirchhoff plates we set up a framework for analyzing mixed finite element methods for the plate problem using a mesh dependent energy norm which applies both to the kirchhoff and to the mindlin reissner formulation of the problem the analysis techniques are applied to some low order finite element schemes where three degrees of freedom are associated to each vertex of a triangulation of the domain the schemes proceed from the mindlin reissner formulation with modified shear energy

capacity bounds for the three dimensional 0 1 run length limited channel the capacity c sub 0 1 sup 8 of a three dimensional 0 1 run length constrained channel is shown to satisfy 0 522501741838 spl les c sub 0 1 sup 8 spl les 0 526880847825

new bounds on the capacity of multidimensional run length constraints we examine the well known problem of determining the capacity of multidimensional run length limited constrained systems by recasting the problem which is essentially a combinatorial counting problem into a probabilistic setting we are able to derive new lower and upper bounds on the capacity of 0 k rll systems these bounds are better than all previously known analytical bounds for k 2 and are tight asymptotically thus we settle the open question what is the rate at which the capacity of 0 k rll systems converges to 1 as k we also provide the first nontrivial upper bound on the capacity of general d k rll systems

convex duality and nonlinear optimal control problems in nonlinear optimal control can be reformulated as convex optimization problems over a vector space of linear functionals in this way methods of convex analysis can be brought to bear on the task of characterizing solutions to such problems the result is a necessary and sufficient condition of optimality that generalizes well known sufficient conditions referred to as verification theorems in dynamic programming as a byproduct we obtain a representation of the minimum cost in terms of the upper envelope of subsolutions to the hamilton jacobi equation it is a striking illustration of the wide range of problems to which convex analysis and in particular convex duality is applicable the approach applied to parametric problems in the calculus of variations was pioneered by l c young lectures on the calculus of variations and optimal control theory w b saunders philadelphia pa 1969 as recent work has shown however it is equally fruitful when applied in optimal control thi

improved lower bounds on capacities of symmetric 2d constraints using rayleigh quotients a method for computing lower bounds on capacities of two dimensional 2d constraints having a symmetric presentation in either the horizontal or the vertical direction is presented the method is a generalization of the method of calkin and wilf siam j discrete math 1998 previous best lower bounds on capacities of certain constraints are improved using the method it is also shown how this method as well as their method for computing upper bounds on the capacity can be applied to constraints which are not of finite type additionally capacities of two families of multidimensional constraints are given exactly

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

a semidefinite programming approach to the generalized problem of moments we consider the generalized problem of moments gpm from a computational point of view and provide a hierarchy of semidefinite programming relaxations whose sequence of optimal values converges to the optimal value of the gpm we then investigate in detail various examples of applications in optimization probability financial economics and optimal control which all can be viewed as particular instances of the gpm

accurate lower bounds on 2 d constraint capacities from corner transfer matrices we analyse the capacity of several 2 d constraint families the exclusion coloring parity and charge model families using baxter s corner transfer matrix formalism combined with the corner transfer matrix renormalization group method of nishino and okunishi we calculate very tight lower bounds and estimates on the growth rates of these models our results strongly improve previous known lower bounds and lead to the surprising conjecture that the capacity of the even and charge 3 constraints are identical

reconstruction of conditional expectations from product moments with applications in this paper it is shown under conditions associated with the moment problem that a sequence of product moments uniquely determines a conditional expectation of two random variables then a numerical procedure is derived to reconstruct a conditional expectation in terms of a sequence of its product moments

variational formulation and isogeometric analysis for fourth order boundary value problems of gradient elastic bar and plane strain stress problems abstract the fourth order boundary value problems of one parameter gradient elastic bar and plane strain stress models are formulated in a variational form within an h 2 sobolev space setting for both problems the existence and uniqueness of the solution is established by proving the continuity and coercivity of the associated symmetric bilinear form for completeness the full sets of boundary conditions of the problems are derived and in particular the new types of boundary conditions featured by the gradient elastic models are given the additional attributes singly and doubly by utilizing the continuity and coercivity of the continuous problems corresponding error estimates are formulated for conforming galerkin formulations finally numerical results with isogeometric c p 1 continuous discretizations for nurbs basis functions of order p 2 confirm the theoretical results and illustrate the essentials of both static and vibration problems