polyhedral properties for the intersection of two knapsacks we address the question to what extent polyhedral knowledge about individual knapsack constraints suffices or lacks to describe the convex hull of the binary solutions to their intersection it turns out that the sign patterns of the weight vectors are responsible for the types of combinatorial valid inequalities appearing in the description of the convex hull of the intersection in particular we introduce the notion of an incomplete set inequality which is based on a combinatorial principle for the intersection of two knapsacks we outline schemes to compute nontrivial bounds for the strength of such inequalities w r t the intersection of the convex hulls of the initial knapsacks an extension of the inequalities to the mixed case is also given this opens up the possibility to use the inequalities in an arbitrary simplex tableau

rigorous results for the number of convex polygons on the square and honeycomb lattices the generating functions for the number of convex polygons on the square and honeycomb lattices are derived rigorously these functions were found by guttmann and enting 1988 their calculation is based on the series expansions up to the 64th order and is nonrigorous the asymptotic form of the mean squared radius of gyration of n step convex polygons on the square lattice is determined and the critical exponent v is 1

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

geometry of the modulational instability iii homoclinic orbits for the periodic sine gordon equation in this paper the homoclinic geometric structure of the integrable sine gordon equation under periodic boundary conditions is developed specifically focus is given to orbits homoclinic to n tori simple examples of such homoclinic orbits are constructed and a physical interpretation of these states is given a labeling is provided which identifies and catalogues all such orbits these orbits are related in a one to one manner to linearized instabilities explicit formulas for all homoclinic orbits are given in terms of backlund transformations

symbolic computation on the darboux transformation for a generalized variable coefficient higher order nonlinear schrodinger equation from fiber optics considering the propagation of ultrashort pulse in the realistic fiber optics a generalized variable coefficient higher order nonlinear schrodinger equation is investigated in this paper under certain constraints a new 3 3 lax pair for this equation is obtained through the ablowitz kaup newell segur procedure furthermore with symbolic computation the darboux transformation and nth iterated potential transformation formula for such a model are explicitly derived the corresponding features of ultrashort pulse in inhomogeneous optical fibers are graphically discussed by the one and two soliton like solutions

bc opt a branch and cut code for mixed integer programs a branch and cut mixed integer programming system called bc opt is described incorporating most of the valid inequalities that have been used or suggested for such systems namely lifted 0 1 knapsack inequalities 0 1 gub knapsack and integer knapsack inequalities flowcover and continuous knapsack inequalities path inequalities for fixed charge network flow structure and gomory mixed integer cuts the principal development is a set of interface routines allowing these cut routines to generate cuts for new subsets or aggregations of constraints

minimum cost capacity installation for multicommodity network flows consider a directed graphg v a and a set of traffic demands to be shipped between pairs of nodes inv capacity has to be installed on the edges of this graph in integer multiples of a base unit so that traffic can be routed in this paper we consider the problem of minimum cost installation of capacity on the arcs to ensure that the required demands can be shipped simultaneously between node pairs we study two different approaches for solving problems of this type the first one is based on the idea of metric inequalities see onaga and kakusho on feasibility conditions of multicommodity flows in networks ieee transactions on circuit theory ct 18 4 1971 425 429 and uses a formulation with only a variables the second uses an aggregated multicommodity flow formulation and has v a variables we first describe two classes of strong valid inequalities and use them to obtain a complete polyhedral description of the associated polyhedron for the complete graph on three nodes next we explain our solution methods for both of the approaches in detail and present computational results our computational experience shows that the two formulations are comparable and yield effective algorithms for solving real life problems 1998 the mathematical programming society inc published by elsevier science b v

bright n soliton solutions in terms of the triple wronskian for the coupled nonlinear schrodinger equations in optical fibers the coupled nonlinear schrodinger nls equations are usually used to describe the dynamics of two component solitons in optical fibers via the darboux transformation the coupled nls equations of the manakov type are found to have triple wronskian solutions proof is finished by virtue of some new triple wronskian identities by solving the zero potential lax pair the triple wronskian solutions give the bright n soliton solutions which are characterized by 3n complex parameters to obtain an understanding of the asymptotic behavior of the bright n soliton solutions with arbitrary n some algebraic properties of the triple wronskian are analyzed and an algebraic procedure is presented to derive the explicit expressions of the asymptotic solitons as t

lifted inequalities for 0 1 mixed integer programming superlinear lifting we study the mixed 0 1 knapsack polytope which is defined by a single knapsack constraint that contains 0 1 and bounded continuous variables through the lifting of continuous variables fixed at their upper bounds we introduce the concept of a superlinear inequality and show that in this case lifting is significantly simpler than for general inequalities we use the superlinearity theory together with the traditional lifting of 0 1 variables to describe families of facets of the mixed 0 1 knapsack polytope finally we show that superlinearity results can be extended to nonsuperlinear inequalities when the coefficients of the variables fixed at their upper bounds are large

an extension of the wronskian technique for the multicomponent wronskian solution to the vector nonlinear schrodinger equation in this paper the wronskian technique is applied to the vector nonlinear schrodinger equation with arbitrary m components which arises from some applications in the multimode fibers photorefractive materials and bose einstein condensates via the iterative algorithm based on the darboux transformation the m 1 component wronskian solution is generated from the zero solution the verification of the solution is finished by using the m 1 component wronskian notation and new determinantal identities with a set of n linearly independent solutions of the zero potential lax pair the m 1 component wronskian solution is found to be the representation of the bright n soliton solution which contains m 1 n parameters for characterizing the asymptotic behavior of the generic bright n soliton solution an algebraic procedure is derived to obtain the explicit expressions of asymptotic solitons as t

diversification strategies in local search for a nonbifurcated network loading problem abstract this paper examines a variant of the network loading problem a network design problem found in the telecommunications industry in this problem facilities of fixed capacity must be installed on the edges of an undirected network to carry the flow from a central vertex to a set of demand vertices the objective is to minimize the total installation costs in this work the nonbifurcated version of the problem is considered where the demand at any given vertex must be satisfied through a single path the proposed heuristics alternate between a construction phase and a local search phase each new construction phase except the first one is part of a diversification strategy aimed at providing a new starting point for the following local search phase different diversification strategies are tested and compared on large scale instances with up to 500 vertices

continuous knapsack sets with divisible capacities we study two continuous knapsack sets mathcal y ge y and mathcal y le y with n n integer one unbounded continuous and m m bounded continuous variables in either ge or le form when the coefficients of the integer variables are integer and divisible we show in both cases that the convex hull is the intersection of the bound constraints and 2 m 2m polyhedra arising as the convex hulls of continuous knapsack sets with a single unbounded continuous variable the latter convex hulls are completely described by an exponential family of partition inequalities and a polynomial size extended formulation is known in the ge case we also provide an extended formulation for the le case it follows that given a specific objective function optimization over both mathcal y ge y and mathcal y le y can be carried out by solving m m polynomial size linear programs a further consequence of these results is that the coefficients of the continuous variables all take the values 0 or 1 after scaling in any non trivial facet defining inequality of the convex hull of such sets

the 0 1 knapsack problem with a single continuous variable constraints arising in practice often contain many 0 1 variables and one or a small number of continuous variables existing knapsack separation routines cannot be used on such constraints here we study such constraint sets and derive valid inequalities that can be used as cats for such sets as well for more general mixed 0 1 constraints specifically we investigate the polyhedral structure of the knapsack problem with a single continuous variable called the mixed 0 1 knapsack problem first different classes of facet defining inequalities are derived based on restriction and lifting the order of lifting particularly of the continuous variable plays an important role secondly we show that the flow cover inequalities derived for the single node flow set consisting of are flows into and out of a single node with binary variable lower and upper bounds on each are can be obtained from valid inequalities for the mixed 0 1 knapsack problem thus the separation heuristic we derive for mixed knapsack sets can also be used to derive cuts for more general mixed 0 1 constraints initial computational results on a variety of problems are presented

the number of convex polygons on the square and honeycomb lattices a subset of the set of self avoiding polygons sap embeddable on the square lattice which display the property of convexity is defined an algorithm for their enumeration is developed and from the available series coefficients the exact generating function is found the singularity structure appears similar to that of the unsolved sap problem but with different critical exponents and critical points the enumeration of convex polygons allows the extension of the existing series for the square lattice sap by one term for the honeycomb lattice similar results have been obtained despite a less natural definition of convexity

on splittable and unsplittable flow capacitated network design arc set polyhedra we study the polyhedra of splittable and unsplittable single arc set relaxations of multicommodity flow capacitated network design problems we investigate the optimization problems over these sets and the separation and lifting problems of valid inequalities for them in particular we give a linear time separation algorithm for the residual capacity inequalities 19 and show that the separation problem of c strong in equalities 7 is np hard but can be solved over the subspace of fractional variables only we introduce two classes of inequalities for the unsplittable flow problems we present a summary of computational experiments with a branch and cut algorithm for multicommodity flow capacitated network design problems to test the effectiveness of the results presented here empirically

a recursive procedure to generate all cuts for 0 1 mixed integer programs we study several ways of obtaining valid inequalities for mixed integer programs we show how inequalities obtained from a disjunctive argument can be represented by superadditive functions and we show how the superadditive inequalities relate to gomory s mixed integer cuts we also show how all valid inequalities for mixed 0 1 programs can be generated recursively from a simple subclass of the disjunctive inequalities

sequence independent lifting for mixed integer programs with variable upper bounds we investigate the convex hull of the set defined by a single inequality with continuous and binary variables with variable upper bound constraints we extend the traditional flow cover inequality and show that it is valid for a restriction of the set in which some variables are fixed we also give conditions under which this inequality is facet defining and when it is not we show how it can be lifted to obtain valid inequalities for the entire set using sequence independent lifting in general computing the lifting function is np hard but under an additional restriction on the cover we obtain a closed form finally we show how these results imply and extend known results about the single node fixed charge flow polyhedron

effective integration of the nonlinear vector schrodinger equation abstract a comprehensive algebro geometric integration of the two component nonlinear vector schrodinger equation manakov system is developed the allied spectral variety is a trigonal riemann surface which is described explicitly and the solutions of the equations are given in terms of functions of the surface the final formulae are effective in the sense that all entries like transcendental constants in exponentials winding vectors etc are expressed in terms of the prime form of the curve and well algorithmized operations on them that made the result available for direct calculations in applied problems implementing the manakov system the simplest solutions in jacobian functions are given as a particular case of general formulae and are discussed in detail

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

optimal allocation of surgery blocks to operating rooms under uncertainty the allocation of surgeries to operating rooms ors is a challenging combinatorial optimization problem there is also significant uncertainty in the duration of surgical procedures which further complicates assignment decisions in this paper we present stochastic optimization models for the assignment of surgeries to ors on a given day of surgery the objective includes a fixed cost of opening ors and a variable cost of overtime relative to a fixed length of day we describe two types of models the first is a two stage stochastic linear program with binary decisions in the first stage and simple recourse in the second stage the second is its robust counterpart in which the objective is to minimize the maximum cost associated with an uncertainty set for surgery durations we describe the mathematical models bounds on the optimal solution and solution methodologies including an easy to implement heuristic numerical experiments based on real data from a large health care provider are used to contrast the results for the two models and illustrate the potential for impact in practice based on our numerical experimentation we find that a fast and easy to implement heuristic works fairly well on average across many instances we also find that the robust method performs approximately as well as the heuristic is much faster than solving the stochastic recourse model and has the benefit of limiting the worst case outcome of the recourse problem

design of capacitated multicommodity networks with multiple facilities this paper addresses the problem of designing a multicommodity network using several facilities with different costs and capacities the problem is addressed in a special context of designing private telecommunications networks using fractional t1 services the algorithm starts from any given solution of the problem and gradually improves it by solving a series of subproblems arriving at a local minimum the subproblem is defined over a subset of links called the subnetwork by using one of the links as the base link it is shown that the subproblem can be formulated as a multiple choice knapsack problem that is solved by dynamic programming computational results and lower bounds are reported on problems of up to 20 nodes and up to 3 facilities on most problems the algorithm produces solutions within about 5 of lower bound on the average although it was not possible to compute lower bounds for larger problems heuristic solutions and running times are reported for problems of up to 99 nodes and four facilities

on the capacitated lot sizing and continuous 0 1 knapsack polyhedra abstract we consider the single item capacitated lot sizing problem a well known production planning model that often arises in practical applications and derive new classes of valid inequalities for it we first derive new easily computable valid inequalities for the continuous 0 1 knapsack problem which has been introduced recently and has been shown to provide a useful relaxation of mixed 0 1 integer programs we then show that by studying an appropriate continuous 0 1 knapsack relaxation of the capacitated lot sizing problem it is possible not only to derive new classes of valid inequalities for the capacitated lot sizing problem but also to derive and or strengthen several known classes

darboux transformation and explicit solutions for the integrable sixth order kdv equation for nonlinear waves abstract korteweg de vries kdv type equations can describe some physical phenomena in fluids nonlinear optics quantum mechanics plasmas etc in this paper with the aid of symbolic computation the integrable sixth order kdv equation is investigated darboux transformation dt with an arbitrary parameter is presented explicit solutions are derived with the dt relevant properties are graphically illustrated which might be helpful to understand some physical processes in fluids plasmas optics and quantum mechanics

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

an improved method for darboux transformation for the coupled inhomogeneous nonlinear schrodinger system from plasma physics and nonlinear optics abstract the coupled inhomogeneous nonlinear schrodinger type system which can be used to control soliton propagation and interaction in certain plasmas and optical fibers is investigated an improved method for darboux transformation dt is presented in more general forms by constructing an improved riccati type backlund transformation r bt with the n th iterated r bt or the n th iterated dt which is a compact representation for the n soliton like solutions and can generate a series of analytic solutions from a pair of the seed solutions through algebraic manipulations the analytic one two soliton like solutions are provided with the choice of parameters for the soliton solutions the dynamical characteristics of the influences of the inhomogeneous parameters on the propagation of the soliton pulses are discussed graphically

exact solution of the convex polygon perimeter and area generating function an explicit expression is derived for the three variable generating function p x y z sigma m or 1 sigma n or 1 sigma r or 1 x2ny2mzrcn m r where cn m r is the number of convex polygons with horizontal width n vertical height m and area r

lifted inequalities for 0 1 mixed integer programming basic theory and algorithms we study the mixed 0 1 knapsack polytope which is defined by a single knapsack constraint that contains 0 1 and bounded continuous variables we develop a lifting theory for the continuous variables in particular we present a pseudo polynomial algorithm for the sequential lifting of the continuous variables and we discuss its practical use

on the facets of the mixed integer knapsack polyhedron we study the mixed integer knapsack polyhedron that is the convex hull of the mixed integer set defined by an arbitrary linear inequality and the bounds on the variables we describe facet defining inequalities of this polyhedron that can be obtained through sequential lifting of inequalities containing a single integer variable these inequalities strengthen and or generalize known inequalities for several special cases we report computational results on using the inequalities as cutting planes for mixed integer programming

mixed 0 1 programming by lift and project in a branch and cut framework we investigate the computational issues that need to be addressed when incorporating general cutting planes for mixed 0 1 programs into a branch and cut framework the cuts we use are of the lift and project variety some of the issues addressed have a theoretical answer but others are of an experimental nature and are settled by comparing alternatives on a set of test problems the resulting code is a robust solver for mixed 0 1 programs we compare it with several existing codes on a wide range of test problems it performs as well as or better than some of the best currently available mixed integer programming codes

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

soliton like solutions of a generalized variable coefficient higher order nonlinear schrodinger equation from inhomogeneous optical fibers with symbolic computation for the long distance communication and manufacturing problems in optical fibers the propagation of subpicosecond or femtosecond optical pulses can be governed by the variable coefficient nonlinear schrodinger equation with higher order effects such as the third order dispersion self steepening and self frequency shift in this paper we firstly determine the general conditions for this equation to be integrable by employing the painleve analysis based on the obtained 3 3 lax pair we construct the darboux transformation for such a model under the corresponding constraints and then derive the nth iterated potential transformation formula by the iterative process of darboux transformation through the one and two soliton like solutions we graphically discuss the features of femtosecond solitons in inhomogeneous optical fibers

darboux transformation for a two component derivative nonlinear schrodinger equation in this paper we consider a two component derivative nonlinear schrodinger equation and present a simple darboux transformation for it by iterating this darboux transformation we construct a compact representation for the n soliton solutions

n step mingling inequalities new facets for the mixed integer knapsack set the n step mixed integer rounding mir inequalities of kianfar and fathi math program 120 2 313 346 2009 are valid inequalities for the mixed integer knapsack set that are derived by using periodic n step mir functions and define facets for group problems the mingling and 2 step mingling inequalities of atamturk and gunluk math program 123 2 315 338 2010 are also derived based on mir and they incorporate upper bounds on the integer variables it has been shown that these inequalities are facet defining for the mixed integer knapsack set under certain conditions and generalize several well known valid inequalities in this paper we introduce new classes of valid inequalities for the mixed integer knapsack set with bounded integer variables which we call n step mingling inequalities for positive integer n these inequalities incorporate upper bounds on integer variables into n step mir and therefore unify the concepts of n step mir and mingling in a single class of inequalities furthermore we show that for each n the n step mingling inequality defines a facet for the mixed integer knapsack set under certain conditions for n 2 we extend the results of atamturk and gunluk on facet defining properties of 2 step mingling inequalities for n 3 we present new facets for the mixed integer knapsack set as a special case we also derive conditions under which the n step mir inequalities define facets for the mixed integer knapsack set in addition we show that n step mingling can be used to generate new valid inequalities and facets based on covers and packs defined for mixed integer knapsack sets

dynamic knapsack sets and capacitated lot sizing a dynamic knapsack set is a natural generalization of the 0 1 knapsack set with a continuous variable studied recently for dynamic knapsack sets a large family of facet defining inequalities called dynamic knapsack inequalities are derived by fixing variables to one and then lifting surprisingly such inequalities have the simultaneous lifting property and for small instances provide a significant proportion of all the facet defining inequalities we then consider single item capacitated lot sizing problems and propose the joint study of three related sets the first models the discrete lotsizing problem the second the continuous lot sizing problem with wagnerwhitin costs and the third the continuous lot sizing problem with arbitrary costs the first set that arises is precisely a dynamic knapsack set the second an intersection of dynamic knapsack sets and the unrestricted problem can be viewed as both a relaxation and a restriction of the second it follows that the dynamic knapsack inequalities and their generalizations provide strong valid inequalities for all three sets some limited computation results are reported as an initial test of the effectiveness of these inequalities on capacitated lot sizing problems

single commodity robust network design with finite and hose demand sets we study a single commodity robust network design problem srnd defined on an undirected graph our goal is to determine minimum cost capacities such that any traffic demand from a given uncertainty set can be satisfied by a feasible single commodity flow we consider two ways of representing the uncertainty set either as a finite list of scenarios or as a polytope we propose a branch and cut algorithm to derive optimal solutions to srnd built on a capacity based integer linear programming formulation it is strengthened with valid inequalities derived as 0 frac 1 2 0 12 chvatal gomory cuts since the formulation contains exponentially many constraints we provide practical separation algorithms extensive computational experiments show that our approach is effective in comparison to existing approaches from the literature as well as to solving a flow based formulation by a general purpose solver

exact solution of the row convex polygon perimeter generating function an explicit expression is obtained for the perimeter generating function g y sigma n or 2 any2n for row convex polygons on the square lattice where an is the number of 2n step row convex polygons an asymptotic expression for an approximately a mu nn 3 2 is obtained where mu 3 22 and a are given the authors also show that the generating function is an algebraic function and that it satisfies an inhomogeneous linear differential equation of degree three

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

on the polyhedral structure of a multi item production planning model with setup times we present and study a mixed integer programming model that arises as a substructure in many industrial applications this model generalizes a number of structured mip models previously studied and it provides a relaxation of various capacitated production planning problems and other fixed charge network flow problems we analyze the polyhedral structure of the convex hull of this model as well as of a strengthened lp relaxation among other results we present valid inequalities that induce facets of the convex hull under certain conditions we also discuss how to strengthen these inequalities by using known results for lifting valid inequalities for 0 1 continuous knapsack problems

on the backlund gauge transformation and homoclinic orbits of a coupled nonlinear schrodinger system abstract the backlund gauge transformation for a system of coupled nls nonlinear schrodinger equations with a degenerate associated spectral operator is derived from an algebraic perspective extending aspects of other results m boiti tu guizhang il nuovo cimento 71b 1982 253 264 d h sattinger v d zurkowski physica d 26 1 3 1987 225 250 that apply in the context of non degenerate spectral operators moreover we demonstrate how the backlund gauge transformation can be used to explicitly construct the entire unstable manifold via superpositions of homoclinic orbits of a plane wave solution with both self phase instabilities and coupling instabilities this work builds on the results of ercolani et al n ercolani m g forest d w mclaughlin physica d 18 1986 472 474 n ercolani m g forest d w mclaughlin physica d 43 2 3 1990 349 384 for the sine gordon equation and forest et al m g forest d w mclaughlin d j muraki o c wright j nonlinear sci in press m g forest s p sheu o c wright phys lett a in press for the integrable coupled nls system

network design using cut inequalities we study the network loading problem with and without bifurcations we use a relaxation based on the cut condition for multicommodity flows we use a solution of the bifurcated case to derive a solution to the nonbifurcated problem a standard procedure is to aggregate the problem into a backbone network we applied this method to backbone networks coming from practical instances we obtained feasible solutions and bounds for the gap from optimality

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

algebraic languages and polyominoes enumeration abstract in this paper the use of algebraic languages theory in solving an open problem in combinatorics is shown by constructing a bijection between convex polyominoes and words of an algebraic language and by solving the corresponding algebraic system we prove that the number of convex polyominoes with perimeter 2n 8 is 2n 11 4 n 4 2n 1 2n n

gauge theory of ba cklund transformations ii abstract a class of gauge transformations is constructed for hamiltonian hierarchies of completely integrable systems on semi simple lie algebras these transformations create or annihilate poles in the scattering data hence create or annihilate solitons in the potential q in that sense they generalize backlund transformations to which they reduce precisely for the 2 2 akns systems to each backlund gauge transformation is associated an automorphism in the weyl group of the lie algebra two proofs are given to show that these backlund gauge transformations preserve the nonlinear evolution equation one proof uses the transformation of the scattering data under the backlund transformation the second proof is purely gauge theoretic

the full positive flows of manakov hierarchy hamiltonian structures and conservation laws based on four sets of lenard recursion sequences and zero curvature equation we derive the full positive flows of the manakov hierarchy associated with a 3x3 matrix spectral problem from which some new nonlinear evolution equations are proposed with the help of the darboux transformation soliton solutions of two new nonlinear evolution equations in the manakov hierarchy are constructed as two special reductions the full positive flows of the coupled modified korteweg de vries hierarchy and the sasa satsuma hierarchy are deduced in which some new nonlinear evolution equations are included and then we construct the hamiltonian structures of the manakov hierarchy and infinite conservation laws of several nonlinear evolution equations

cutting planes for integer programs with general integer variables we investigate the use of cutting planes for integer programs with general integer variables we show how cutting planes arising from knapsack inequalities can be generated and lifted as in the case of 0 1 variables we also explore the use of gomory s mixed integer cuts we address both theoretical and computational issues and show how to embed these cutting planes in a branch and bound framework we compare results obtained by using our cut generation routines in two existing systems with a commercially available branch and bound code on a range of test problems arising from practical applications 1998 the mathematical programming society inc published by elsevier science b v

using cuts for mixed integer knapsack sets to generate cuts for mixed integer polyhedral conic sets based on a bijective mapping between two mixed integer sets we introduce a new perspective on developing cuts for the mixed integer polyhedral conic mipc set by establishing a one to one correspondence between the cuts for this set and those for a related mixed integer knapsack mik set the face facet defining properties of the corresponding cuts are identical for their respective sets we further show that the cut generation approach for the mipc set resulting from this new perspective always produces cuts that dominate those generated based on any of the two individual mik constraints corresponding to the mipc constraint our computational results show this dominance can be quite significant as a special case of this new perspective the conic mir inequality of atamturk and narayanan for the mipc set and its properties can be directly derived from the mir inequality for the mik set and its properties we also generalize these cuts to the n step conic mir inequalities which are directly derived form the n step mir inequalities for the mik set

design of survivable networks using three and four partition facets this paper considers the problem of designing a multicommodity network with single facility type subject to the requirement that under failure of any single edge the network should permit a feasible flow of all traffic we study the polyhedral structure of the problem by considering the multigraph obtained by shrinking the nodes but not the edges in a k partition of the original graph a key theorem is proved according to which a facet of the k node problem defined on the multigraph resulting from a k partition is also facet defining for the larger problem under a mild condition after reviewing the prior work on two partition inequalities we develop two classes of three partition inequalities and a large number of inequality classes based on four partitions proofs of facet defining status for some of these are provided while the rest are stated without proof computational results show that the addition of three and four partition inequalities results in substantial increase in the bound values compared to those possible with two partition inequalities alone problems of 35 nodes and 80 edges with fully dense traffic matrices have been solved optimally within a few minutes of computer time

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

perimeter generating function for row convex polygons on the rectangular lattice the perimeter generating function derived recently by brak et al 1990 for row convex polygons on the square lattice is generalized to the rectangular lattice

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

mingling mixed integer rounding with bounds mixed integer rounding mir is a simple yet powerful procedure for generating valid inequalities for mixed integer programs when used as cutting planes mir inequalities are very effective for mixed integer programming problems with unbounded integer variables for problems with bounded integer variables however cutting planes based on lifting techniques appear to be more effective this is not surprising as lifting techniques make explicit use of the bounds on variables whereas the mir procedure does not in this paper we describe a simple procedure which we call mingling for incorporating variable bound information into mir by explicitly using the variable bounds the mingling procedure leads to strong inequalities for mixed integer sets with bounded variables we show that facets of mixed integer knapsack sets derived earlier by superadditive lifting techniques can be obtained by the mingling procedure in particular the mingling inequalities developed in this paper subsume the continuous cover and reverse continuous cover inequalities of marchand and wolsey math program 85 15 33 1999 as well as the continuous integer knapsack cover and pack inequalities of atamturk math program 98 145 175 2003 ann oper res 139 21 38 2005 in addition mingling inequalities give a generalization of the two step mir inequalities of dash and gunluk math program 105 29 53 2006 under some conditions

polyhedral results for the edge capacity polytope network loading problems occur in the design of telecommunication networks in many different settings for instance bifurcated or non bifurcated routing also called splittable and unsplittable can be considered in most settings the same polyhedral structures return a better understanding of these structures therefore can have a major impact on the tractability of polyhedral guided solution methods in this paper we investigate the polytopes of the problem restricted to one arc edge of the network the undirected directed edge capacity problem for the non bifurcated routing case as an example one of the basic variants of network loading is described including an integer linear programming formulation as the edge capacity problems are relaxations of this network loading problem their polytopes are intimately related we give conditions under which the inequalities of the edge capacity polytopes define facets of the network loading polytope we describe classes of strong valid inequalities for the edge capacity polytopes and we derive conditions under which these constraints define facets for the diverse classes the complexity of lifting projected variables is stated the derived inequalities are tested on i the edge capacity problem itself and ii the described variant of the network loading problem the results show that the inequalities substantially reduce the number of nodes needed in a branch and cut approach moreover they show the importance of the edge subproblem for solving network loading problems

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

lifting two integer knapsack inequalities in this paper we discuss the derivation of strong valid inequalities for mixed integer knapsack sets based on lifting of valid inequalities for basic knapsack sets with two integer variables and one continuous variable the basic polyhedra can be described in polynomial time we use superadditive valid lifting functions in order to obtain sequence independent lifting most of these superadditive functions and valid inequalities are not obtained in polynomial time

lifting inequalities a framework for generating strong cuts for nonlinear programs in this paper we introduce the first generic lifting techniques for deriving strong globally valid cuts for nonlinear programs the theory is geometric and provides insights into lifting based cut generation procedures yielding short proofs of earlier results in mixed integer programming using convex extensions we obtain conditions that allow for sequence independent lifting in nonlinear settings paving a way for efficient cut generation procedures for nonlinear programs this sequence independent lifting framework also subsumes the superadditive lifting theory that has been used to generate many general purpose strong cuts for integer programs we specialize our lifting results to derive facet defining inequalities for mixed integer bilinear knapsack sets finally we demonstrate the strength of nonlinear lifting by showing that these inequalities cannot be obtained using a single round of traditional integer programming cut generation techniques applied on a tight reformulation of the problem

on the 0 1 knapsack polytope this paper deals with the 0 1 knapsack polytope in particular we introduce the class ofweight inequalities this class of inequalities is needed to describe the knapsack polyhedron when the weights of the items lie in certain intervals a generalization of weight inequalities yields the so called weight reduction principle and the class of extended weight inequalities the latter class of inequalities includes minimal cover and l k configuration inequalities the properties of lifted minimal cover inequalities are extended to this general class of inequalities

cutting planes for mixed integer knapsack polyhedra an algorithm for generating cutting planes for mixed integer knapsack polyhedra is described the algorithm represents an exact separation procedure and is based on a general methodology proposed by one of the authors in an earlier paper computational results are presented 1998 the mathematical programming society inc published by elsevier science b v

maximizing a class of submodular utility functions given a finite ground set n and a value vector a in mathbb r n we consider optimization problems involving maximization of a submodular set utility function of the form h s f left sum i in s a i right s subseteq n where f is a strictly concave increasing differentiable function this utility function appears frequently in combinatorial optimization problems when modeling risk aversion and decreasing marginal preferences for instance in risk averse capital budgeting under uncertainty competitive facility location and combinatorial auctions these problems can be formulated as linear mixed 0 1 programs however the standard formulation of these problems using submodular inequalities is ineffective for their solution except for very small instances in this paper we perform a polyhedral analysis of a relevant mixed integer set and by exploiting the structure of the utility function h strengthen the standard submodular formulation significantly we show the lifting problem of the submodular inequalities to be a submodular maximization problem with a special structure solvable by a greedy algorithm which leads to an easily computable strengthening by subadditive lifting of the inequalities computational experiments on expected utility maximization in capital budgeting show the effectiveness of the new formulation

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

a semi discrete integrable multi component coherently coupled nonlinear schrodinger system a new integrable semi discrete version is proposed for the multi component coherently coupled nonlinear schrodinger equation the integrability of the semi discrete system is confirmed by existence of lax pair and infinite number of conservation laws with the aid of gauge transformations explicit formulas for n fold darboux transformations are derived whereby some physically important solutions of the system are presented furthermore the theory of the semi discrete system including lax pair darboux transformations exact solutions and infinite number of conservation laws are shown for their continuous counterparts in the continuous limit