bounding the gap between the mccormick relaxation and the convex hull for bilinear functions we investigate how well the graph of a bilinear function b 0 1 n rightarrow mathbb r b 0 1 n r can be approximated by its mccormick relaxation in particular we are interested in the smallest number c such that the difference between the concave upper bounding and convex lower bounding functions obtained from the mccormick relaxation approach is at most c times the difference between the concave and convex envelopes answering a question of luedtke namazifar and linderoth we show that this factor c cannot be bounded by a constant independent of n more precisely we show that for a random bilinear function b we have asymptotically almost surely c geqslant sqrt n 4 c n 4 on the other hand we prove that c leqslant 600 sqrt n c 600n which improves the linear upper bound proved by luedtke namazifar and linderoth in addition we present an alternative proof for a result of misener smadbeck and floudas characterizing functions b for which the mccormick relaxation is equal to the convex hull

an fptas for optimizing a class of low rank functions over a polytope we present a fully polynomial time approximation scheme fptas for optimizing a very general class of non linear functions of low rank over a polytope our approximation scheme relies on constructing an approximate pareto optimal front of the linear functions which constitute the given low rank function in contrast to existing results in the literature our approximation scheme does not require the assumption of quasi concavity on the objective function for the special case of quasi concave function minimization we give an alternative fptas which always returns a solution which is an extreme point of the polytope our technique can also be used to obtain an fptas for combinatorial optimization problems with non linear objective functions for example when the objective is a product of a fixed number of linear functions we also show that it is not possible to approximate the minimum of a general concave function over the unit hypercube to within any factor unless p np we prove this by showing a similar hardness of approximation result for supermodular function minimization a result that may be of independent interest

convex extensions and envelopes of lower semi continuous functions we define a convex extension of a lower semi continuous function to be a convex function that is identical to the given function over a pre specified subset of its domain convex extensions are not necessarily constructible or unique we identify conditions under which a convex extension can be constructed when multiple convex extensions exist we characterize the tightest convex extension in a well defined sense using the notion of a generating set we establish conditions under which the tightest convex extension is the convex envelope then we employ convex extensions to develop a constructive technique for deriving convex envelopes of nonlinear functions finally using the theory of convex extensions we characterize the precise gaps exhibited by various underestimators of x y over a rectangle and prove that the extensions theory provides convex relaxations that are much tighter than the relaxation provided by the classical outer linearization of bilinear terms

image compression using address vector quantization a novel vector quantization scheme called the address vector quantizer a vq is proposed it is based on exploiting the interblock correlation by encoding a group of blocks together using an address codebook the address codebook consists of a set of address codevectors where each codevector represents a combination of addresses indexes each element of this codevector is an address of an entry in the lbg codebook representing a vector quantized block the address codebook consists of two regions one is the active addressable region and the other is the inactive nonaddressable region during the encoding process the codevectors in the address codebook are reordered adaptively in order to bring the most probable address codevectors into the active region when encoding an address codevector the active region of the address codebook is checked and if such an address combination exist its index is transmitted to the receiver otherwise the address of each block is transmitted individually the quality snr value of the images encoded by the proposed a vq method is the same as that of a memoryless vector quantizer but the bit rate would be reduced by a factor of approximately two when compared to a memoryless vector quantizer

convex envelopes generated from finitely many compact convex sets we consider the problem of constructing the convex envelope of a lower semi continuous function defined over a compact convex set we formulate the envelope representation problem as a convex optimization problem for functions whose generating sets consist of finitely many compact convex sets in particular we consider nonnegative functions that are products of convex and component wise concave functions and derive closed form expressions for the convex envelopes of a wide class of such functions several examples demonstrate that these envelopes reduce significantly the relaxation gaps of widely used factorable relaxation techniques

on finitely terminating branch and bound algorithms for some global optimization problems global optimization algorithms are typically terminated with an epsilon approximate solution after a finite number of iterations this paper shows how existing infinitely convergent branch and bound algorithms can be augmented to guarantee finite termination with an exact global solution for problems having extreme point solutions

adaptive vector quantization with codebook updating based on locality and history we propose two techniques that are applicable to any adaptive vector quantization avq systems the first one is called the locality based codebook updating when performing a codebook updating we update the operational codebook using not only the current input vector but also the codewords at all positions within a selected neighboring area called the locality while the operational codebook is organized in a cache manner this technique is rationalized by the high correlation cross neighboring vectors that facilitates a more efficient coding of the indices of the codewords chosen from the codebook the second technique is called the history aid which makes use of the information of previously coded vectors to quantize the current input vector if it is used to update the operational codebook a more effective avq system is obtained by combining together the history aid and the locality based updating extensive simulations are carried out to demonstrate the improved results achieved by our avq systems particularly when the operational codebook size is relatively small the improvement over a benchmark avq system the generalized threshold replenishment gtr is drastic for example when the size is 32 testing on a nonstationary signal containing frames from different video sequences ordered in the concatenating or interleaving format shows that the combination of history aid and locality based updating offers more than 4 db gain over gtr at 0 5 bpp

convex envelopes for edge concave functions deterministic global optimization algorithms frequently rely on the convex underestimation of nonconvex functions in this paper we describe the structure of the polyhedral convex envelopes of edge concave functions over polyhedral domains using geometric arguments an algorithm for computing the facets of the convex envelope over hyperrectangles in 3 is described sufficient conditions are described under which the convex envelope of a sum of edge concave functions may be shown to be equivalent to the sum of the convex envelopes of these functions

existence of s asymptotically ω periodic solutions to abstract integro differential equations we study the existence of solutions for a class of abstract equations we construct an exponentially stable resolvent operator solutions s asymptotically periodic are obtained we obtained existence results with less restrictions on the constants we provide a concrete example for the abstract results the main aim of this work is to study the existence of s asymptotically periodic solutions for a class of abstract integro differential equations modeled in the following form d dt x t 0 t n t s x s ds ax t 0 t b t s x s ds f t x t t 0 x 0 x 0 x where a b t for t 0 are closed linear operators defined on a common domain d a which is dense in x n t for t 0 are bounded linear operators on x and f 0 i x x is an appropriate function the existence results are obtained by applying the theory of exponentially stable resolvent operators we also discuss an application of these results

entropy constrained vector quantization an iterative descent algorithm based on a lagrangian formulation for designing vector quantizers having minimum distortion subject to an entropy constraint is discussed these entropy constrained vector quantizers ecvqs can be used in tandem with variable rate noiseless coding systems to provide locally optimal variable rate block source coding with respect to a fidelity criterion experiments on sampled speech and on synthetic sources with memory indicate that for waveform coding at low rates about 1 bit sample under the squared error distortion measure about 1 6 db improvement in the signal to noise ratio can be expected over the best scalar and lattice quantizers when block entropy coded with block length 4 even greater gains are made over other forms of entropy coded vector quantizers for pattern recognition it is shown that the ecvq algorithm is a generalization of the k means and related algorithms for estimating cluster means in that the ecvq algorithm estimates the prior cluster probabilities as well experiments on multivariate gaussian distributions show that for clustering problems involving classes with widely different priors the ecvq outperforms the k means algorithm in both likelihood and probability of error

a piecewise linearization framework for retail shelf space management models abstract managing shelf space is critical for retailers to attract customers and optimize profits this article develops a shelf space allocation optimization model that explicitly incorporates essential in store costs and considers space and cross elasticities a piecewise linearization technique is used to approximate the complicated nonlinear space allocation model the approximation reformulates the non convex optimization problem into a linear mixed integer programming mip problem the mip solution not only generates near optimal solutions for large scale optimization problems but also provides an error bound to evaluate the solution quality consequently the proposed approach can solve single category shelf space management problems with as many products as are typically encountered in practice and with more complicated cost and profit structures than currently possible by existing methods numerical experiments show the competitive accuracy of the proposed method compared with the mixed integer nonlinear programming shelf space model several extensions of the main model are discussed to illustrate the flexibility of the proposed methodology

convex envelopes of some quadratic functions over the n dimensional unit simplex in this paper we discuss convex envelopes over the unit simplex for some quadratic functions the discussion exploits the combinatorics related to quadratic functions over the unit simplex after recalling the definition of the underlying convex graph of a quadratic function i e the graph whose nodes are the vertices of the unit simplex and whose edges correspond to the edges of the unit simplex along which the function is strictly convex it is shown that the analytical formula of the convex envelope for quadratic functions whose underlying graph is triangle free is the maximum of a finite number of functions each of which is related to a spanning forest of the graph the special case of an underlying star graph is discussed and computational experiments based on a decomposition of a general quadratic function into a sum of quadratic functions with an underlying star graph are presented

s asymptotically ω periodic and asymptotically ω periodic solutions to semi linear cauchy problems with non dense domain abstract in this work we study the existence and uniqueness of s asymptotically periodic and asymptotically periodic solutions to a first order differential equation with linear part dominated by a hille yosida operator with non dense domain applications to partial differential equations fractional integro differential and neutral differential equations are given

a new two level linear relaxed bound method for geometric programming problems in this paper a new two level linear relaxed bound method is proposed for solving the global solution of geometric programming problems and its convergent properties is proved and a numerical example is used to illustrate the effectiveness of the presented algorithm the bound technique in this algorithm is different from the other ones the two level relaxed linear programming problems of geometric programming problems are given without additional new variables and constraints by making use of the linear approximation of power functions and the new formulas for product to be unequal with sum

relaxations of factorable functions with convex transformable intermediates we propose to strengthen standard factorable relaxations of global optimization problems through the use of functional transformations of intermedi ate expressions in particular we exploit convex transformability of the component functions of factorable programs as a tool in the generation of bounds we define suit able forms of transforming functions and assess theoretically and computationally the sharpness of the resulting relaxations in comparison to existing schemes

pooling problem alternate formulations and solution methods the pooling problem which is fundamental to the petroleum industry describes a situation in which products possessing different attribute qualities are mixed in a series of pools in such a way that the attribute qualities of the blended products of the end pools must satisfy given requirements it is well known that the pooling problem can be modeled through bilinear and nonconvex quadratic programming in this paper we investigate how best to apply a new branch and cut quadratic programming algorithm to solve the pooling problem to this effect we consider two standard models one is based primarily on flow variables and the other relies on the proportion of flows entering pools a hybrid of these two models is proposed for general pooling problems comparison of the computational properties of flow and proportion models is made on several problem instances taken from the literature moreover a simple alternating procedure and a variable neighborhood search heuristic are developed to solve large instances and compared with the well known method of successive linear programming solution of difficult test problems from the literature is substantially accelerated and larger ones are solved exactly or approximately

semidefinite relaxations for quadratically constrained quadratic programming a review and comparisons at the intersection of nonlinear and combinatorial optimization quadratic programming has attracted significant interest over the past several decades a variety of relaxations for quadratically constrained quadratic programming qcqp can be formulated as semidefinite programs sdps the primary purpose of this paper is to present a systematic comparison of sdp relaxations for qcqp using theoretical analysis it is shown that the recently developed doubly nonnegative relaxation is equivalent to the shor relaxation when the latter is enhanced with a partial first order relaxation linearization technique these two relaxations are shown to theoretically dominate six other sdp relaxations a computational comparison reveals that the two dominant relaxations require three orders of magnitude more computational time than the weaker relaxations while providing relaxation gaps averaging 3 as opposed to gaps of up to 19 for weaker relaxations on 700 randomly generated problems with up to 60 variables an sdp relaxation derived from lagrangian relaxation after the addition of redundant nonlinear constraints to the primal achieves gaps averaging 13 in a few cpu seconds

convex relaxations for nonconvex quadratically constrained quadratic programming matrix cone decomposition and polyhedral approximation we present a decomposition approximation method for generating convex relaxations for nonconvex quadratically constrained quadratic programming qcqp we first develop a general conic program relaxation for qcqp based on a matrix decomposition scheme and polyhedral piecewise linear underestimation by employing suitable matrix cones we then show that the convex conic relaxation can be reduced to a semidefinite programming sdp problem in particular we investigate polyhedral underestimations for several classes of matrix cones including the cones of rank 1 and rank 2 matrices the cone generated by the coefficient matrices the cone of positive semidefinite matrices and the cones induced by rank 2 semidefinite inequalities we demonstrate that in general the new sdp relaxations can generate lower bounds at least as tight as the best known sdp relaxations for qcqp moreover we give examples for which tighter lower bounds can be generated by the new sdp relaxations we also report comparison results of different convex relaxation schemes for nonconvex qcqp with convex quadratic linear constraints nonconvex quadratic constraints and 0 1 constraints

on stepanov almost periodic semigroups and cosine functions of operators abstract in this paper we consider stepanov asymptotically almost periodic sp a a p mild solutions of first and second order abstract differential equations specifically we prove that a c0 semigroup is sp a a p if and only if it is asymptotically almost periodic and that a c0 cosine function of operators is sp a a p if and only if it is almost periodic

kronecker product gain shape vector quantization for multispectral and hyperspectral image coding this paper proposes a new vector quantization based vq based technique for very low bit rate encoding of multispectral images we rely on the assumption that the shape of a generic spatial block does not change significantly from band to band as is the case for high spectral resolution imagery in such a hypothesis it is possible to accurately quantize a three dimensional 3 d block composed of homologous two dimensional 2 d blocks drawn from several bands as the kronecker product of a spatial shape codevector and a spectral gain codevector with significant computation saving with respect to straight vq an even higher complexity reduction is obtained by representing each 3 d block in its minimum square error kronecker product form and by quantizing the component shape and gain vectors for the block sizes considered this encoding strategy is over 100 times more computationally efficient than unconstrained vq and over ten times more computationally efficient than direct gain shape vq the proposed technique is obviously suboptimal with respect to vq but the huge complexity reduction allows one to use much larger blocks than usual and to better exploit both the statistical and psychovisual redundancy of the image numerical experiments show fully satisfactory results whenever the shape invariance hypothesis turns out to be accurate enough as in the case of hyperspectral images in particular for a given level of complexity and image quality the compression ratio is up to five times larger than that provided by ordinary vq and also larger than that provided by other techniques specifically designed for multispectral image coding

cutting planes for optimization of convex functions over nonconvex sets we derive linear inequality characterizations for sets of the form mathop rm conv x q in mathbb r d times mathbb r q ge q x x in mathbb r d mathop rm int p where q is convex and differentiable and p subset mathbb r d we show that in several cases our characterization leads to polynomial time separation algorithms that operate in the original space of variables in particular when q is a positive definite quadratic and p is a polyhedron or an ellipsoid

a branch and cut algorithm for nonconvex quadratically constrained quadratic programming we present a branch and cut algorithm that yields in finite time a globally e optimal solution with respect to feasibility and optimality of the nonconvex quadratically constrained quadratic programming problem the idea is to estimate all quadratic terms by successive linearizations within a branching tree using reformulation linearization techniques rlt to do so four classes of linearizations cuts depending on one to three parameters are detailed for each class we show how to select the best member with respect to a precise criterion the cuts introduced at any node of the tree are valid in the whole tree and not only within the subtree rooted at that node in order to enhance the computational speed the structure created at any node of the tree is flexible enough to be used at other nodes computational results are reported that include standard test problems taken from the literature some of these problems are solved for the first time with a proof of global optimality

note on solving linear complementarity problems as jointly constrained bilinear programs abstract we investigate refinements of an existing nonconvex programming algorithm that exploits special structure of linear complementarity problems we construct a linear program having an n n working basis which is equivalent to the subproblems generated by the algorithm that generally require 3n 3n basis matrices

minimal sets of almost periodic motions les conditions de nemytskii stepanov et deysach sell pour un mouvement x sont equivalentes a la presque periodicite asymptotique de x r la presque periodicite asymptotique de x r n est pas necessaire afin que x soit un ensemble minimal de mouvements presque periodiques

asymptotic almost periodicity and motions of semigroups of operators abstract in this survey we present an overview of recent work in resolving certain questions that arise in a qualitative study of solutions to the abstract cauchy problem focusing on motions of c 0 semigroups of operators and their asymptotic behavior we address 1 the characterization of asymptotically almost periodic motions and 2 the characterization of those asymptotically almost periodic functions for which this property carries over to their integral moreover we take up the problem raised by nemytskii and stepanov as to when the limit set of a motion of a dynamical system is a minimal set of almost periodic motions

asymptotically periodic solutions of abstract differential equations abstract in this paper we introduce the concept of s asymptotically periodicity in the stepanov sense we study some properties of the space consisting of s asymptotically periodic functions and we use this notion to establish the existence of s asymptotically periodic solutions to linear and semi linear first order abstract differential equations in particular we characterize the strongly continuous semigroups of bounded linear operators defined on reflexive spaces which are s asymptotically periodic as those semigroups which are the direct sum of an periodic and a strongly stable semigroup

generalized bilinear programming part i models applications and linear programming relaxation abstract the evolution of the bilinear programming problem is reviewed and a new more general model is discussed the model involves two decision vectors and reduces to a linear program when one of the decision vectors is fixed the class of problems under consideration contains traditional bilinear programs general quadratic programs and bilinearly constrained and quadratically constrained extensions of these problems we describe how several important applications from the literature including the multiple modular design problem can be modeled as generalized bilinear programs finally we derive a linear programming relaxation that can be used as a subproblem in algorithmic solution schemes based on outer approximation and branch and bound

nonlinear separation of data via mixed 0 1 integer and linear programming abstract this paper presents a new mathematical programming based learning methodology for separation of two types of data specifically we develop a new l 1 norm error distance metric and use it to develop a mixed 0 1 integer and linear programming milp model that optimizes the interplay of user provided discriminant functions including kernel functions for support vector machines to implement a nonlinear nonconvex and or disjoint decision boundary for the best separation of data at hand with the concurrent optimization of discriminant functions the milp based learning can be used for finding the optimal and least complex classification rule for noise free data and for implementing a most robust classification rule for real life data with noise with extensive experiments on separation of two dimensional artificial datasets that are clean and noisy we graphically illustrate the aforementioned advantages of the new milp based learning methodology with experiments on real life benchmark datasets from the uc irvine repository of machine learning databases in comparison with the multisurface method and the support vector machines we demonstrate the advantage of using and concurrently optimizing more than a single discriminant function for a robust separation of real life data hence the utility of the proposed methodology in supervised learning

on convex envelopes for bivariate functions over polytopes in this paper we discuss convex envelopes for bivariate functions satisfying suitable assumptions over polytopes we first propose a technique to compute the value and a supporting hyperplane of the convex envelope over a general two dimensional polytope through the solution of a three dimensional convex subproblem with continuously differentiable constraint functions then for quadratic functions as well as for some polynomial and rational ones again satisfying suitable assumptions we show how the same computations can be carried out through the solution of a single semidefinite problem

explicit convex and concave envelopes through polyhedral subdivisions in this paper we derive explicit characterizations of convex and concave envelopes of several nonlinear functions over various subsets of a hyper rectangle these envelopes are obtained by identifying polyhedral subdivisions of the hyper rectangle over which the envelopes can be constructed easily in particular we use these techniques to derive in closed form the concave envelopes of concave extendable supermodular functions and the convex envelopes of disjunctive convex functions

globally optimal algorithms for stratified autocalibration we present practical algorithms for stratified autocalibration with theoretical guarantees of global optimality given a projective reconstruction we first upgrade it to affine by estimating the position of the plane at infinity the plane at infinity is computed by globally minimizing a least squares formulation of the modulus constraints in the second stage this affine reconstruction is upgraded to a metric one by globally minimizing the infinite homography relation to compute the dual image of the absolute conic diac the positive semidefiniteness of the diac is explicitly enforced as part of the optimization process rather than as a post processing step for each stage we construct and minimize tight convex relaxations of the highly non convex objective functions in a branch and bound optimization framework we exploit the inherent problem structure to restrict the search space for the diac and the plane at infinity to a small fixed number of branching dimensions independent of the number of views chirality constraints are incorporated into our convex relaxations to automatically select an initial region which is guaranteed to contain the global minimum experimental evidence of the accuracy speed and scalability of our algorithm is presented on synthetic and real data

a simplicial branch and bound algorithm for solving quadratically constrained quadratic programs we propose a branch and bound algorithm for solving nonconvex quadratically constrained quadratic programs the algorithm is novel in that branching is done by partitioning the feasible region into the cartesian product of two dimensional triangles and rectangles explicit formulae for the convex and concave envelopes of bilinear functions over triangles and rectangles are derived and shown to be second order cone representable the usefulness of these new relaxations is demonstrated both theoretically and computationally

deriving convex hulls through lifting and projection we consider convex hull descriptions for certain sets described by inequality constraints over a hypercube and propose a lifting and projection technique to construct them the general procedure obtains the convex hulls as an intersection of semi infinite families of linear inequalities each derived using lifting techniques that are interpreted using convexification tools we demonstrate that differentiability and concavity of certain perturbation functions help reduce the number of inequalities needed for this characterization each family of inequalities yields a few linear nonlinear constraints fully characterized in the space of the original problem variables when the projection problems are amenable to a closed form solution in particular we illustrate the complete procedure by constructing the convex hulls of the subsets of a compact hypercube defined by the constraints x 1 b 1 x 2 b 2 ge x 3 and x 1 x 2 b 2 le x 3 where b 1 b 2 ge 1 as a consequence we obtain a closed form description of the convex hull of the bilinear equality x 1x 2 x 3 in the presence of variable bounds as an intersection of a few linear and nonlinear inequalities this explicit characterization hitherto unknown can improve relaxation techniques for factorable functions which utilize this equality to relax products of functions with known relaxations

data separation via a finite number of discriminant functions a global optimization approach abstract this paper presents a mixed 0 1 integer and linear programming milp model for separation of data via a finite number of non linear and non convex discriminant functions the milp model concurrently optimizes the parameters of the user provided individual discriminant functions to implement a decision boundary for an optimal separation of data under analysis the performance of the milp based classification of data is illustrated on randomly generated two dimensional datasets and extensively tested on six well studied datasets in data mining research in comparison with three well established supervised learning methodologies namely the multisurface method the logical analysis of data and the support vector machines numerical results from these experiments show that the new milp based classification of data is an effective and useful methodology for supervised learning

existence and uniqueness of pseudo almost periodic solutions to some abstract partial neutral functional differential equations and applications the paper considers the existence and uniqueness of pseudo almost periodic mild solutions to some classes of first order partial neutral functional differential equations upon making some suitable assumptions existence and uniqueness results are obtained applications include both a partial integro differential equation arising in control systems and the scalar reaction diffusion equation with delay

error bounds for monomial convexification in polynomial optimization convex hulls of monomials have been widely studied in the literature and monomial convexifications are implemented in global optimization software for relaxing polynomials however there has been no study of the error in the global optimum from such approaches we give bounds on the worst case error for convexifying a monomial over subsets of this implies additive error bounds for relaxing a polynomial optimization problem by convexifying each monomial separately our main error bounds depend primarily on the degree of the monomial making them easy to compute since monomial convexification studies depend on the bounds on the associated variables in the second part we conduct an error analysis for a multilinear monomial over two different types of box constraints as part of this analysis we also derive the convex hull of a multilinear monomial over

lower bound functions for polynomials relaxation techniques for solving nonlinear systems and global optimisation problems require bounding from below the nonconvexities that occur in the constraints or in the objective function by affine or convex functions in this paper we consider such lower bound functions in the case of problems involving multivariate polynomials they are constructed by using bernstein expansion an error bound exhibiting quadratic convergence in the univariate case and some numerical examples are given

nonexistence of periodic solutions and s asymptotically periodic solutions in fractional difference equations nonexistence of periodic solution is proved for periodic right hand side existence of s asymptotically periodic solutions is discussed existence and uniqueness results are obtained for s asymptotically periodic solution nontrivial estimations of a ratio of gamma functions are used in this paper it is shown that a fractional difference equation with a periodic right hand side can not possess a periodic solution but it can have an s asymptotically periodic solution sufficient conditions are proved for the existence of a unique s asymptotically periodic solution an example is also given illustrating the obtained results

existence of s asymptotically ω periodic solutions for fractional order functional integro differential equations with infinite delay abstract we study s asymptotically periodic solutions of the abstract fractional equation u 1 a u f t u t 1 2 where a is a linear operator of sectorial type 0

global optimization of mixed integer quadratically constrained quadratic programs miqcqp through piecewise linear and edge concave relaxations we propose a deterministic global optimization approach whose novel contributions are rooted in the edge concave and piecewise linear underestimators to address nonconvex mixed integer quadratically constrained quadratic programs miqcqp to epsilon global optimality the facets of low dimensional n 3 edge concave aggregations dominating the termwise relaxation of miqcqp are introduced at every node of a branch and bound tree concave multivariable terms and sparsely distributed bilinear terms that do not participate in connected edge concave aggregations are addressed through piecewise linear relaxations extensive computational studies are presented for point packing problems standard and generalized pooling problems and examples from globallib meeraus globallib http www gamsworld org global globallib htm

s asymptotically ω positive periodic solutions for a class of neutral fractional differential equations in this paper we investigate the existence of the s asymptotically positive periodic solutions to a class of semilinear neutral caputo fractional differential equations with infinite delay given by d t x t f t x t a x t g t x t t 0 x 0 b the function is considered in a banach space x for 0 1 here a denotes the infinitesimal generator of an analytic semigroup t t t 0

jointly constrained bilinear programs and related problems an overview abstract linear complementarity quadratic programming and separably constrained bilinear programming problems are all special cases of jointly constrained bilinear programs in this paper we review current theory and procedures for bilinear programs and related problems the key notion of convex envelopes that drives the general procedure is explored and some minor extensions to known results are presented extensions and directions for future research are also discussed

design of planar articulated mechanisms using branch and bound this paper considers an optimization model and a solution method for the design of two dimensional mechanical mechanisms the mechanism design problem is modeled as a nonconvex mixed integer program which allows the optimal topology and geometry of the mechanism to be determined simultaneously the underlying mechanical analysis model is based on a truss representation allowing for large displacements for mechanisms undergoing large displacements elastic stability is of major concern we derive conditions modeled by nonlinear matrix inequalities which guarantee that a stable equilibrium is found and that buckling is prevented the feasible set of the design problem is described by nonlinear differentiable and non differentiable constraints as well as nonlinear matrix inequalities to solve the mechanism design problem a branch and bound method based on convex relaxations is developed to guarantee convergence of the method two different types of convex relaxations are derived the relaxations are strengthened by adding valid inequalities to the feasible set and by solving bound contraction sub problems encouraging computational results indicate that the branch and bound method can reliably solve mechanism design problems of realistic size to global optimality

jointly constrained biconvex programming this paper presents a branch and bound algorithm for minimizing the sum of a convex function in x a convex function in y and a bilinear term in x and y over a closed set such an objective function is called biconvex with biconcave functions similarly defined the feasible region of this model permits joint constraints in x and y to be expressed the bilinear programming problem becomes a special case of the problem addressed in this paper we prove that the minimum of a biconcave function over a nonempty compact set occurs at a boundary point of the set and not necessarily an extreme point the algorithm is proven to converge to a global solution of the nonconvex program we discuss extensions of the general model and computational experience in solving jointly constrained bilinear programs for which the algorithm has been implemented

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

a polyhedral study of binary polynomial programs we study the polyhedral convex hull of a mixed integer set x1d4ae defined by a collection of multilinear equations over the unit hypercube such sets appear frequently in the factorable reformulation of mixed integer nonlinear optimization problems in particular the set x1d4ae represents the feasible region of a linearized unconstrained binary polynomial optimization problem we define an equivalent hypergraph representation of the mixed integer set x1d4ae which enables us to derive several families of facet defining inequalities structural properties and lifting operations for its convex hull in the space of the original variables our theoretical developments extend several well known results from the boolean quadric polytope and the cut polytope literature paving a way for devising novel optimization algorithms for nonconvex problems containing multilinear sub expressions

vector predictive coding of speech at 16 kbits s vector quantization in its simplest form may be regarded as a generalization of pcm independent quantization of each sample of a waveform to what might be called vector pcm where a block of consecutive samples a vector is simultaneously quantized as one unit in theory a performance arbitrarily close to the ultimate rate distortion limit is achievable with waveform vector quantization if the dimension of the vector k is large enough the main obstacle in effectively using vector quantization is complexity a vector quantizer of dimension k operating at a rate of r bits sample requires a number of computations on the order of k2 kr and a memory of the same order however a low dimensional vector quantizer dimensions 4 8 achieves a remarkable improvement over scalar quantization pcm consequently using the vector quantizer as a building block and imbedding it with other waveform data compression techniques may lead to the development of a new and powerful class of waveform coding systems this paper proposes and analyzes a waveform coding system adaptive vector predictive coding avpc in which a low dimensionality vector quantizer is used in an adaptive predictive coding scheme in the encoding process a locally generated prediction of the current input vector is subtracted from the current vector and the resulting error vector is coded by a vector quantizer each frame consisting of many vectors is classified into one of m statistical types this classification determines which one of m fixed predictors and of m vector quantizers will be used for encoding the current frame

methods for global concave minimization a bibliographic survey the global concave minimization problem is to find the constrained global minimum of a concave function since such a function may have many local minima finding the global minimum is a computationally difficult problem in this bibliographic survey which includes most of the recent papers on constrained global concave minimization we have attempted to briefly summarize the main ideas in each paper these recent papers include those concerned with large scale global concave minimization and bilinear programming

generalized cost measure based address predictive vector quantization address predictive vector quantization apvq exploits the interblock dependency by jointly encoding the addresses of the codewords associated with spatially close blocks it profiles the same image quality as memoryless vq for a much lesser bit rate br and the same computational complexity in the generalized cost measure based apvq the two steps of the encoding process namely vq and predictive address encoding are carried out jointly by minimizing a generalized cost measure which takes into account both the br and the distortion computer simulations show that a significant improvement can be obtained with respect to apvq in terms of both br and distortion compared with memoryless vq a bit rate reduction of almost 60 is obtained for the same image quality

some results on the strength of relaxations of multilinear functions we study approaches for obtaining convex relaxations of global optimization problems containing multilinear functions specifically we compare the concave and convex envelopes of these functions with the relaxations that are obtained with a standard relaxation approach due to mccormick the standard approach reformulates the problem to contain only bilinear terms and then relaxes each term independently we show that for a multilinear function having a single product term this approach yields the convex and concave envelopes if the bounds on all variables are symmetric around zero we then review and extend some results on conditions when the concave envelope of a multilinear function can be written as a sum of concave envelopes of its individual terms finally for bilinear functions we prove that the difference between the concave upper bounding and convex lower bounding functions obtained from the mccormick relaxation approach is always within a constant of the difference between the concave and convex envelopes these results along with numerical examples we provide give insight into how to construct strong relaxations of multilinear functions

linear programming based converses for finite blocklength lossy joint source channel coding a linear programming lp based framework is presented for obtaining converses for finite blocklength lossy joint source channel coding problems the framework applies for any loss criterion generalizes certain previously known converses and also extends to multi terminal settings the finite blocklength problem is posed equivalently as a nonconvex optimization problem and using a lift and project like method a close but tractable lp relaxation of this problem is derived lower bounds on the original problem are obtained by the construction of feasible points for the dual of the lp relaxation a particular application of this approach leads to new converses which recover and improve on the converses of kostina and verdu for finite blocklength lossy joint source channel coding and lossy source coding for finite blocklength channel coding the lp relaxation recovers the converse of polyanskiy poor and verdu and leads to a new improvement on the converse of wolfowitz showing thereby that our lp relaxation is asymptotically tight with increasing blocklengths for channel coding lossless source coding and joint source channel coding with the excess distortion probability as the loss criterion using a duality based argument a new converse is derived for finite blocklength joint source channel coding for a class of source channel pairs employing this converse the lp relaxation is also shown to be tight for all blocklengths for the minimization of the expected average symbolwise hamming distortion of a q ary uniform source over a q ary symmetric memoryless channel for any q in mathbb n the optimization formulation and the lift and project method are extended to networked settings and demonstrated by obtaining an improvement on a converse of zhou et al for the successive refinement problem for successively refinable source distortion measure triplets

on s asymptotically ω periodic functions on banach spaces and applications abstract this paper is devoted to the study of the class of continuous and bounded functions f 0 x for which exists 0 such that lim t f t f t 0 in the sequel called s asymptotically periodic functions we discuss qualitative properties and establish some relationships between this type of functions and the class of asymptotically periodic functions we also study the existence of s asymptotically periodic mild solutions of the first order abstract cauchy problem in banach spaces

pseudo s asymptotically periodic solutions of second order abstract cauchy problems in this work we discuss the existence of pseudo s asymptotically periodic mild solutions for a second order abstract cauchy problem in particular we show that pseudo s asymptotically periodic cosine functions of operators are periodic the paper is completed with an application to the nonautonomous wave equation

efficient and safe global constraints for handling numerical constraint systems numerical constraint systems are often handled by branch and prune algorithms that combine splitting techniques local consistencies and interval methods this paper first recalls the principles of tt quad a global constraint that works on a tight and safe linear relaxation of quadratic subsystems of constraints then it introduces a generalization of tt quad to polynomial constraint systems it also introduces a method to get safe linear relaxations and shows how to compute safe bounds of the variables of the linear constraint system different linearization techniques are investigated to limit the number of generated constraints tt quadsolver a new branch and prune algorithm that combines tt quad local consistencies and interval methods is introduced tt quadsolver has been evaluated on a variety of benchmarks from kinematics mechanics and robotics on these benchmarks it outperforms classical interval methods as well as constraint satisfaction problem solvers and it compares well with state of the art optimization solvers

global optimization algorithms for linearly constrained indefinite quadratic problems abstract in this paper we consider the problem of finding the constrained global optimum of an indefinite quadratic function since such a function may have many local optima finding the global optimum is a computationally difficult problem we give an overview of the most important methods used which include benders decomposition concave programming approaches enumerative techniques and bilinear programming

linear quadratic and bilinear programming approaches to the linear complementarity problem abstract traditional pivoting procedures for solving the linear complementarity problem can only guarantee convergence for problems having well defined structures recently optimization procedures based on linear quadratic and bilinear programming have been developed to extend the class of problems that can be solved efficiently these latter approaches are the focus of this paper the strengths and weaknesses of each of the approaches are discussed the linear programming approach advanced by mangasarian is the most efficient once an appropriate objective function is found this requires the solution of a system of linear and bilinear equations that is easily solvable only in some cases extensions to this approach due to shiau show some promise but are still limited to special cases of the general problem the quadratic programming approaches discussed here are restricted to specialized procedures for the complementarity problem one proposed by cheng is based on the levitin poljak gradient projection method and the other due to cirina is based on karush kuhn tucker theory both are only successful on some problems the two bilinear programming algorithms discussed are the most general for any problem they are guaranteed to find at least one solution or conclude that none exist one is a specialization of the recent biconvex programming algorithm of al khayyal and falk and the other is an entirely new implicit enumeration procedure

computationally efficient identification of global arx parameters with guaranteed stability identification of stable parametric models from input output data of a process stable is an essential task in system identification for a stable process the identified parametric model may be unstable due to one or more of the following reasons 1 presence of noise in the measurements 2 plant disturbances 3 finite sample effects 4 over under modeling of the process and 5 nonlinear distortions therefore it is essential to impose stability conditions on the parameters during model estimation in this technical note we develop a computationally efficient approach for the identification of global arx parameters with guaranteed stability the computational advantage of the proposed approach is derived from the fact that a series of computationally tractable quadratic programming qp problems are solved to identify the globally optimal parameters the importance of identifying globally optimal stable model parameters is high lighted through illustrative examples this does not seem to have been discussed much in the literature

weighted sp pseudo s asymptotic periodicity and applications to volterra integral equations abstract this paper is related to the function space formed by weighted sp pseudo s asymptotic periodicity and their applications initially the translation invariance and completeness of the function space are investigated additionally the composition theorem and convolution operator generated by lebesgue integrable functions are presented finally existence and uniqueness of solutions with weighted sp pseudo s asymptotic periodicity for two classes of volterra equations are proved by using the results obtained above and some concrete examples are given the methods mainly include minkowski s inequality convolution inequality contraction mapping principle and especially the generalized minkowski s inequality our results extend some known results on asymptotic periodicity

asymptotic periodicity for hyperbolic evolution equations and applications we study the existence and uniqueness of pseudo s asymptotically periodic mild solutions for hyperbolic evolution equations we show concrete applications of our methods

two stage quadratic integer programs with stochastic right hand sides we consider two stage quadratic integer programs with stochastic right hand sides and present an equivalent reformulation using value functions we propose a two phase solution approach the first phase constructs value functions of quadratic integer programs in both stages the second phase solves the reformulation using a global branch and bound algorithm or a level set approach we derive some basic properties of value functions of quadratic integer programs and utilize them in our algorithms we show that our approach can solve instances whose extensive forms are hundreds of orders of magnitude larger than the largest quadratic integer programming instances solved in the literature

asymptotic periodicity for some evolution equations in banach spaces abstract this work deals with the existence and uniqueness of pseudo almost periodic and asymptotically periodic mild solutions to some evolution equations in banach spaces

a study of vector quantization for noisy channels several issues related to vector quantization for noisy channels are discussed an algorithm based on simulated annealing is developed for assigning binary codewords to the vector quantizer code vectors it is shown that this algorithm could result in dramatic performance improvements as compared to randomly selected codewords a modification of the simulated annealing algorithm for binary codeword assignment is developed for the case where the bits in the codeword are subjected to unequal error probabilities resulting from unequal levels of error protection an algorithm for the design of an optimal vector quantizer for a noisy channel is briefly discussed and its robustness under channel mismatch conditions is studied numerical results for a stationary first order gauss markov source and a binary symmetric channel are provided it is concluded that the channel optimized vector quantizer design algorithm if used carefully can result in a fairly robust system with no additional delay the case in which the communication channel is nonstationary as in mobile radio channels is studied and some preliminary ideas for quantizer design are presented

pseudo gray coding a pseudo gray code is an assignment of n bit binary indexes to 2 points in a euclidean space so that the hamming distance between two points corresponds closely to the euclidean distance pseudo gray coding provides a redundancy free error protection scheme for vector quantization vq of analog signals when the binary indexes are used as channel symbols on a discrete memoryless channel and the points are signal codevectors binary indexes are assigned to codevectors in a way that reduces the average quantization distortion introduced in the reproduced source vectors when a transmitted index is corrupted by channel noise a globally optimal solution to this problem is generally intractable due to an inherently large computational complexity a locally optimal solution the binary switching algorithm is introduced based on the objective of minimizing a useful upper bound on the average system distortion the algorithm yields a significant reduction in average distortion and converges in reasonable running times the sue of pseudo gray coding is motivated by the increasing need for low bit rate vq based encoding systems that operate on noisy channels such as in mobile radio speech communications

outer approximation algorithms for separable nonconvex mixed integer nonlinear programs a rigorous decomposition approach to solve separable mixed integer nonlinear programs where the participating functions are nonconvex is presented the proposed algorithms consist of solving an alternating sequence of relaxed master problems mixed integer linear program and two nonlinear programming problems nlps a sequence of valid nondecreasing lower bounds and upper bounds is generated by the algorithms which converge in a finite number of iterations a primal bounding problem is introduced which is a convex nlp solved at each iteration to derive valid outer approximations of the nonconvex functions in the continuous space two decomposition algorithms are presented in this work on finite termination the first yields the global solution to the original nonconvex minlp and the second finds a rigorous bound to the global solution convergence and optimality properties and refinement of the algorithms for efficient implementation are presented finally numerical results are compared with currently available algorithms for example problems illuminating the potential benefits of the proposed algorithm

solving mixed integer bilinear problems using milp formulations in this paper we examine a mixed integer linear programming reformulation for mixed integer bilinear problems where each bilinearterm involves the product of a nonnegative integer variable and a nonnegative continuous variable this reformulation is obtained by first replacing a general integer variable with its binary expansion and then using mccormick envelopes to linearize the resulting product of continuous and binary variables we present the convex hull of the underlying mixed integer linear set the effectiveness of this reformulation and associated facet defining inequalities are computationally evaluated on five classes of instances

global optimization of mixed integer nonlinear programs a theoretical and computational study this work addresses the development of an efficient solution strategy for obtaining global optima of continuous integer and mixed integer nonlinear programs towards this end we develop novel relaxation schemes range reduction tests and branching strategies which we incorporate into the prototypical branch and bound algorithm in the theoretical algorithmic part of the paper we begin by developing novel strategies for constructing linear relaxations of mixed integer nonlinear programs and prove that these relaxations enjoy quadratic convergence properties we then use lagrangian linear programming duality to develop a unifying theory of domain reduction strategies as a consequence of which we derive many range reduction strategies currently used in nonlinear programming and integer linear programming this theory leads to new range reduction schemes including a learning heuristic that improves initial branching decisions by relaying data across siblings in a branch and bound tree finally we incorporate these relaxation and reduction strategies in a branch and bound algorithm that incorporates branching strategies that guarantee finiteness for certain classes of continuous global optimization problems in the computational part of the paper we describe our implementation discussing wherever appropriate the use of suitable data structures and associated algorithms we present computational experience with benchmark separable concave quadratic programs fractional 0 1 programs and mixed integer nonlinear programs from applications in synthesis of chemical processes engineering design just in time manufacturing and molecular design

generalized bilinear programming an application in farm management abstract in this paper we analyze a model which arises from considering manure and fertilizer aspects in the farm management modelling for cattle husbandry modelling these aspects may lead to a multi extremal problem with bilinear terms solution techniques for this problem are discussed and a specific branch and bound procedure is outlined an example illustrates the multi extremal structure of the problem and the solution methods mentioned

compression of multispectral images by spectral classification and transform coding this paper presents a new technique for the compression of multispectral images which relies on the segmentation of the image into regions of approximately homogeneous land cover the rationale behind this approach is that within regions of the same land cover the pixels have stationary statistics and are characterized by mostly linear dependency contrary to what usually happens for unsegmented images therefore by applying conventional transform coding techniques to homogeneous groups of pixels the proposed algorithm is able to effectively exploit the statistical redundancy of the image thereby improving the rate distortion performance the proposed coding strategy consists of three main steps first each pixel is classified by vector quantizing its spectral response vector so that both a reliable classification and a minimum distortion encoding of each vector are obtained then the classification map is entropy encoded and sent as side information finally the residual vectors are grouped according to their classes and undergo karhunen loeve transforming in the spectral domain and discrete cosine transforming in the spatial domain numerical experiments on a six band thematic mapper image show that the proposed technique outperforms the conventional transform coding technique by 1 to 2 db at all rates of interest

lifted inequalities for 0 mathord 1 mixed integer bilinear covering sets in this paper we study 0 mathord 1 mixed integer bilinear covering sets we derive several families of facet defining inequalities via sequence independent lifting techniques we then show that these sets have a polyhedral structure that is similar to that of a certain fixed charge single node flow set as a result we also obtain new facet defining inequalities for the single node flow set that generalize well known lifted flow cover inequalities from the integer programming literature

simultaneous convexification of bilinear functions over polytopes with application to network interdiction we study the simultaneous convexification of graphs of bilinear functions g k x y y intercal a k x over x in ensuremath xi x in 0 1 n e x geq f and y in delta m y in r m 1 intercal y leq 1 we propose a constructive procedure to obtain a linear description of the convex hull of the resulting set this procedure can be applied to derive convex and concave envelopes of certain bilinear functions to study unary expansions of integer variables in mixed integer bilinear sets and to obtain convex hulls of sets with complementarity constraints exploiting the structure of xi the procedure naturally yields stronger linearizations for bilinear terms in a variety of practical settings in particular we demonstrate the effectiveness of the approach by strengthening the traditional dual formulation of network interdiction problems and report encouraging preliminary numerical results

asymptotically almost periodic solutions of abstract differential equations abstract in this paper we are concerned with the problem of existence and uniqueness of asymptotically almost periodic mild solutions of nonhomogeneous first order abstract differential equations essentially we obtain existence of asymptotically almost periodic solutions when the semigroup is asymptotically almost periodic or when the forcing term is an asymptotically stepanov almost periodic function

a new framework to relax composite functions in nonlinear programs in this paper we devise new relaxations for composite functions which improve the prevalent factorable relaxations without introducing additional variables by exploiting the inner function structure we outer approximate inner functions using arbitrary under and over estimators and then convexify the outer function over a polytope p which models the ordering relationships between the inner functions and their estimators and utilizes bound information on the inner functions as well as on the estimators we show that there is a subset q of p with significantly simpler combinatorial structure such that the separation problem of the graph of the outer function over p is polynomially equivalent via a fast combinatorial algorithm to that of its graph over q we specialize our study to consider the product of two inner functions with one non trivial underestimator for each inner function for the corresponding polytope p we show that there are eight valid inequalities besides the four mccormick inequalities which improve the factorable relaxation finally we show that our results generalize to simultaneous convexification of a vector of outer functions

jointly constrained semidefinite bilinear programming with an application to dobrushin curves we propose a branch and bound algorithm for minimizing a bilinear functional of the form f x y mathrm tr x otimes y q mathrm tr ax mathrm tr by of pairs of hermitian matrices x y restricted by joint semidefinite programming constraints the functional is parametrized by self adjoint matrices q a and b this problem generalizes that of a bilinear program where x and y belong to polyhedra the algorithm converges to a global optimum and yields upper and lower bounds on its value in every step various problems in quantum information theory can be expressed in this form as an example application we compute dobrushin curves of quantum channels giving upper bounds on classical coding with energy constraints

a note on the solution of bilinear programming problems by reduction to concave minimization we present a new algorithm for solving the bilinear programming problem by reduction to concave minimization this algorithm is finite does not assume the boundedness of the constraint set and uses an efficient procedure for checking whether a concave function is bounded below on a given halfline some preliminary computational experience with a computer code for implementing the algorithm on a microcomputer is also reported

quasiconvex relaxations based on interval arithmetic abstract interval analysis provides a tool for i forward error analysis ii estimating and controlling rounding and approximation errors automatically and iii proving existence and uniqueness of solutions in this context the terms self validating methods inclusion methods orverification methods are in use in this paper we present a new self validating method for solving global constrained optimization problems this method is based on the construction of quasiconvex lower bound and quasiconcave upper bound functions of a given function the latter defined by an arithmetical expression no further assumptions about the nonlinearities of the given function are necessary these lower and upper bound functions are rigorous by using the tools of interval arithmetic in its easiest form they are constructed by taking appropriate linear and or quadratical estimators which yield quasiconvex quasiconcave bound functions we show how these bound functions can be used to define rigorous quasiconvex relaxations for constrained global optimization problems and nonlinear systems these relaxations can be incorporated in a branch and bound framework yielding a self validating method

on quasi periodic properties of fractional sums and fractional differences of periodic functions this article is devoted to the study of discrete fractional calculus our goal is to investigate quasi periodic properties of fractional order sums and differences of periodic functions using riemann liouville and caputo type definitions we study concepts close to the well known idea of periodic function such as asymptotically periodicity or s asymptotically periodicity we use basic tools of discrete fractional calculus boundedness of sums and differences of fractional order of a given periodic function is also investigated