hub location as the minimization of a supermodular set function this paper highlights how a general class of hub location problems can be modeled as the minimization of a real valued supermodular set function well known problems such as uncapacitated hub location p hub median and hub arc location among others are shown to be particular cases of this class two integer programming formulations are introduced and compared one uses path based variables frequently employed in hub location whereas the other exploits properties of supermodular functions in addition several worst case bounds for a greedy and a local improvement heuristic are obtained for the general class and for some particular cases in which sharper bounds can be devised computational experiments are performed to compare both formulations when used with a general purpose solver computational results obtained on benchmark instances confirm the superiority of the supermodular formulation

linear programming for the 0 1 quadratic knapsack problem abstract in this paper we consider the quadratic knapsack problem which consists in maximizing a positive quadratic pseudo boolean function subject to a linear capacity constraint we propose a new method for computing an upper bound this method is based on the solution of a continuous linear program constructed by adding to a classical linearization of the problem some constraints rebundant in 0 1 variables but nonredundant in continuous variables the obtained upper bound is better than the bounds given by other known methods we also propose an algorithm for computing a good feasible solution this algorithm is an elaboration of the heuristic methods proposed by chaillou hansen and mahieu and by gallo hammer and simeone the relative error between this feasible solution and the optimum solution is generally less than 1 we show how these upper and lower bounds can be efficiently used to determine the values of some variables at the optimum finally we propose a branch and bound algorithm for solving the quadratic knapsack problem and report extensive computational tests

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

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

computational experience with a bundle approach for semidefinite cutting plane relaxations of max cut and equipartition we propose a dynamic version of the bundle method to get approximate solutions to semidefinite programs with a nearly arbitrary number of linear inequalities our approach is based on lagrangian duality where the inequalities are dualized and only a basic set of constraints is maintained explicitly this leads to function evaluations requiring to solve a relatively simple semidefinite program our approach provides accurate solutions to semidefinite relaxations of the max cut and the equipartition problem which are not achievable by direct approaches based only on interior point methods

second order cone relaxations for binary quadratic polynomial programs several types of relaxations for binary quadratic polynomial programs can be obtained using linear second order cone or semidefinite techniques in this paper we propose a general framework to construct conic relaxations for binary quadratic polynomial programs based on polynomial programming using our framework we re derive previous relaxation schemes and provide new ones in particular we present three relaxations for binary quadratic polynomial programs the first two relaxations based on second order cone and semidefinite programming represent a significant improvement over previous practical relaxations for several classes of nonconvex binary quadratic polynomial problems from a practical point of view due to the computational cost semidefinite based relaxations for binary quadratic polynomial problems can be used only to solve small to midsize instances to improve the computational efficiency for solving such problems we propose a third relaxation based purely on second order cone programming computational tests on different classes of nonconvex binary quadratic polynomial problems including quadratic knapsack problems show that the second order cone based relaxation outperforms the semidefinite based relaxations that are proposed in the literature in terms of computational efficiency and it is comparable in terms of bounds

when is the space of minimal usco cusco maps a topological vector space abstract we will define a vector structure on the space of minimal usco resp cusco maps from a locally compact space to a stegall locally convex space we will prove that these spaces equipped with the topology of uniform convergence on compacta are locally convex spaces

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

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

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

computational experience with an interior point cutting plane algorithm there has been a great deal of success in the last 20 years with the use of cutting plane algorithms to solve specialized integer programming problems generally these algorithms work by solving a sequence of linear programming relaxations of the integer programming problem and they use the simplex algorithm to solve the relaxations in this paper we describe experiments using a predictor corrector interior point method to solve the relaxations for some problems the interior point code requires considerably less time than a simplex based cutting plane algorithm

an efficient maximization algorithm with implications in min max predictive control in this technical note an algorithm for binary quadratic programs defined by matrices with band structure is proposed it was shown in the article by t alamo d m de la pentildea d limon and e f camacho ldquoconstrained min max predictive control modifications of the objective function leading to polynomial complexity rdquo ieee tran autom control vol 50 pp 710 714 may 2005 that this class of problems arise in robust model predictive control when min max techniques are applied although binary quadratic problems belongs to a class of np complete problems the computational burden of the proposed maximization algorithm for band matrices is polynomial with the dimension of the optimization variable and exponential with the band size computational results and comparisons on several hundred test problems demonstrate the efficiency of the algorithm

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

an unconstrained minimization method for solving low rank sdp relaxations of the maxcut problem in this paper we consider low rank semidefinite programming lrsdp relaxations of combinatorial quadratic problems that are equivalent to the maxcut problem using the gramian representation of a positive semidefinite matrix the lrsdp problem can be formulated as the nonconvex nonlinear programming problem of minimizing a quadratic function with quadratic equality constraints for the solution of this problem we propose a continuously differentiable exact merit function that exploits the special structure of the constraints and we use this function to define an efficient and globally convergent algorithm finally we test our code on an extended set of instances of the maxcut problem and we report comparisons with other existing codes

solving max cut to optimality by intersecting semidefinite and polyhedral relaxations we present a method for finding exact solutions of max cut the problem of finding a cut of maximum weight in a weighted graph we use a branch and bound setting that applies a dynamic version of the bundle method as bounding procedure this approach uses lagrangian duality to obtain a nearly optimal solution of the basic semidefinite max cut relaxation strengthened by triangle inequalities the expensive part of our bounding procedure is solving the basic semidefinite relaxation of the max cut problem which has to be done several times during the bounding process we review other solution approaches and compare the numerical results with our method we also extend our experiments to instances of unconstrained quadratic 0 1 optimization and to instances of the graph equipartition problem the experiments show that our method nearly always outperforms all other approaches in particular for dense graphs where linear programming based methods fail our method performs very well exact solutions are obtained in a reasonable time for any instance of size up to n 100 independent of the density for some problems of special structure we can solve even larger problem classes we could prove optimality for several problems of the literature where to the best of our knowledge no other method is able to do so

solving quadratic 0 1 problems by semidefinite programs and cutting planes we present computational experiments for solving quadratic 0 1 problems our approach combines a semidefinite relaxation with a cutting plane technique and is applied in a branch and bound setting our experiments indicate that this type of approach is very robust and allows to solve many moderately sized problems having say less than 100 binary variables in a routine manner 1998 the mathematical programming society inc published by elsevier science b v

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

solutions for an operator equation under the conditions of pairs of paralleled lower and upper solutions abstract in this paper under the condition of two pairs of strict lower and upper solutions and using the concept of e 1 b limit increasing operator some multiplicity results for an operator equation are obtained by the method of the fixed point index

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

finding quasi core with simulated stacked neural networks studying networks is promising for diverse applications we are often interested in exploring significant substructures in different types of real life networks finding cliques which denote a complete subgraph of a graph is one such important problem in network analysis interestingly many real life networks often contain a significant number of almost quasi complete subgraphs which are not entirely complete due to the presence of noise considering these networks as weighted adds further challenges to the problem finding quasi complete subgraphs in weighted graphs has never been formally addressed in this paper we propose a stacked neural network model for finding out the largest quasi complete module core in weighted graphs we show the effectiveness of the proposed approach on dimacs graphs we also highlight its utility in analyzing scientific collaboration networks social networks and biological networks

adaptive memory tabu search for binary quadratic programs recent studies have demonstrated the effectiveness of applying adaptive memory tabu search procedures to combinatorial optimization problems in this paper we describe the development and use of such an approach to solve binary quadratic programs computational experience is reported showing that the approach optimally solves the most difficult problems reported in the literature for challenging problems of limited size which are capable of being approached by exact procedures we find optimal solutions considerably faster than the best reported exact method moreover we demonstrate that our approach is significantly more efficient and yields better solutions than the best heuristic method reported to date finally we give outcomes for larger problems that are considerably more challenging than any currently reported in the literature

cluster analysis and mathematical programming given a set of entities cluster analysis aims at finding subsets called clusters which are homogeneous and or well separated as many types of clustering and criteria for homogeneity or separation are of interest this is a vast field a survey is given from a mathematical programming viewpoint steps of a clustering study types of clustering and criteria are discussed then algorithms for hierarchical partitioning sequential and additive clustering are studied emphasis is on solution methods i e dynamic programming graph theoretical algorithms branch and bound cutting planes column generation and heuristics

performance of simulated annealing based heuristic for the unconstrained binary quadratic programming problem abstract the unconstrained binary quadratic programming problem bqp is known to be np hard and has many practical applications this paper presents a simulated annealing sa based heuristic for the bqp the new sa heuristic for the bqp is based on a simple 1 opt local search heuristic and designed with a simple cooling schedule but the multiple annealing processes are adopted to show practical performances of the sa we test on publicly available benchmark instances of large size ranging from 500 to 2500 variables and compare them with other heuristics such as multi start local search the previous sa tabu search and genetic algorithm incorporating the 1 opt local search computational results indicate that our sa leads to high quality solutions with short times and is more effective than the competitors particularly for the largest benchmark set furthermore the values of new best known solutions found by the sa for several large instances are also reported

optimal toll design a lower bound framework for the asymmetric traveling salesman problem we propose a framework of lower bounds for the asymmetric traveling salesman problem tsp based on approximating the dynamic programming formulation with different basis vector sets we discuss how several well known tsp lower bounds correspond to intuitive basis vector choices and give an economic interpretation wherein the salesman must pay tolls as he travels between cities we then introduce an exact reformulation that generates a family of successively tighter lower bounds all solvable in polynomial time we show that the base member of this family yields a bound greater than or equal to the well known held karp bound obtained by solving the linear programming relaxation of the tsp s integer programming arc based formulation

strong uniform continuity abstract let b be an ideal of subsets of a metric space x d this paper considers a strengthening of the notion of uniform continuity of a function restricted to members of b which reduces to ordinary continuity when b consists of the finite subsets of x and agrees with uniform continuity on members of b when b is either the power set of x or the family of compact subsets of x the paper also presents new function space topologies that are well suited to this strengthening as a consequence of the general theory we display necessary and sufficient conditions for continuity of the pointwise limit of a net of continuous functions

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

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

essentially smooth lipschitz functions in this paper we address some of the most fundamental questions regarding the differentiability structure of locally lipschitz functions defined on separable banach spaces for example we examine the relationship between integrability d representability and strict differentiability in addition to this we show that on any separable banach space there is a significant family of locally lipschitz functions that are integrable d representable and possess desirable differentiability properties we also present some striking applications of our results to distance functions

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

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

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

minimal usco and minimal cusco maps and compactness abstract we prove a nice generalization of the arzela ascoli theorem from continuous functions to minimal usco cusco maps into metric spaces let x be a locally compact space y d be a metric space k y be the space of all nonempty compact subsets of y and mu x y be the space of minimal usco maps from x to y the family e mu x y is compact in k y x equipped with the topology uc of uniform convergence on compact sets if and only if e is closed in mu x y uc pointwise bounded and densely equicontinuous the same result holds also for compact subsets of mc x y uc the space of minimal cusco maps from x to a banach space y

simulated annealing for the unconstrained quadratic pseudo boolean function abstract in this paper we develop a simulated annealing sa based heuristic for the unconstrained quadratic pseudo boolean function an algorithm that solves the problem in o n 2 at each temperature of the cooling schedule is given the performance of sa based heuristic is compared with existing bounding algorithms for this problem computational results and comparisons on several hundred test problems demonstrate the efficiency of our heuristic in terms of solution quality and computational time a new set of hard test problems with their best solution is provided to facilitate future comparison

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

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

the data correcting algorithm for the minimization of supermodular functions the data correcting dc algorithm is a recursive branch and bound type algorithm in which the data of a given problem instance are heuristically corrected at each branching in such a way that the new instance will be as close as possible to polynomially solvable and the result satisfies a prescribed accuracy the difference between optimal and current solution in this paper the dc algorithm is applied to determining exact or approximate global minima of supermodular functions the working of the algorithm is illustrated by an instance of the simple plant location spl problem computational results obtained for the quadratic cost partition problem qcp show that the dc algorithm outperforms a branch and cut algorithm not only for sparse graphs but also for nonsparse graphs with density more than 40 often with speeds 100 times faster

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

the maximum capture problem with random utilities problem formulation and algorithms abstract a model for the optimal location of new facilities in a competitive market is introduced under the hypothesis that customers behavior can be modeled by random utility functions it means that the company that wished to locate uses a random utility model to forecast the market share of a location therefore the company cannot forecast the behavior of every customer in a deterministic fashion but is able to embed him by a probability distribution three formulations are proposed to compute upper bounds of the objective function and compared in a numerical simulation a branch and bound method is developed and tested on examples with up to 50 potential locations and a variable neighborhood search heuristic is proposed to solve larger instances

three positive fixed points of nonlinear operators on ordered banach spaces we generalize the fixed point theorem of leggett williams which is a theorem giving conditions that imply the existence of three fixed points of an operator defined on a cone in a banach space we then show how to apply our theorem to prove the existence of three positive solutions to a second order discrete boundary value problem 2001 elsevier science ltd all rights reserved

multiple solutions for nonlinear operator equations under the condition of two pairs of paralleled lower and upper solutions in this paper under the condition of two pairs of paralleled lower and upper solutions some new multiple solution theorems are obtained by using the fixed point index theory the theoretical results are applied to the nonlinear hammerstein integral equation

an application of combinatorial optimization to statistical physics and circuit layout design we study the problem of finding ground states of spin glasses with exterior magnetic field and the problem of minimizing the number of vias holes on a printed circuit board or contacts on a chip subject to pin preassignments and layer preferences the former problem comes up in solid state physics and the latter in very large scale integrated vlsi circuit design and in printed circuit board design both problems can be reduced to the max cut problem in graphs based on a partial characterization of the cut polytope we design a cutting plane algorithm and report on computational experience with it our method has been used to solve max cut problems on graphs with up to 1 600 nodes

minimization of a quadratic pseudo boolean function abstract we present a branch and bound algorithm for minimizing a quadratic pseudo boolean function f x at each node of the search tree the lower bound is computed in three phases and is equal to b 1 b 2 b 3 computation of b 1 is based upon roof duality b 2 uses the characterization of some positive quadratic posiforms associated with the directed cycles of the implication graph of aspvall plass and tarjan and b 3 is computed by searching in a posiform of degree 4 some subfunctions which cannot be equal to zero these subfunctions are found by using the notion of implication between literals computational results on several hundred test problems with up to 100 variables demonstrate the efficiency of this lower bound

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

multiple solutions for nonlinear operators and applications in this paper some new multiple solutions theorems are obtained by introducing some new concepts such as the o bounded cone the abstract results obtained here are applied to ordinary differential equations and nonlinear systems of differential equations

laplacian eigenvalues and the maximum cut problem we introduce and study an eigenvalue upper bound g on the maximum cut mc g of a weighted graph the function g has several interesting properties that resemble the behaviour of mc g the following results are presented we show that is subadditive with respect to amalgam and additive with respect to disjoint sum and 1 sum we prove that g is never worse that 1 131 mc g for a planar or more generally a weakly bipartite graph with nonnegative edge weights we give a dual characterization of g and show that g is computable in polynomial time with an arbitrary precision

generalization for amann s and leggett williams three solution theorems and applications abstract in this paper some new three solution theorems are obtained moreover the famous amann s and leggett williams three solution theorems in nonlinear functional analysis can be seen as their special cases namely these two famous theorems are united so they are both improved and that some new three solution theorems are applied to ordinary differential equations and some new existence results are obtained

arzela s theorem and strong uniform convergence on bornologies abstract in 1883 arzela 1983 1984 2 gave a necessary and sufficient condition via quasi uniform convergence for the pointwise limit of a sequence of real valued continuous functions on a compact interval to be continuous arzela s work paved the way for several outstanding papers a milestone was the p s alexandroff convergence introduced in 1948 to tackle the question for a sequence of continuous functions from a topological space not necessarily compact to a metric space in 2009 in the realm of metric spaces beer and levi 2009 10 found another necessary and sufficient condition through the novel notion of strong uniform convergence on finite sets we offer a direct proof of the equivalence of arzela alexandroff and beer levi conditions the proof reveals the internal gear of these important convergences and sheds more light on the problem we also study the main properties of the topology of strong uniform convergence of functions on bornologies initiated in beer and levi 2009 10

maximization of submodular functions theory and enumeration algorithms abstract submodular functions are powerful tools to model and solve either to optimality or approximately many operational research problems including problems defined on graphs after reviewing some long standing theoretical results about the structure of local and global maxima of submodular functions cherenin s selection rules and his dichotomy algorithm we revise the above mentioned theory and show that our revision is useful for creating new non binary branching algorithms and finding either approximation solutions with guaranteed accuracy or exact ones

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 cutting plane algorithm for the linear ordering problem the linear ordering problem is an np hard combinatorial optimization problem with a large number of applications including triangulation of input output matrices archaeological senation minimizing total weighted completion time in one machine scheduling and aggregation of individual preferences in a former paper we have investigated the facet structure of the 0 1 polytope associated with the linear ordering problem here we report on a new algorithm that is based on these theoretical results the main part of the algorithm is a cutting plane procedure using facet defining inequalities this procedure is combined with various heuristics and branch and bound techniques our computational results compare favorably with the results of existing codes in particular we could triangulate all input output matrices of size up to 60 60 available to us within acceptable time bounds

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