a branch and cut algorithm for scheduling of projects with variable intensity activities in this paper we study a resource constrained project scheduling problem in which the resource usage of each activity may vary over time proportionally to its varying intensity we formalize the problem by means of a mixed integer linear program prove that feasible solution existence is np complete in the strong sense and propose a branch and cut algorithm for finding optimal solutions to this end we provide a complete description of the polytope of feasible intensity assignments to two variable intensity activities connected by a precedence constraint along with a fast separation algorithm a computational evaluation confirms the effectiveness of our method on various benchmark instances

solving multi item lot sizing problems with an mip solver using classification and reformulation based on research on the polyhedral structure of lot sizing models over the last 20 years we claim that there is a nontrivial fraction of practical lot sizing problems that can now be solved by nonspecialists just by taking an appropriate a priori reformulation of the problem and then feeding the resulting formulation into a commercial mixed integer programming solver this claim uses the fact that many multi item problems decompose naturally into a set of single item problems with linking constraints and that there is now a large body of knowledge about single item problems to put this knowledge to use we propose a classification of lot sizing problems in large part single item and then indicate in a set of tables what is known about a particular problem class and how useful it might be specifically we indicate for each class i whether a tight extended formulation is known and its size ii whether one or more families of valid inequalities are known defining the convex hull of solutions and the complexity of the corresponding separation algorithms and iii the complexity of the corresponding optimization algorithms which would be useful if a column generation or lagrangian relaxation approach was envisaged three distinct multi item lot sizing instances are then presented to demonstrate the approach and comparative computational results are presented finally we also use the classification to point out what appear to be some of the important open questions and challenges

wavelets and filter banks theory and design the wavelet transform is compared with the more classical short time fourier transform approach to signal analysis then the relations between wavelets filter banks and multiresolution signal processing are explored a brief review is given of perfect reconstruction filter banks which can be used both for computing the discrete wavelet transform and for deriving continuous wavelet bases provided that the filters meet a constraint known as regularity given a low pass filter necessary and sufficient conditions for the existence of a complementary high pass filter that will permit perfect reconstruction are derived the perfect reconstruction condition is posed as a bezout identity and it is shown how it is possible to find all higher degree complementary filters based on an analogy with the theory of diophantine equations an alternative approach based on the theory of continued fractions is also given these results are used to design highly regular filter banks which generate biorthogonal continuous wavelet bases with symmetries

finite sums of products of functions in single variables abstract a characterization is given of smooth functions h of k variables that admit decompositions into a finite sum of products of k functions of single variables only this sufficient and necessary condition is given in terms of certain ordinary linear differential equations formed from a given h which also serve for the construction of the functions in single variables occuring in such a decomposition conditions for the existence of special decompositions are also given including the case of three variables when h x y t n i 1 i x g i y h i t

l stability for linear multiresolution algorithms a new explicit approach part iii the 2 d case abstract this paper is devoted to the stability in the infinity norm for the mallat s multiresolution transform in 2 d explicit error bounds are presented by using some appropriate reformulations of the successive convolutions some contraction properties and some alternative normalizations the two new things in our study is the use of only basic rules and the derivation of explicit bounds in particular it can be understood for a wide part of the scientific community moreover the results should be useful in mathematical medical physical biological and engineering applications

harmonic analysis of signed ruelle transfer operators abstract motivated by wavelet analysis we prove that there is a one to one correspondence between the following data i solutions to r h h where r is a certain non positive ruelle transfer operator ii operators that intertwine a certain class of representations of the c algebra a n on two unitary generators u v subject to the relation uvu 1 v n this correspondence enables us to give a criterion for the biorthogonality of a pair of scaling functions and calculate all solutions of the equation r h h in some concrete cases

aggregation and mixed integer rounding to solve mips in this paper we discuss the use of mixed integer rounding mir inequalities to solve mixed integer programs mir inequalities are essentially gomory mixed integer cuts however as we wish to use problem structure we insist that mir inequalities be generated from constraints or simple aggregations of constraints of the original problem this idea is motivated by the observation that several strong valid inequalities based on specific problem structure can be derived as mir inequalities here we present and test a separation routine for such mir inequalities that includes a heuristic row aggregation procedure to generate a single knapsack plus continuous variables constraint complementation of variables and finally the generation of an mir inequality inserted in a branch and cut system the results suggest that such a routine is a useful additional tool for tackling a variety of mixed integer programming problems

facets and algorithms for capacitated lot sizing the dynamic economic lot sizing model which lies at the core of numerous production planning applications is one of the most highly studied models in all of operations research and yet capacitated multi item versions of this problem remain computationally elusive we study the polyhedral structure of an integer programming formulation of a single item capacitated version of this problem and use these results to develop solution methods for multi item applications in particular we introduce a set of valid inequalities for the problem and show that they define facets of the underlying integer programming polyhedron computational results on several single and multiple product examples show that these inequalities can be used quite effectively to develop an efficient cutting plane branch and bound procedure moreover our results show that in many instances adding certain of these inequalities a priori to the problem formulation and avoiding the generation of cutting planes can be equally effective

refinement type equations and grincevicjus series abstract we consider l 1 solutions of the following refinement type equations f x n z c n 1 f k x n n z c n 1 f k x n where k 2 is an integer and for all n z reals c n 1 c n 1 are non negative with n z c n 1 c n 1 k and n z log n c n 1 c n 1 necessary and sufficient conditions for the existence of non trivial l 1 solutions in several special cases are given

geometry of pdes i integral bordism groups in pdes we improve some our previous theorems on the calculation of integral bordism groups of formally integrable and completely integrable pdes emphasizing the role played by singular solutions and weak solutions some applications to interesting pdes defined on finite dimensional manifolds are also considered

infinite convolution products and refinable distributions on lie groups sufficient conditions for the convergence in distribution of an infinite convolution product p1 p2 of measures on a connected lie group 5 with respect to left invariant haar measure are derived these conditions are used to construct distributions that satisfy to where t is a refinement operator constructed from a measure t and a dilation automorphism a the existence of a implies 5 is nilpotent and simply connected and the exponential map is an analytic homeomorphism furthermore there exists a unique minimal compact subset ic c 5 such that for any open set u containing ic and for any distribution f on g with compact support there exists an integer n u f such that n n u f implies supp tnf c u if p is supported on an a invariant uniform subgroup r then t is related by an intertwining operator to a transition operator w on c r necessary and sufficient conditions for tnf to converge to e l2 and for the f translates of to be orthogonal or to form a riesz basis are characterized in terms of the spectrum of the restriction of w to functions supported on q k c cn r

wavelets properties and approximate solution of a second kind integral equation abstract in this paper we show that under some conditions a wavelet basis of l 2 r can be used as a tool for the uniform approximation in the space c r l 2 r 0 where c r denotes the holder space of exponent as a result of this property we give a numerical application of wavelets this application is a wavelet based method for the numerical solution of a redholm equation of the second kind with solution lying in c 0 r the holder space of compactly supported functions with holder exponent 1 2

subset coefficient reduction cuts for 0 1 mixed integer programming we describe a method for generating cuts for mixed integer 0 1 programs these cuts are designed to tighten an integer program prior to applying linear programming based branch and bound algorithms the method involves two basic ideas subset selection and coefficient reduction coefficient reduction is a process of reducing the coefficients of the 0 1 variables subset selection is combined with coefficient reduction by applying the coefficient reduction process to the coefficients of a subset of variables from the constraints in the problem formulation the paper exploits these two simple ideas to derive a broad class of cuts for integer programs with both 0 1 and continuous variables it also reports on the use of this methodology in solving a variety of fixed charge problems

orthogonal and biorthogonal fir hexagonal filter banks with sixfold symmetry recently hexagonal image processing has attracted attention the hexagonal lattice has several advantages in comparison with the rectangular lattice the conventionally used lattice for image sampling and processing for example a hexagonal lattice needs less sampling points it has better consistent connectivity it has higher symmetry and its structure is plausible to human vision systems the multiresolution analysis method has been used for hexagonal image processing since the hexagonal lattice has high degree of symmetry it is desirable that the hexagonal filter banks designed for multiresolution hexagonal image processing also have high order of symmetry which is pertinent to the symmetry structure of the hexagonal lattice the orthogonal or prefect reconstruction pr hexagonal filter banks that are available in the literature have only threefold symmetry in this paper we investigate the construction of orthogonal and pr finite impulse response fir hexagonal filter banks with sixfold symmetry we obtain block structures of 7 size refinement seven channel two dimensional orthogonal and pr fir hexagonal filter banks with sixfold rotational symmetry radic7 refinement orthogonal and biorthogonal wavelets based on these block structures are constructed in this paper we also consider fir hexagonal filter banks with axial line symmetry and we present a block structure of fir hexagonal filter banks with pseudo sixfold axial symmetry

biorthogonal spline type wavelets let f be an orthonormal scaling function with approximation degree p 1 and let b nbe the b spline of order n then spline type scaling functions defined by n b n n 1 2 possess higher approximation order p n 1 and compact support the corresponding biorthogonal wavelet functions are also constructed this technique is extended to the case of biorthogonal scaling function system as an application of the method supplied in this paper one can easily construct a sequence of pairs of biorthogonal spline type scaling functions from one pair of biorthogonal scaling functions or an orthonormal scaling function in particular if both the method and the lifting scheme of sweldens see 1 are applied then all pairs of biorthogonal spline type scaling functions shown in references 2 and 3 can be constructed from the haar scaling function

theoretical challenges towards cutting plane selection while many classes of cutting planes are at the disposal of integer programming solvers our scientific understanding is far from complete with regards to cutting plane selection i e the task of selecting a portfolio of cutting planes to be added to the lp relaxation at a given node of the branch and bound tree in this paper we review the different classes of cutting planes available known theoretical results about their relative strength important issues pertaining to cut selection and discuss some possible new directions to be pursued in order to accomplish cutting plane selection in a more principled manner finally we review some lines of work that we undertook to provide a preliminary theoretical underpinning for some of the issues related to cut selection

convergence of nonstationary cascade algorithms a nonstationary multiresolution of l 2 mathbb r s is generated by a sequence of scaling functions phi k in l 2 mathbb r s k in mathbb z we consider phi k that is the solution of the nonstationary refinement equations phi k m sum j h k 1 j phi k 1 m cdot j k in mathbb z where h k is finitely supported for each k and m is a dilation matrix we study various forms of convergence in l 2 mathbb r s of the corresponding nonstationary cascade algorithm phi k n m sum j h k 1 j phi k 1 n 1 m cdot j as k or n tends to infty it is assumed that there is a stationary refinement equation at infty with filter sequence h and that sum k h k j h j infty the results show that the convergence of the nonstationary cascade algorithm is determined by the spectral properties of the transition operator associated with h

a geometric approach of the generalized d alembert equation abstract by using a geometric approach we prove that the set of solutions of the generalized d alembert equation n log f x 1 x n 0 considered in domain of the x1 xn space r n is larger than the set of the functions that can be represented in the form as f x1 xn f1 x2 xn fn x1 xn 1

convergence of multidimensional cascade algorithm a necessary and sufficient condition on the spectrum of the restricted transition operator is given for the convergence in l 2 bbb r d of the multidimensional cascade algorithm for any starting function phi 0 whose shifts form a partition of unity

on the extended crystal pde s stability i the n d alembert extended crystal pde s abstract our recent results on extended crystal pde s and geometric theory on pde s stability are applied to the generalized n d alembert pde s d a n n 2 we prove that these are extended crystal pde s for any n 2 for suitable n d a n becomes an extended 0 crystal pde and also a 0 crystal pde an equation having all the same smooth solutions of d a n but without unstabilities at finite time is obtained for each n 2

biorthogonal bases of compactly supported wavelets orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide we show here that under fairly general conditions exact reconstruction schemes with synthesis filters different from the analysis filters give rise to two dual riesz bases of compactly supported wavelets we give necessary and sufficient conditions for biorthogonality of the corresponding scaling functions and we present a sufficient condition for the decay of their fourier transforms we study the regularity of these biorthogonal bases we provide several families of examples all symmetric corresponding to linear phase filters in particular we can construct symmetric biorthogonal wavelet bases with arbitrarily high preassigned regularity we also show how to construct symmetric biorthogonal wavelet bases close to a nonsymmetric orthonormal basis

compactly supported orthogonal and biorthogonal sqrt 5 refinement wavelets with 4 fold symmetry recently radic5 refinement hierarchical sampling has been studied and radic5 refinement has been used for surface subdivision compared with other refinements such as the dyadic or quincunx refinement radic5 refinement has a special property that the nodes in a refined lattice form groups of five nodes with these five nodes having different x and y coordinates this special property has been shown to be very useful to represent adaptively and render complex and procedural geometry when radic5 refinement is used for multiresolution data processing radic5 refinement filter banks and wavelets are required while the construction of 2 d nonseparable bi orthogonal wavelets with the dyadic or quincunx refinement has been studied by many researchers the construction of bi orthogonal wavelets with radic5 refinement has not been investigated the main goal of this paper is to construct compactly supported orthogonal and biorthogonal wavelets with radic5 refinement in this paper we obtain block structures of orthogonal and biorthogonal radic5 refinement fir filter banks with 4 fold rotational symmetry we construct compactly supported orthogonal and biorthogonal wavelets based on these block structures

new modeling approaches for the design of local access transport area networks abstract this paper is concerned with the design of a local access and transport area lata network that arises in the context of providing a special access service by a local exchange facility the problem includes locating suitable hub facilities having adequate multiplexing capacity and routing the demand traffic from various clients end offices through their hubs or alternatively through a special point of presence node in order to minimize total costs these costs involve various traffic distribution and equipment component costs where the latter typically exhibit nonlinear economies of scale we develop a model for this problem and apply the reformulation linearization technique rlt to construct various enhanced tightened versions of the proposed model we also design a hybrid heuristic procedure that combines a limited run of an exact method with a lagrangian dual relaxation procedure extensive computational results are provided to study the relative tightness resulting from the various formulations and their effect on providing good quality feasible solutions

the multi item capacitated lot sizing problem with setup times and shortage costs we address a multi item capacitated lot sizing problem with setup times and shortage costs that arises in real world production planning problems demand cannot be backlogged but can be totally or partially lost the problem is np hard a mixed integer mathematical formulation is presented our approach in this paper is to propose some classes of valid inequalities based on a generalization of miller et al a j miller g l nemhauser m w p savelsbergh on the polyhedral structure of a multi item production planning model with setup times mathematical programming 94 2003 375 405 and marchand and wolsey h marchand l a wolsey the 0 1 knapsack problem with a single continuous variable mathematical programming 85 1999 15 33 results we also describe fast combinatorial separation algorithms for these new inequalities we use them in a branch and cut framework to solve the problem some experimental results showing the effectiveness of the approach are reported

sequence independent lifting for mixed knapsack problems with gub constraints in this paper we consider the semi continuous knapsack problem with generalized upper bound constraints on binary variables we prove that generalized flow cover inequalities are valid in this setting and under mild assumptions are facet defining inequalities for the entire problem we then focus on simultaneous lifting of pairs of variables the associated lifting problem naturally induces multidimensional lifting functions and we prove that a simple relaxation in a restricted domain is a superadditive function furthermore we also prove that this approximation is under extra requirements the optimal lifting function we then analyze the separation problem in two phases first finding a seed inequality and second select the inequality to be added in the first step we evaluate both exact and heuristic methods the second step is necessary because the proposed lifting procedure is simultaneous from where our class of lifted inequalities might contain an exponential number of these we choose a strategy of maximizing the resulting violation finally we test this class of inequalities using instances arising from electrical planning problems our tests show that the proposed class of inequalities is strong in the sense that the addition of these inequalities closes on average 57 70 of the root integrality gap and 97 70 of the relative gap while adding less than three cuts on average

l stability for linear multiresolution algorithms a new explicit approach part ii the cases of symlets coiflets biorthogonal wavelets and supercompact multiwavelets abstract in amat and moncayo s amat m moncayo l stability for linear multiresolution algorithms a new explicit approach part i the basic rules and the daubechies case appl math comput 206 1 2008 74 91 was introduced a direct procedure to obtain an explicit computation of the error bounds for the mallat s multiresolution transform associated to orthogonal wavelet filters the general stability framework was applied to the specific case of daubechies filters in this paper we extend our approach giving the bounds related to others well known and used wavelet families as symlets coiflets biorthogonal wavelets and supercompact multiwavelets

mixing mixed integer inequalities mixed integer rounding mir inequalities play a central role in the development of strong cutting planes for mixed integer programs in this paper we investigate how known mir inequalities can be combined in order to generate new strong valid inequalities given a mixed integer region s and a collection of valid base mixed integer inequalities we develop a procedure for generating new valid inequalities for s the starting point of our procedure is to consider the mir inequalities related with the base inequalities for any subset of these mir inequalities we generate two new inequalities by combining or mixing them we show that the new inequalities are strong in the sense that they fully describe the convex hull of a special mixed integer region associated with the base inequalities we discuss how the mixing procedure can be used to obtain new classes of strong valid inequalities for various mixed integer programming problems in particular we present examples for production planning capacitated facility location capacitated network design and multiple knapsack problems we also present preliminary computational results using the mixing procedure to tighten the formulation of some difficult integer programs finally we study some extensions of this mixing procedure

cutting plane proofs in polynomial space following chvatal cutting planes may be viewed as a proof system for establishing that a given system of linear inequalities has no integral solution we show that such proofs may be carried out in polynomial workspace

generalized hyers ulam rassias stability of n sesquilinear quadratic mappings on banach modules over c algebras assume that x is a left banach module over a unital c algebra a it is shown that almost every n sesquilinear quadratic mapping h xxxxx n a is an n sesquilinear quadratic mapping when h rx y z 1 z n h x ry z 1 z n h x y rz 1 z 2 z n h x y z 1 z 2 rz n rh x y z 1 z 2 z n r 0 r 1 holds for all x y z 1 z email protected x moreover we prove the generalized hyers ulam rassias stability of an n sesquilinear quadratic mapping on a left banach module over a unital c algebra

necessary and sufficient conditions for constructing orthonormal wavelet bases this paper proves a previous conjecture of the author characterizing sequences h l2 z that yield orthonormal wavelet bases of l2 r in terms of the multiplicity of the eigenvalue 1 of an operator associated to h the proof utilizes a result of cohen characterizing these sequences in terms of the real zeros of their fourier transforms the mapping from sequences to wavelets is shown to define a continuous mapping from a subset of l2 z into l2 r related conjectures are discussed

strong formulations for mixed integer programs valid inequalities and extended formulations we examine progress over the last fifteen years in finding strong valid inequalities and tight extended formulations for simple mixed integer sets lying both on the easy and hard sides of the complexity frontier most progress has been made in studying sets arising from knapsack and single node flow sets and a variety of sets motivated by different lot sizing models we conclude by citing briefly some of the more intriguing new avenues of research

lot sizing with constant batches formulation and valid inequalities we consider the classical lot sizing problem with constant production capacities lcc and a variant in which the capacity in each period is an integer multiple of some basic batch size lcb we first show that the classical dynamic program for lcc simplifies for lcb leading to an on2 min n c algorithm where n is the number of periods and c the batch size using this new algorithm we show how to formulate both problems as linear programs with on3 variables and constraints a class of so called k l s i inequalities are described for lcb which capture both the dynamic nature of the problem as well as the capacity aspects for lcb we prove that these inequalities are the only facet defining inequalities of a certain form for lcc we show that these inequalities include all the known classes of valid inequalities finally we discuss several open questions and possible extensions

orthogonal and biorthogonal sqrt 3 refinement wavelets for hexagonal data processing the hexagonal lattice was proposed as an alternative method for image sampling the hexagonal sampling has certain advantages over the conventionally used square sampling hence the hexagonal lattice has been used in many areas a hexagonal lattice allows radic3 dyadic and radic7 refinements which makes it possible to use the multiresolution multiscale analysis method to process hexagonally sampled data the radic3 refinement is the most appealing refinement for multiresolution data processing due to the fact that it has the slowest progression through scale and hence it provides more resolution levels from which one can choose this fact is the main motivation for the study of radic3 refinement surface subdivision and it is also the main reason for the recommendation to use the radic3 refinement for discrete global grid systems however there is little work on compactly supported radic3 refinement wavelets in this paper we study the construction of compactly supported orthogonal and biorthogonal radic3 refinement wavelets in particular we present a block structure of orthogonal fir filter banks with twofold symmetry and construct the associated orthogonal radic3 refinement wavelets we study the sixfold axial symmetry of perfect reconstruction biorthogonal fir filter banks in addition we obtain a block structure of sixfold symmetric radic3 refinement filter banks and construct the associated biorthogonal wavelets

polyhedral description of the integer single node flow set with constant bounds we consider the integer single node flow problem with constant lower and upper bounds on the variables this problem arises as subproblem of more general mixed integer problems such as lotsizing problems a complete description of the convex hull of the feasible solution set is given this description is based on families of valid inequalities whose coefficients are obtained from the coefficients of the facet defining inequalities for the convex hull of integer knapsack sets with two variables all the coefficients can be computed in polynomial time in the size of the input data we present polynomial separation algorithms and an extended formulation

a study of the lot sizing polytope the lot sizing polytope is a fundamental structure contained in many practical production planning problems here we study this polytope and identify facet defining inequalities that cut off all fractional extreme points of its linear programming relaxation as well as liftings from those facets we give a polynomial time combinatorial separation algorithm for the inequalities when capacities are constant we also report computational experiments on solving the lot sizing problem with varying cost and capacity characteristics

mra frame wavelets with certain regularities associated with the refinable generators of shift invariant spaces in this paper we will start the discussion with the refinable generators of the shift invariant si spaces in l2 r that possess the largest possible regularities and required vanishing moments for the pseudo scaling generators the corresponding mra frame wavelets with certain regularities are constructed in addition the stability of the refinable si spaces and the corresponding complementary spaces biorthogonality of the si spaces and the approximation property of the spaces are also discussed

a technique for the construction of compactly supported biorthogonal wavelets of l2 rn n 2 abstract in this paper a technique for the concrete construction of compactly supported biorthogonal wavelet bases of l2 rn is given this technique does not depend on the dimension n and it gives rise to non separable multidimensional wavelet bases of special interest is the study of the stability of the constructed wavelet frames it is assumed that the dilation matrices associated with the wavelet bases are given by d 2in

a family of orthonormal wavelet bases with dilation factor 4 abstract in this paper we study a method for the construction of orthonormal wavelet bases with dilation factor 4 more precisely for any integer m 0 we construct an orthonormal scaling filter m m that generates a mother scaling function m associated with the dilation factor 4 the computation of the different coefficients of m m 2 is done by the use of a simple iterative method also this work shows how this construction method provides us with a whole family of compactly supported orthonormal wavelet bases with arbitrary high regularity a first estimate of m the asymptotic regularity of m is given by m 0 25 m examples are provided to illustrate the results of this work

valid linear inequalities for fixed charge problems many problems in the operations research management science literature can be formulated with both zero one and continuous variables however the exact optimization of such mixed zero one models remains a computational challenge in this paper we propose to study mixed problems from a mathematical point of view that is similar in spirit to recent research on purely combinatorial problems that has investigated systems of defining linear inequalities or facets of the underlying polytope at least two numerical studies have validated this line of research computationally and the advances in the problem solving capabilities are considerable we expect that similar gains are possible for the mixed zero one problem more precisely we consider the mixed integer programs whose feasible region x is composed of i a simple additive constraint in the continuous variables xj for j 1 2 n and ii constraints 0 xj mjyj defined by binary variables yj for j 1 2 n this type of feasible region arises in a variety of mixed integer problems and particularly in network problems with fixed charges on the arcs we derive two classes of facet defining linear inequalities of the convex hull of x and show that the second of these classes gives a complete description of the convex hull when mj m for all j we also develop methods to detect violated inequalities from these classes so that these facets can be used as cutting planes to strengthen the formulations of certain mixed integer problems

the construction of multivariate periodic wavelet bi frames abstract this paper addresses periodic wavelet bi frames associated with general expansive matrices periodization is an important method to obtain periodic wavelets from wavelets on r d mep and moep provide us with criteria for the construction of wavelet bi frames on r d based on periodization techniques mep and moep periodic wavelet bi frames associated with the dyadic matrix have been constructed however the problem of constructing periodic wavelet bi frames associated with general expansive matrices is still open the geometry of a general expansive matrix is much more complicated than the dyadic matrix in this paper with the help of quasi norms mep and moep we construct periodic wavelet bi frames associated with general expansive matrices

bend minimization in planar orthogonal drawings using integer programming we consider the problem of minimizing the number of bends in a planar orthogonal graph drawing while the problem can be solved via network flow for a given planar embedding of a graph it is np hard if we consider all planar embeddings our approach for biconnected graphs combines a new integer linear programming ilp formulation for the set of all embeddings of a planar graph with the network flow formulation of the bend minimization problem for fixed embeddings we report on extensive computational experiments with two benchmark sets containing a total of more than 12 000 graphs where we compared the performance of our ilp based algorithm with a heuristic and a previously published branch bound algorithm for solving the same problem our new algorithm is significantly faster than the previously published approach for the larger graphs of the benchmark graphs derived from industrial applications and almost twice as fast for the benchmark graphs from the artificially generated set of hard instances

l stability for linear multiresolution algorithms a new explicit approach part i the basic rules and the daubechies case abstract this general study is motivated by recent experiments showing that a multiresolution scheme without control in the infinity norm can produce numerical artifacts this class of stability for the mallat s multiresolution transform associated to orthogonal wavelet filters that belong to the class of linear multiresolution algorithms is revisited explicit error bounds in the infinity norm are presented by using an appropriate reformulation of the successive convolutions of a vector and assuming a contraction property in the case of the decomposition an alternative normalization is necessary we apply our general stability framework to the specific case of daubechies filters the knowledge of explicit error bounds has some advantages in real problems as industry applications medical pathologies or fbi fingerprint compression our workable bounds give the level of compression necessary to recover the signal with a reconstruction error smaller than a prefixed tolerance our study presents two important aspects the first one is the fact that we give precise error bounds and the second one is that we only use basic rules that can be understood for a wide part of the scientific community moreover the results should be useful in mathematical medical physical biological and engineering applications

lifted euclidean inequalities for the integer single node flow set with upper bounds abstract in this paper we discuss the polyhedral structure of the integer single node flow set with two possible values for the upper bounds on the arc flows such mixed integer sets arise as substructures in complex mixed integer programs for real application problems this work builds on results for the integer single node flow polytope with two arcs given by agra and constantino 2006a valid inequalities are extended to a new family the lifted euclidean inequalities and a complete description of the convex hull is given all the coefficients of the facet defining inequalities can be computed in polynomial time we report on some computational experimentations for three problems an inventory distribution problem a facility location problem and a multi item production planning model

lp based approximation algorithms for capacitated facility location in the capacitated facility location problem with hard capacities we are given a set of facilities mathcal f and a set of clients mathcal d in a common metric space each facility i has a facility opening cost f i and capacity u i that specifies the maximum number of clients that may be assigned to this facility we want to open some facilities from the set mathcal f and assign each client to an open facility so that at most u i clients are assigned to any open facility i the cost of assigning client j to facility i is given by the distance c ij and our goal is to minimize the sum of the facility opening costs and the client assignment costs the only known approximation algorithms that deliver solutions within a constant factor of optimal for this np hard problem are based on local search techniques it is an open problem to devise an approximation algorithm for this problem based on a linear programming lower bound or indeed to prove a constant integrality gap for any lp relaxation we make progress on this question by giving a 5 approximation algorithm for the special case in which all of the facility costs are equal by rounding the optimal solution to the standard lp relaxation one notable aspect of our algorithm is that it relies on partitioning the input into a collection of single demand capacitated facility location problems approximately solving them and then combining these solutions in a natural way

compactly supported tight wavelet frames and orthonormal wavelets of exponential decay with a general dilation matrix tight wavelet frames and orthonormal wavelet bases with a general dilation matrix have applications in many areas in this paper for any d d dilation matrix m we demonstrate in a constructive way that we can construct compactly supported tight m wavelet frames and orthonormal m wavelet bases in l2 rd of exponential decay which are derived from compactly supported m refinable functions such that they can have both arbitrarily high smoothness and any preassigned order of vanishing moments this paper improves several results in battle comm math phys 110 1987 601 bownik j fourier anal appl 7 2001 489 grochenig and ron proc amer math soc 126 1998 1101 lemarie j math pures appl 67 1988 227 and strichartz constr approx 9 1993 327

solving mixed integer programming problems using automatic reformulation in this paper we describe computational experience in solving mixed 0 1 programming problems using strong valid inequalities as cutting planes in particular we report on the solution to optimality of 18 medium to large size problems including production planning problems with setup costs and capacity constraints multilevel distribution planning problems drainage and heating system design problems and electricity generator scheduling problems the solution approach uses the theory of strong valid inequalities that we developed in a series of earlier papers here we report specifically on the implementation of an experimental system mpsarx which consists of the sciconic mathematical programming system and an automatic reformulation executor arx that use this theory and on the results obtained with this system

valid inequalities for separable concave constraints with indicator variables we study valid inequalities for optimization models that contain both binary indicator variables and separable concave constraints these models reduce to a mixed integer linear program milp when the concave constraints are ignored or to a nonconvex global optimization problem when the binary restrictions are ignored in algorithms designed to solve these problems to global optimality cutting planes to strengthen the relaxation are traditionally obtained using valid inequalities for the milp only we propose a technique to obtain valid inequalities that are based on both the milp constraints and the concave constraints we begin by characterizing the convex hull of a four dimensional set consisting of a single binary indicator variable a single concave constraint and two linear inequalities using this analysis we demonstrate how valid inequalities for the single node flow set and for the lot sizing polyhedron can be tilted to give valid inequalities that also account for separable concave functions of the arc flows we present computational results demonstrating the utility of the new inequalities for nonlinear transportation problems and for lot sizing problems with concave costs to our knowledge this is one of the first works that simultaneously convexifies both nonconvex functions and binary variables to strengthen the relaxations of practical mixed integer nonlinear programs

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

chva tal closures for mixed integer programming problems chvatal introduced the idea of viewing cutting planes as a system for proving that every integral solution of a given set of linear inequalities satisfies another given linear inequality this viewpoint has proven to be very useful in many studies of combinatorial and integer programming problems the basic ingredient in these cutting plane proofs is that for a polyhedronp and integral vectorw if max wx x p wx integer t thenwx t is valid for all integral vectors inp we consider the variant of this step where the requirement thatwx be integer may be replaced by the requirement that bar wx be integer for some other integral vector bar w the cutting plane proofs thus obtained may be seen either as an abstraction of gomory s mixed integer cutting plane technique or as a proof version of a simple class of the disjunctive cutting planes studied by balas and jeroslow our main result is that for a given polyhedronp the set of vectors that satisfy every cutting plane forp with respect to a specified subset of integer variables is again a polyhedron this allows us to obtain a finite recursive procedure for generating the mixed integer hull of a polyhedron analogous to the process of repeatedly taking chvatal closures in the integer programming case these results are illustrated with a number of examples from combinatorial optimization our work can be seen as a continuation of that of nemhauser and wolsey on mixed integer cutting planes

eigenvalues of spl darr 2 h and convergence of the cascade algorithm this paper is about the eigenvalues and eigenvectors of spl darr 2 h the ordinary fir filter h is a convolution with a vector h h o h n which is the impulse response the operator spl darr 2 downsamples the output y h x keeping the even numbered components y 2n where h is represented by a constant diagonal matrix this is a toeplitz matrix with h k on its kth diagonal the odd numbered rows are removed in spl darr 2 h the result is a double shift between rows yielding a block toeplitz matrix with 1 spl times 2 blocks iteration of the filter is governed by the eigenvalues if the transfer function h z spl sigma h k z sup k has a zero of order p at z 1 corresponding to spl omega spl pi then spl darr 2 h has p special eigenvalues 1 2 1 4 1 2 sup p we show how each additional zero at spl pi divides all eigenvalues by 2 and creates a new eigenvector for spl lambda 1 2 this eigenvector solves the dilation equation spl phi t 2 spl sigma h k spl phi 2t k at the integers t n the left eigenvectors show how 1 t t sup p 1 can be produced as combinations of spl phi t k the dilation equation is solved by the cascade algorithm which is an infinite iteration of m spl darr 2 2h convergence in l sup 2 is governed by the eigenvalues of t spl darr 2 2hh sup t corresponding to the response 2h z h z sup 1 we find a simple proof of the necessary and sufficient condition for convergence

modeling and solving the two facility capacitated network loading problem this paper studies a topical and economically significant capacitated network design problem that arises in the telecommunications industry in this problem given point to point communication demand in a network must be met by installing loading capacitated facilities on the arcs loading a facility incurs an arc specific and facility dependent cost this paper develops modeling and solution approaches for loading facilities to satisfy the given demand at minimum cost we consider two approaches for solving the underlying mixed integer program a lagrangian relaxation strategy and a cutting plane approach that uses three classes of valid inequalities that we identify for the problem we show that a linear programming formulation that includes these inequalities always approximates the value of the mixed integer program at least as well as the lagrangian relaxation bound our computational results on a set of prototypical telecommunication data show that including these inequalities considerably improves the gap between the integer programming formulation and its linear programming relaxation from an average of 25 to an average of 8 these results show that strong cutting planes can be an effective modeling and algorithmic tool for solving problems of the size that arise in the telecommunications industry

the flow set with partial order the flow set with partial order is a mixed integer set described by a budget on total flow and a partial order on the arcs that may carry positive flow this set is a common substructure of resource allocation and scheduling problems with precedence constraints and robust network flow problems under demand capacity uncertainty we give a polyhedral analysis of the convex hull of the flow set with partial order unlike for the flow set without partial order cover type inequalities based on partial order structure are a function of a lifting sequence we study the lifting sequences and describe structural results on the lifting coefficients for general and simpler special cases we show that all lifting coefficients can be computed in polynomial time by solving maximum weight closure problems in general for the special case of induced minimal covers we give a sequence dependent characterization of the lifting coefficients we prove however if the partial order is defined by an arborescence then lifting is sequence independent and all lifting coefficients can be computed in linear time moreover if the partial order is defined by a path total order then the coefficients can be expressed explicitly we also give a complete polyhedral description of the flow set with partial order for the polynomially solvable total order case we show that finding an optimal lifting order for a given induced minimal cover and a given fractional solution is a submodular optimization problem which is solved greedily finally we present preliminary computational results with a cutting plane algorithm based on the lifting and separation results

effective heuristics for multiproduct partial shipment models motivated by real applications we consider the problem of shipping products to multiple customers from limited inventory after formulating the optimization problems under different restrictions on partial shipments we find that commercially available packages applied directly are unsatisfactory as are simple greedy approaches we develop a scheme of heuristics that enables the user to select a good balance between computation time and effectiveness a detailed computational study of one and two period industrial sized problems indicates that these heuristics are computationally practical and generate solutions that are on average within 34 of the optimum

the confined primal integral a measure to benchmark heuristic minlp solvers against global minlp solvers it is a challenging task to fairly compare local solvers and heuristics against each other and against global solvers how does one weigh a faster termination time against a better quality of the found solution in this paper we introduce the confined primal integral a new performance measure that rewards a balance of speed and solution quality it emphasizes the early part of the solution process by using an exponential decay thereby it avoids that the order of solvers can be inverted by choosing an arbitrarily large time limit we provide a closed analytic formula to compute the confined primal integral a posteriori and an incremental update formula to compute it during the run of an algorithm for the latter we show that we can drop one of the main assumptions of the primal integral namely the knowledge of a fixed reference solution to compare against furthermore we prove that the confined primal integral is a transitive measure when comparing local solves with different final solution values finally we present a computational experiment where we compare a local minlp solver that uses certain classes of cutting planes against a solver that does not both versions show very different tendencies w r t average running time and solution quality and we use the confined primal integral to argue which of the two is the preferred setting

biorthogonal partners and applications two digital filters h z and f z are said to be biorthogonal partners of each other if their cascade h z f z satisfies the nyquist or zero crossing property biorthogonal partners arise in many different contexts such as filterbank theory exact and least squares digital interpolation and multiresolution theory they also play a central role in the theory of equalization especially fractionally spaced equalizers in digital communications we first develop several theoretical properties of biorthogonal partners we also develop conditions for the existence of biorthogonal partners and fir biorthogonal pairs and establish the connections to the riesz basis property we then explain how these results play a role in many of the above mentioned applications

fixed charge transportation problem facets of the projection polyhedron in this paper we consider the well known fixed charge transportation problem to send any flow from source si to destination tj we incur a unit variable shipping cost of cij and a fixed cost fij here we study the structure of the projection polyhedron of this problem in the space of 0 1 variables associated with fixed charges and we develop several classes of valid inequalities and derive conditions under which they are facet defining in some cases if the conditions are not satisfied we show how they can be lifted to define facets several heuristics for generating and adding these facets are presented using these results we develop a computationally effective algorithm for solving the problem the computational results clearly indicate the usefulness of this approach

the convex hull of two core capacitated network design problems the network loading problem nlp is a specialized capacitated network design problem in which prescribed point to point demand between various pairs of nodes of a network must be met by installing loading a capacitated facility we can load any number of units of the facility on each of the arcs at a specified arc dependent cost the problem is to determine the number of facilities to be loaded on the arcs that will satisfy the given demand at minimum cost

assembly system design a branch and cut approach this paper addresses the single product assembly system design problem asdp which seeks to minimize total cost by optimally integrating design selecting the machine type to locate at each activated station and operating issues assigning tasks to observe precedence relationships and cycle time restrictions we propose an effective branch and cut approach for solving single product asdps adapting inequalities known to be valid for embedded line balancing structures to form inequalities that are valid for the asdp the implementation alsoinvolves a specialized preprocessor a heuristic separation procedures and an enumeration scheme computational tests establish benchmark results for this first implementation of cutting planes for solving the asdp

lifted flow cover inequalities for mixed 0 1 integer programs we investigate strong inequalities for mixed 0 1 integer programs derived from flow cover inequalities flow cover inequalities are usually not facet defining and need to be lifted to obtain stronger inequalities however because of the sequential nature of the standard lifting techniques and the complexity of the optimization problems that have to be solved to obtain lifting coefficients lifting of flow cover inequalities is computationally very demanding we present a computationally efficient way to lift flow cover inequalities based on sequence independent lifting techniques and give computational results that show the effectiveness of our lifting procedures

a rough estimate for the spectral radius of the sampling operator abstract let a k e i k be a bounded measurable function on the unit circle t given m n n the sampling operator s m n is a bounded linear operator on l 2 t whose matrix with respect to the standard basis e k z z k k z is given by a mi nj i j z in proc ams 129 11 2001 3285 a formula for the l 2 spectral radius r of s m n is obtained in terms of the asymptotic behavior of the supnorms of some continuous functions of when is continuous and positive on t where m pt n qt t g c d m n in this paper we shall establish an inequality that provide upper and lower bounds for r in terms of the parameter m n and a positive eigenvalue of s m n c t with maximal module where s m n c t is the restriction of s m n on c t the space of continuous functions on t we will also compute the actual value of r in some nontrivial cases

a polyhedral study of integer variable upper bounds we study the polyhedron of the single node capacitated network design model with integer variable upper bounds we first give a characterization of valid inequalities that is useful in proving the validity of several classes of inequalities next we derive several classes of valid inequalities and we give conditions for them to be facet defining sequence independent lifting is used to obtain additional facets we conclude by reporting computational results with a branch and cut algorithm

tightening simple mixed integer sets with guaranteed bounds in this paper we study how to reformulate knapsack sets and simple mixed integer sets in order to obtain provably tight polynomially large formulations

sums of products of functions of x times functions of y we study functions u of the form u x y nk 1ak x bk y mostly by means of two functional determinants wn u and dn u a sample result is that if u has the form on an open subset of a region g on which u is real analytic then u must have this form with possibly different ak and bk on all of g we also study the representability of u x y in the form on a region g in which u is either n times differentiable or is merely continuous the question whether the limit of functions of the form is again of the form is treated moreover we study those functions of the form that are solutions of some classical partial differential equations

valid inequalities for the single item capacitated lot sizing problem with step wise costs this paper presents a new class of valid inequalities for the single item capacitated lot sizing problem with step wise production costs ls sw we first provide a survey of different optimization methods proposed to solve ls sw then flow cover and flow cover inequalities derived from the single node flow set are described in order to generate the new class of valid inequalities the single node flow set can be seen as a generalization of some valid relaxations of ls sw a new class of valid inequalities we call mixed flow cover is derived from the integer flow cover inequalities by a lifting procedure the lifting coefficients are sequence independent when the batch sizes v and the production capacities p are constant and if v divides p when the restriction of the divisibility is removed the lifting coefficients are shown to be sequence independent we identify some cases where the mixed flow cover inequalities are facet defining a cutting plane algorithm is proposed for these three classes of valid inequalities the exact separation algorithm proposed for the constant capacitated case runs in polynomial time finally some computational results are given to compare the performance of the different optimization methods including the new class of valid inequalities

computational experience with a difficult mixed integer multicommodity flow problem the following problem arises in the study of lightwave networks given a demand matrix containing amounts to be routed between corresponding nodes we wish to design a network with certain topological features and in this network route all the demands so that the maximum load total flow on any edge is minimized as we show even small instances of this combined design routing problem are extremely intractable we describe computational experience with a cutting plane algorithm for this problem

lot sizing with inventory bounds and fixed costs polyhedral study and computation we investigate the polyhedral structure of the lot sizing problem with inventory bounds we consider two models one with linear cost on inventory the other with linear and fixed costs on inventory for both models we identify facet defining inequalities that make use of the inventory bounds explicitly and give exact separation algorithms we also describe a linear programming formulation of the problem when the order and inventory costs satisfy the wagner whitin nonspeculative property we present computational experiments that show the effectiveness of the results in tightening the linear programming relaxations of the lot sizing problem with inventory bounds and fixed costs

lp rounding approximation algorithms for stochastic network design we study the steiner tree problem and the single cable single sink network design problem under a two stage stochastic model with recourse and finitely many scenarios in these models some edges are purchased in a first stage when only probabilistic information about the second stage is available in the second stage one of a finite number of specified scenarios is realized which results in the set of terminals becoming known and the opportunity to purchase additional edges under an inflated cost function to augment the first stage solution we provide constant factor approximation algorithms for these problems by rounding the linear relaxation of ip formulations of the problems our algorithms involve solving the linear relaxation first followed by a primal dual routine that is guided by the lp solution we also show that because our bounds are local the cost of each component is bounded by its cost in the lp solution we are able to obtain bounds that guard against a form of downside risk

capacitated facility location valid inequalities and facets we examine the polyhedral structure of the convex hull of feasible solutions of the capacitated facility location problem in particular we derive necessary and sufficient conditions for a family of effective capacity inequalities to be facet defining and further results on a more general family called submodular inequalities