refined proximity and sensitivity results in linearly constrained convex separable integer programming abstract we define a new measure of the structure of a linear constraint matrix and establish some propertie we then use the measure to refine proximity and sensitivity results in the literature in particular we refine bounds on the distance between continuous and discrete optima of linearly constrained convex separable programming problems and on the size of the feasible region for a linear program we apply the results to improve the computational complexity of hochbaum and shanthikumar s algorithm for integer nonlinear programming reducing the computational effort for certain problems from exponential to polynomial

on the stable set polytope of a series parallel graph we give a short proof of chvatal s conjecture that the nontrivial facets of the stable set polytope of a series parallel graph all come from edges and odd holes

on decomposability of multilinear sets we consider the multilinear set mathcal s defined as the set of binary points x y satisfying a collection of multilinear equations of the form y i prod i in i x i i in mathcal i where mathcal i denotes a family of subsets of 1 ldots n of cardinality at least two such sets appear in factorable reformulations of many types of nonconvex optimization problems including binary polynomial optimization a great simplification in studying the facial structure of the convex hull of the multilinear set is possible when mathcal s is decomposable into simpler multilinear sets mathcal s j j in j namely the convex hull of mathcal s can be obtained by convexifying each mathcal s j separately in this paper we study the decomposability properties of multilinear sets utilizing an equivalent hypergraph representation for multilinear sets we derive necessary and sufficient conditions under which mathcal s is decomposable into mathcal s j j in j based on the structure of pair wise intersection hypergraphs our characterizations unify and extend the existing decomposability results for the boolean quadric polytope finally we propose a polynomial time algorithm to optimally decompose a multilinear set into simpler subsets our proposed algorithm can be easily incorporated in branch and cut based global solvers as a preprocessing step for cut generation

on reduced semidefinite programs for second order moment bounds with applications we show that the complexity of computing the second order moment bound on the expected optimal value of a mixed integer linear program with a random objective coefficient vector is closely related to the complexity of characterizing the convex hull of the points 1 atopwithdelims varvec x 1 atopwithdelims varvec x varvec x in mathcal x 1x1x x x where mathcal x x is the feasible region in fact we can replace the completely positive programming formulation for the moment bound on mathcal x x with an associated semidefinite program provided we have a linear or a semidefinite representation of this convex hull as an application of the result we identify a new polynomial time solvable semidefinite relaxation of the distributionally robust multi item newsvendor problem by exploiting results from the boolean quadric polytope for mathcal x x described explicitly by a finite set of points our formulation leads to a reduction in the size of the semidefinite program we illustrate the usefulness of the reduced semidefinite programming bounds in estimating the expected range of random variables with two applications arising in random walks and best worst choice models

optimal design of a distributed network with a two level hierarchical structure abstract this paper deals with the topological design of a network with a two level hierarchical structure where the embedded backbone network is full meshed and the local networks attached to it are of star type the whole problem unlike the conventional approaches partitioning it into two subproblems is directly handled in such a general setting that both a backbone network and local access networks are to be simultaneously determined we formulate this problem as a quadratic 0 1 programming model and also propose an equivalent linearized version of the model based on the observation that the linearized model is a variant of the well established uncapacitated facility location problem a dual based solution procedure is developed and shown to be efficient from computing experiments which were done on a wide variety of problems with up to 50 backbone nodes and 50 user nodes

separation and relaxation for cones of quadratic forms let p subseteq mathfrak r n be a pointed polyhedral cone in this paper we study the cone mathcal c rm cone xx t x in p of quadratic forms understanding the structure of mathcal c is important for globally solving np hard quadratic programs over p we establish key characteristics of mathcal c and construct a separation algorithm for mathcal c provided one can optimize with respect to a related cone over the boundary of p this algorithm leads to a nonlinear representation of mathcal c and a class of tractable relaxations for mathcal c each of which improves a standard polyhedral semidefinite relaxation of mathcal c the relaxation technique can further be applied recursively to obtain a hierarchy of relaxations and for constant recursive depth the hierarchy is tractable we apply this theory to two important cases p is the nonnegative orthant in which case mathcal c is the cone of completely positive matrices and p is the homogenized cone of the box 0 1 n through various results and examples we demonstrate the strength of the theory for these cases for example we achieve for the first time a separation algorithm for 5 5 completely positive matrices

on nonconvex quadratic programming with box constraints nonconvex quadratic programming with box constraints is a fundamental mathcal np hard global optimization problem recently some authors have studied a certain family of convex sets associated with this problem we prove several fundamental results concerned with these convex sets we determine their dimension characterize their extreme points and vertices show their invariance under certain affine transformations and show that various linear inequalities induce facets we also show that the sets are closely related to the boolean quadric polytope a fundamental polytope in the field of polyhedral combinatorics finally we give a classification of valid inequalities and show that this yields a finite recursive procedure to check the validity of any proposed inequality

on the cut polytope the cut polytopep c g of a graphg v e is the convex hull of the incidence vectors of all edge sets of cuts ofg we show some classes of facet defining inequalities ofp c g we describe three methods with which new facet defining inequalities ofp c g can be constructed from known ones in particular we show that inequalities associated with chordless cycles define facets of this polytope moreover for these inequalities a polynomial algorithm to solve the separation problem is presented we characterize the facet defining inequalities ofp c g ifg is not contractible tok 5 we give a simple characterization of adjacency inp c g and prove that for complete graphs this polytope has diameter one and thatp c g has the hirsch property a relationship betweenp c g and the convex hull of incidence vectors of balancing edge sets of a signed graph is studied

experiments in quadratic 0 1 programming we present computational experience with a cutting plane algorithm for 0 1 quadratic programming without constraints our approach is based on a reduction of this problem to a max cut problem in a graph and on a partial linear description of the cut polytope

lower bounding procedure for the asymmetric quadratic traveling salesman problem abstract in this paper we consider the asymmetric quadratic traveling salesman problem aqtsp given a directed graph and a function that maps every pair of consecutive arcs to a cost the problem consists in finding a cycle that visits every vertex exactly once and such that the sum of the costs is minimal we propose an extended linear programming formulation that has a variable for each cycle in the graph since the number of cycles is exponential in the graph size we propose a column generation approach moreover we apply a particular reformulation linearization technique on a compact representation of the problem and compute lower bounds based on lagrangian relaxation we compare our new bounds with those obtained by some linearization models proposed in the literature computational results on some set of benchmarks used in the literature show that our lower bounding procedures are very promising

connections between semidefinite relaxations of the max cut and stable set problems we describe links between a recently introduced semidefinite relaxation for the max cut problem and the well known semidefinite relaxation for the stable set problem underlying the lovasz s theta function it turns out that the connection between the convex bodies defining the semidefinite relaxations mimics the connection existing between the corresponding polyhedra we also show how the semidefinite relaxations can be combined with the classical linear relaxations in order to obtain tighter relaxations

algorithmic and modeling insights via volumetric comparison of polyhedral relaxations this is mostly a survey on some mathematical results concerning volumes of polytopes of interest in non convex optimization our motivation is in geometrically comparing relaxations in the context of mixed integer linear and nonlinear optimization with the goal of gaining algorithmic and modeling insights we consider relaxations of fixed charge formulations vertex packing polytopes boolean quadric polytopes and relaxations of graphs of monomials on box domains besides surveying the area we do give a few new results and we provide many directions for further work

distribution free inventory risk pooling in a multi location newsvendor we study a multilocation newsvendor network when the only information available on the joint distribution of demands are the values of its mean vector and covariance matrix we adopt a distribution

nonlinear 0 1 programming i linearization techniques any real valued nonlinear function in 0 1 variables can be rewritten as a multilinear function we discuss classes of lower and upper bounding linear expressions for multilinear functions in 0 1 variables for any multilinear inequality in 0 1 variables we define an equivalent family of linear inequalities this family contains the well known system of generalized covering inequalities as well as other linear equivalents of the multilinear inequality that are more compact i e of smaller cardinality in a companion paper 7 we discuss dominance relations between various linear equivalents of a multilinear inequality and describe a class of algorithms for multilinear 0 1 programming based on these results

roof duality complementation and persistency in quadratic 0 1 optimization the paper is concerned with the primal problem of maximizing a given quadratic pseudo boolean function four equivalent problems are discussed the primal the complementation the discrete rhys lp and the weighted stability problem of a sam graph each of them has a relaxation the roof dual the quadratic complementation the continuous rhys lp and the fractional weighted stability problem of a sam graph the main result is that the four gaps associated with the four relaxations are equal furthermore a solution to any of these problems leads at once to solutions of the other three equivalent ones the four relaxations can be solved in polynomial time by transforming them to a bipartite maximum flow problem the optimal solutions of the roof dual define best linear majorantsp x off having the following persistency property if theith coefficient inp is positive negative thenx i 1 0 in every optimum of the primal problem several characterizations are given for the case where these persistency results cannot be used to fix any variable of the primal on the other hand a class of gap free functions properly including the supermodular ones is exhibited

transitive packing a unifying concept in combinatorial optimization this paper attempts to provide a better understanding of the facial structure of polyhedra previously investigated separately it introduces the notion of transitive packing and the transitive packing polytope polytopes that turn out to be special cases of the transitive packing polytope include the node packing acyclic subdigraph bipartite subgraph planar subgraph clique partitioning partition transitive acyclic subdigraph interval order and relatively transitive subgraph polytopes we give cutting plane proofs for several rich classes of valid inequalities of the transitive packing polytope thereby introducing generalized cycle generalized clique generalized antihole generalized antiweb and odd partition inequalities on the one hand these classes subsume several known classes of valid inequalities for several special cases on the other hand they yield many new inequalities for several other special cases for some of the classes we also prove a lower bound on their gomory chvatal rank finally we relate the concept of transitive packing to generalized set packing and covering as well as to balanced and ideal matrices

strict contractions of symmetric cones let v be a euclidean jordan algebra with an associative inner product hji and let be the corresponding symmetric cone then the linear automorphism group h of the symmetric cone acts transitively on it let t v i be the symmetric tube domain associated to the symmetric cone and let g t be the biholomorphic automorphism group of t there exists a conformal compactification m g t p of v with respect to a parabolic subgroup p a lie semigroup that is naturally related to the symmetric cone occurs as the semigroup of compressionsof in the lie group g t fg 2 g t j g g it turns out that admits an ol shanskii decomposition h expc for an ad h invariant convex cone c in the lie algebra g t of g t a triple decomposition h 8 9 and it determines a globally hyperbolic causal order on the homogeneous space g t h defined by ah bh if and only if a 1 b 2 the family of bilinear forms x u v hp x 1 ujvi defines a hinvariant riemannian metric on where the map p is the quadratic representation of the jordan algebra v in 8 koufany showed that the elements of are contractions of for the riemannian metric the main result of this paper is that the elements of the interior of called strict compressions are strict contractions of for the riemannian distance induced by

facets of the clique partitioning polytope a subseta of the edge set of a graphg v e is called a clique partitioning ofg is there is a partition of the node setv into disjoint setsw 1 w k such that eachw i induces a clique i e a complete but not necessarily maximal subgraph ofg and such thata 1 uv u v w i u v given weightsw e for alle e the clique partitioning problem is to find a clique partitioninga ofg such that e a w e is as small as possible this problem known to be hard see wakabayashi 1986 comes up for instance in data analysis and here the underlying graphg is typically a complete graph in this paper we study the clique partitioning polytope of the complete graphk n i e is the convex hull of the incidence vectors of the clique partitionings ofk n we show that triangles 2 chorded odd cycles 2 chorded even wheels and other subgraphs ofk n induce facets of the theoretical results described here have been used to design an empirically efficient cutting plane algorithm with which large real world instances of the clique partitioning problem could be solved these computational results can be found in grotschel and wakabayashi 1989

nonlinear 0 1 programming ii dominance relations and algorithms a nonlinear 0 1 program can be restated as a multilinear 0 1 program which in turn is known to be equivalent to a linear 0 1 program with generalized covering g c inequalities in a companion paper 6 we have defined a family of linear inequalities that contains more compact smaller cardinality linearizations of a multilinear 0 1 program than the one based on the g c inequalities in this paper we analyze the dominance relations between inequalities of the above family in particular we give a criterion that can be checked in linear time for deciding whether a g c inequality can be strengthened by extending the cover from which it was derived we then describe a class of algorithms based on these results and discuss our computational experience we conclude that the g c inequalities can be strengthened most of the time to an extent that increases with problem density in particular the algorithm using the strengthening procedure outperforms the one using only g c inequalities whenever the number of nonlinear terms per constraint exceeds about 12 15 and the difference in their performance grows with the number of such terms

on convex relaxations for quadratically constrained quadratic programming we consider convex relaxations for the problem of minimizing a possibly nonconvex quadratic objective subject to linear and possibly nonconvex quadratic constraints let mathcal f denote the feasible region for the linear constraints we first show that replacing the quadratic objective and constraint functions with their convex lower envelopes on mathcal f is dominated by an alternative methodology based on convexifying the range of the quadratic form genfrac 0 0pt 1 x genfrac 0 0pt 1 x t for x in mathcal f we next show that the use of alpha bb underestimators as computable estimates of convex lower envelopes is dominated by a relaxation of the convex hull of the quadratic form that imposes semidefiniteness and linear constraints on diagonal terms finally we show that the use of a large class of d c difference of convex underestimators is dominated by a relaxation that combines semidefiniteness with rlt constraints

matroid optimisation problems with nested non linear monomials in the objective function recently buchheim and klein discrete appl math 177 34 52 2014 suggested to study polynomial time solvable optimisation problems with linear objective functions combined with exactly one additional quadratic monomial they concentrated on special quadratic spanning tree or forest problems we extend their results to general matroid optimisation problems with a set of nested monomials in the objective function we study polytopes arising from the standard linearisation of the monomials our results provide insight on the polyhedral structure of matroid optimisation problems with arbitrary polynomial objective function with a focus on separation algorithms and strengthened cutting planes extending results by edmonds comb struct appl 69 87 1970 for the matroid polytope we present a complete description for the linearised polytope indeed apart from the standard linearisation one needs appropriately strengthened rank inequalities satisfying certain non separability conditions the separation problem of these extended rank inequalities reduces to a submodular function minimisation problem in the case of exactly one additional non linear monomial we completely characterise the facetial structure of the associated polytope

minimization of half products in this paper a special class of quadratic functions the so called half products are considered it is shown that while the minimization over the set of binary n vectors for half products is np complete an e approximating solution can be found in polynomial time for any e 0

antiweb wheel inequalities and their separation problems over the stable set polytopes a stable set in a graph g is a set of pairwise nonadjacent vertices the problem of finding a maximum weight stable set is one of the most basic hard problems an important approach to this problem is to formulate it as the problem of optimizing a linear function over the convex hull stab g of incidence vectors of stable sets since it is impossible unless co to obtain a concise characterization of stab g as the solution set of a system of linear inequalities it is a more realistic goal to find large classes of valid inequalities with the property that the corresponding separation problem given a point x find if possible an inequality in the class that x violates is efficiently solvable some known large classes of separable inequalities are the trivial edge cycle and wheel inequalities in this paper we give a polynomial time separation algorithm for the t antiweb inequalities of trotter we then introduce an even larger class in fact a sequence of classes of valid inequalities called t antiweb s wheel inequalities this class is a common generalization of the t antiweb inequalities and the wheel inequalities we also give efficient separation algorithms for them

a conservative multi tracer transport scheme for spectral element spherical grids atmospheric models used for practical climate simulation must be capable handling the transport of hundreds of tracers for computational efficiency conservative multi tracer semi lagrangian type transport schemes are appropriate global models based on high order galerkin approach employ highly non uniform spectral element grids and semi lagrangian transport is a challenge on those grids a conservative semi lagrangian scheme spelt spectral element lagrangian transport employing a multi moment compact reconstruction procedure is developed for non uniform quadrilateral grids the scheme is based on a characteristic semi lagrangian method that avoids complex and expensive upstream area computations the spelt scheme has been implemented in the high order method modeling environment homme which is based on a cubed sphere grid with spectral element spatial discretization additionally we show the strong scalability and multi tracer efficiency using several benchmark tests the spelt solution can be made monotonic positivity preserving by combining the flux corrected transport algorithm which is demonstrated on a uniform resolution grid in particular spelt can be efficiently used for non uniform grids and provides accurate and stable results for high resolution meshes

on the copositive representation of binary and continuous nonconvex quadratic programs in this paper we model any nonconvex quadratic program having a mix of binary and continuous variables as a linear program over the dual of the cone of copositive matrices this result can be viewed as an extension of earlier separate results which have established the copositive representation of a small collection of np hard problems a simplification which reduces the dimension of the linear conic program and an extension to complementarity constraints are established and computational issues are discussed

a lagrangian relaxation approach to the edge weighted clique problem the b clique polytope cp n b is the convex hull of the node and edge incidence vectors of all subcliques of size at most b of a complete graph on n nodes including the boolean quadric polytope qp n as a special case and being closely related to the quadratic knapsack polytope it has received considerable attention in the literature in particular the max cut problem is equivalent with optimizing a linear function over qp n n the problem of optimizing linear functions over cp n b has so far been approached via heuristic combinatorial algorithms and cutting plane methods we study the structure of cp n b in further detail and present a new computational approach to the linear optimization problem based on lucena s suggestion of integrating cutting planes into a lagrangian relaxation of an integer programming problem in particular we show that the separation problem for tree inequalities becomes polynomial in our lagrangian framework finally computational results are presented

a decomposition method for quadratic zero one programming this paper proposes a decomposition method to compute a lower bound for unconstrained quadratic zero one minimization first we show that any quadratic function can be expressed as a sum of particular quadratic functions whose minima can be computed by a simple branch and bound algorithm then assuming some hypothesis we prove that among all possible decompositions the best one can be found by a lagrangian decomposition method moreover we show that our algorithm gives at least the roof dual bound and should give better results in practice eventually computational results and comparison with pardalos and rodgers algorithm demonstrate the efficiency of our method for medium size problems up to 100 variables

the boolean quadric polytope some characteristics facets and relatives we study unconstrained quadratic zero one programming problems havingn variables from a polyhedral point of view by considering the boolean quadric polytope qpn inn n 1 2 dimensions that results from the linearization of the quadratic form we show that qpn has a diameter of one descriptively identify three families of facets of qpn and show that qpn is symmetric in the sense that all facets of qpn can be obtained from those that contain the origin by way of a mapping the naive linear programming relaxation qpnlp of qpn is shown to possess the trubin property and its extreme points are shown to be 0 1 2 1 valued furthermore o n3 facet defining inequalities of qpn suffice to cut off all fractional vertices of qpnlp whereas the family of facets described by us has at least o 3n members the problem is also studied for sparse quadratic forms i e when several or many coefficients are zero and conditions are given for the previous results to carry over to this case polynomially solvable problem instances are discussed and it is shown that the known polynomiality result for the maximization of nonnegative quadratic forms can be obtained by simply rounding the solution to the linear programming relaxation in the case that the graph induced by the nonzero coefficients of the quadratic form is series parallel a complete linear description of the associated boolean quadric polytope is given the relationship of the boolean quadric polytope associated to sparse quadratic forms with the vertex packing polytope is discussed as well

applications of cut polyhedra ii abstract in this paper and in its continuation part ii this issue we group within a unified framework many applications of the following polyhedra cut boolean quadric hypermetric and metric polyhedra we treat in particular the following applications l1 and l1 metrics in functional analysis the max cut problem the boole problem and multicommodity flow problems in combinatorial optimization lattice holes in geometry of numbers density matrices of many fermions systems in quantum mechanics we present some other applications in probability theory statistical data analysis and design theory in this first part after introducing the main definitions and operations for cut polyhedra we describe the connections with l1 metrics and with other metric properties and we consider in particular the applications to some classes of metrics arising from graphs normed spaces and lattices

facets for the cut cone i we study facets of the cut conec n i e the cone of dimension 1 2n n 1 generated by the cuts of the complete graph onn vertices actually the study of the facets of the cut cone is equivalent in some sense to the study of the facets of the cut polytope we present several operations on facets and in particular a lifting procedure for constructing facets ofc n 1 from given facets of the lower dimensional conec n after reviewing hypermetric valid inequalities we describe the new class of cycle inequalities and prove the facet property for several subclasses the new class of parachute facets is developed and other known facets and valid inequalities are presented

cutting planes for rlt relaxations of mixed 0 1 polynomial programs the reformulation linearization technique due to sherali and adams can be used to construct hierarchies of linear programming relaxations of mixed 0 1 polynomial programs as one moves up the hierarchy the relaxations grow stronger but the number of variables increases exponentially we present a procedure that generates cutting planes at any given level of the hierarchy by optimally weakening linear inequalities that are valid at any given higher level computational experiments conducted on instances of the quadratic knapsack problem indicate that the cutting planes can close a significant proportion of the integrality gap

tighter linear and semidefinite relaxations for max cut based on the lovasz schrijver lift and project procedure we study how the lift and project method introduced by lovasz and schrijver siam j optim 1 1991 pp 166 190 applies to the cut polytope we show that the cut polytope of a graph can be found in k iterations if there exist k edges whose contraction produces a graph with no k5 minor therefore for a graph g with n ge 4 nodes with stability number alpha g n 4 iterations suffice instead of the m number of edges iterations required in general and under some assumption n alpha g 3 iterations suffice the exact number of needed iterations is determined for small n le 7 by a detailed analysis of the new relaxations if positive semidefiniteness is added to the construction then one finds in one iteration a relaxation of the cut polytope which is tighter than its basic semidefinite relaxation and than another one introduced recently by anjos and wolkowicz discrete appl math to appear we also show how the lovasz schrijver relaxations for the stable set polytope of g can be strengthened using the corresponding relaxations for the cut polytope of the graph g nabla obtained from g by adding a node adjacent to all nodes of g

complete positivity and distance avoiding sets we introduce the cone of completely positive functions a subset of the cone of positive type functions and use it to fully characterize maximum density distance avoiding sets as the optimal solutions of a convex optimization problem as a consequence of this characterization it is possible to reprove and improve many results concerning distance avoiding sets on the sphere and in euclidean space

facets of the bipartite subgraph polytope the bipartite subgraph polytope pbg of a graph g v e is the convex hull of the incidence vectors of all edge sets of bipartite subgraphs of g we show that all complete subgraphs of g of odd order and all so called odd bicycle wheels contained in g induce facets of pbg moreover we describe several methods with which new facet defining inequalities of pbg can be constructed from known ones examples of these methods are contraction of node sets in odd complete subgraphs odd subdivision of edges certain splittings of nodes and subdivision of all edges of a cut using these methods we can construct facet defining inequalities of pbg having coefficients of order v 2

on the relationship between convex bodies related to correlation experiments with dichotomic observables in this paper we explore further the connections between convex bodies related to quantum correlation experiments with dichotomic variables and related bodies studied in combinatorial optimization especially cut polyhedra such a relationship was established in avis et al 2005 j phys a math gen 38 10971 87 with respect to bell inequalities we show that several well known bodies related to cut polyhedra are equivalent to bodies such as those defined by tsirelson 1993 hadronic j suppl 8 329 45 to represent hidden deterministic behaviours quantum behaviours and no signalling behaviours among other things our results allow a unique representation of these bodies give a necessary condition for vertices of the no signalling polytope and give a method for bounding the quantum violation of bell inequalities by means of a body that contains the set of quantum behaviours optimization over this latter body may be performed efficiently by semidefinite programming in the second part of the paper we apply these results to the study of classical correlation functions we provide a complete list of tight inequalities for the two party case with m n dichotomic observables when m 4 n 4 and when min m n 3 and give a new general family of correlation inequalities

some thoughts on combinatorial optimisation abstract a group of young researchers from the esi x summer school hec jouy en josas 1994 give their personal views on the current status of and prospects for combinatorial optimisation several issues are considered and discussed with emphasis on a selected number of techniques heuristics and polyhedral approaches and problems knapsack quadratic 0 1 programming machine scheduling routing and network design

an extended formulation approach to the edge weighted maximal clique problem we consider an extended formulation approach to the edge weighted maximal clique problem the problem is formulated by using additional variables for the set of nodes with the natural variables for the set of edges we show that the proposed formulation is superior to the natural formulation both theoretically and practically by using the projection technique we can also derive new classes of facet defining inequalities for the lower dimensional polytope of the natural variables computational results are reported

exactness of semidefinite relaxations for nonlinear optimization problems with underlying graph structure this work is concerned with finding a global optimization technique for a broad class of nonlinear optimization problems including quadratic and polynomial optimization problems the main objective of this paper is to investigate how the hidden structure of a given real complex valued optimization problem makes it easy to solve to this end three conic relaxations are proposed necessary and sufficient conditions are derived for the exactness of each of these relaxations and it is shown that these conditions are satisfied if the optimization problem is highly structured more precisely the structure of the optimization problem is mapped into a generalized weighted graph where each edge is associated with a weight set extracted from the coefficients of the optimization problem in the real valued case it is shown that the relaxations are all exact if each weight set is sign definite and in addition a condition is satisfied for each cycle of the graph it is also proved that if some of these conditi

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

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

binary positive semidefinite matrices and associated integer polytopes we consider the positive semidefinite psd matrices with binary entries along with the corresponding integer polytopes we begin by establishing some basic properties of these matrices and polytopes then we show that several families of integer polytopes in the literature the cut boolean quadric multicut and clique partitioning polytopes are faces of binary psd polytopes finally we present some implications of these polyhedral relationships in particular we answer an open question in the literature on the max cut problem by showing that the rounded psd inequalities define a polytope

efficient reduction of polynomial zero one optimization to the quadratic case we address the problem of optimizing a polynomial with real coefficients over binary variables we show that a complete polyhedral description of the linearization of such a problem can be derived in a simple way from the polyhedral description of the linearization of some quadratic optimization problem the number of variables in the latter linearization is only slightly larger than in the former if polynomial constraints are present in the original problem then their linearized counterparts carry over to the linearized quadratic problem unchanged if the original problem formulation does not contain any constraints we obtain a reduction to unconstrained quadratic zero one optimization which is equivalent to the well studied max cut problem the separation problem for general unconstrained polynomial zero one optimization thus reduces to the separation problem for the cut polytope this allows us to transfer the entire knowledge gained for the latter polytope by intensive research and in particular the sophisticated separation techniques that have been developed we report preliminary experimental results obtained with a straightforward implementation of this approach

berge acyclic multilinear 0 1 optimization problems abstract the problem of optimizing a multilinear polynomial f in 0 1 variables arises in applications from many different areas we are interested in resolution methods based on reformulating the polynomial problem into an equivalent linear one an approach that attempts to draw benefit from the extensive literature in integer linear programming more precisely we characterize instances for which the classical standard linearization procedure guarantees integer optimal solutions we show that the standard linearization polytope ph is integer if and only if the hypergraph h defined by the higher degree monomials of f is berge acyclic or equivalently when the matrix defining ph is balanced this characterization follows from more general conditions that guarantee integral optimal vertices for a relaxed formulation depending on the sign pattern of the monomials of f

new facets and a branch and cut algorithm for the weighted clique problem abstract we consider a polyhedral approach to the weighted maximal b clique problem given a node and edge weighted complete graph the problem is to find a complete subgraph clique with no more than b nodes such that the sum of the weights of all nodes and edges in the clique is maximal we introduce four new classes of facet defining inequalities for the associated b clique polytope one of these inequality classes constitutes a generalization of the well known tree inequalities the other classes are associated with multistars we use these inequalities together with other classes of facet defining inequalities in a branch and cut algorithm for the problem we give a description of this algorithm including some separation procedures and present the computational results for different sets of test problems the computation times that are obtained indicate that this algorithm is more efficient than previously described algorithms for the problem

unbounded convex sets for non convex mixed integer quadratic programming this paper introduces a fundamental family of unbounded convex sets that arises in the context of non convex mixed integer quadratic programming it is shown that any mixed integer quadratic program with linear constraints can be reduced to the minimisation of a linear function over a face of a set in the family some fundamental properties of the convex sets are derived along with connections to some other well studied convex sets several classes of valid and facet inducing inequalities are also derived

maximizing a submodular function by integer programming polyhedral results for the quadratic case abstract we give a new mixed integer programming mip formulation for the quadratic cost partition problem that is derived from a mip formulation for maximizing a submodular function several classes of valid inequalities for the convex hull of the feasible solutions are derived using the valid inequalities for the node packing polyhedron facet defining conditions and separation algorithms are discussed and computational results are reported

cut polytopes boolean quadratic polytopes and nonnegative quadratic pseudo boolean functions in this paper we present a class of valid inequalities as well as a class of facets for the cut polytope of the complete graph it is shown that many of the known classes of valid inequalities e g triangle hypermetric and cycle inequalities belong to this class by specializing some of the parameters large classes of facets can be defined it is shown that each of these classes contains at least n 3n 3 facets where n 10 denotes the number of vertices and that the corresponding separation problems are polynomially solvable these results are based on a one to one correspondence established between the set of valid inequalities facets for the cut polytope of kn 1 and the set of nonnegative extremal quadratic pseudo boolean functions in n variables

min cut clustering we describe a decomposition framework and a column generation scheme for solving a min cut clustering problem the subproblem to generate additional columns is itself an np hard mixed integer programming problem we discuss strong valid inequalities for the subproblem and describe some efficient solution strategies computational results on compiler construction problems are reported

persistency in 0 1 polynomial programming an optimal solution to the continuous relaxation of a mixed integer 0 1 linear programming problem is defined to be persistent if the set of 0 1 variables realizing binary values retains those same binary values in at least one integer optimum a mixed integer 0 1 linear program is said to possess the persistency property or equivalently to be persistent if every optimal solution to the continuous relaxation is a persistent solution we study the issue of persistency for a special family of mixed integer 0 1 linear programming problems which derive from the linearization strategy of sherali and adams 1990 in particular we first examine the hierarchy of linearizations which result from applying this strategy to the general class of unconstrained 0 1 polynomial programming problems and then consider a class of specially structured constrained problems our study of persistency differs from previous works in that we use the explicit algebraic representation of the convex hull of feasible binary solutions available at the highest level of this hierarchy as a tool for making inferences relative to optimal integer solutions from optimal continuous solutions each level of this hierarchy consists of a mixed integer 0 1 linear program and we provide for any given optimal solution to the continuous relaxation of any member of this hierarchy a set of sufficient conditions on the optimal dual multipliers for the solution to be persistent these sufficient conditions turn out to be satisfied at every optimal solution of the quadratic level and thus allow for the reformulation of any unconstrained 0 1 quadratic programming problem as a persistent mixed integer linear program in a suitably defined higher dimensional space we subsequently address the fa mily of constrained 0 1 polynomial programs and use a projection operation to prove that a special family of 0 1 linear programs possesses the persistency property included in this family is a generalization of the vertex packing problem that permits quadratic objective function coefficients unlike earlier published works on the vertex packing and related problems our proofs do not rely on balinski s 1965 extreme point characterization

the symmetric quadratic traveling salesman problem in the quadratic traveling salesman problem a cost is associated with any three nodes traversed in succession this structure arises e g if the succession of two edges represents energetic conformations a change of direction or a possible change of transportation means in the symmetric case costs do not depend on the direction of traversal we study the polyhedral structure of a linearized integer programming formulation of the symmetric quadratic traveling salesman problem our constructive approach for establishing the dimension of the underlying polyhedron is rather involved but offers a generic path towards proving facetness of several classes of valid inequalities we establish relations to facets of the boolean quadric polytope exhibit new classes of polynomial time separable facet defining inequalities that exclude conflicting configurations of edges and provide a generic strengthening approach for lifting valid inequalities of the usual traveling salesman problem to stronger valid inequalities for the symmetric quadratic traveling salesman problem applying this strengthening to subtour elimination constraints gives rise to facet defining inequalities but finding a maximally violated inequality among these is np complete for the simplest comb inequality with three teeth the strengthening is no longer sufficient to obtain a facet preliminary computational results indicate that the new cutting planes may help to considerably improve the quality of the root relaxation in some important applications

formulating logical implications in combinatorial optimisation abstract when practical problems are formulated as combinatorial optimisation models one must often include logical implications between decisions it is useful to express these implications as linear constraints involving binary variables since linear constraints offer the possibility of using linear programming and branch and bound as an initial solution method often this formulation step of the modelling process seems far from evident to many practitioners students and researchers therefore it is of interest to make simple rules available to clarify and help in this process in this educationally oriented tutorial paper we introduce such a rule lip the logical implication principle it offers an easy and automatic way to translate elementary logical implications involving 0 1 variables into linear constraints based on an extremely simple cutting plane we demonstrate how the rule is extended using complementarity and implicit binary variables and leads to a simple but powerful instrument this is illustrated by several examples of applications in various fields of operational research the paper culminates in the description of a novel set of constraints which fully eliminate all permutation equivalent solutions to numbered clustering problems

analytic extension of the holomorphic discrete series introduction the embedding r c naturally divides c into two half planes each with r as boundary these domains together with their function theory play an important role in the harmonic analysis of r as but one example there is the theorem of paley and wiener that describes any square integrable function on r as a sum of boundary values of holomorphic functions in hardy spaces on these domains much of the theory has natural extensions to r to tube domains in c based on cones and even to more general siegel domains however the dominant influence is abelian harmonic analysis on the other hand if g is a semisimple lie group over r it may often be embedded in its natural complexification gc does any of the abelian

an evolutionary heuristic for quadratic 0 1 programming abstract in this paper we present a heuristic algorithm for the well known unconstrained quadratic 0 1 programming problem the approach is based on combining solutions in a genetic paradigm and incorporates intensification algorithms used to improve solutions and speed up the method extensive computational experiments on instances with up to 500 variables are presented and we compare our approach both with powerful heuristic and exact algorithms from the literature establishing the effectiveness of the method in terms of solutions quality and computing time

a polyhedral study of nonconvex quadratic programs with box constraints by reformulating quadratic programs using necessary optimality conditions we obtain a linear program with complementarity constraints for the case where the only constraints are bounds on the variables we study a relaxation based on a subset of the optimality conditions by characterizing its convex hull we obtain a large class of valid inequalities these inequalities are used in a branch and cut scheme see 13

an application of the lovasz schrijver m k k operator to the stable set problem although the lift and project operators of lovasz and schrijver have been the subject of intense study their m k k operator has received little attention we consider an application of this operator to the stable set problem we begin with an initial linear programming lp relaxation consisting of clique and non negativity inequalities and then apply the operator to obtain a stronger extended lp relaxation we discuss theoretical properties of the resulting relaxation describe the issues that must be overcome to obtain an effective practical implementation and give extensive computational results remarkably the upper bounds obtained are sometimes stronger than those obtained with semidefinite programming techniques