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

ergodic actions of group extensions on von neumann algebras on considere des actions automorphes sur des algebres de von neumann d un groupe localement compact e donne comme une extension topologique o a e g o ou a est abelien compact et second denombrable on cherche a decrire et a classer des actions ergodiques de e quand g est fini

regularity of potential functions of the optimal transportation problem the potential function of the optimal transportation problem satisfies a partial differential equation of monge ampere type in this paper we introduce the notion of a generalized solution and prove the existence and uniqueness of generalized solutions of the problem we also prove the solution is smooth under certain structural conditions on the cost function

lagrangean decomposition a model yielding stronger lagrangean bounds given a mixed integer programming problem with two matrix constraints it is possible to define a lagrangean relaxation such that the relaxed problem decomposes additively into two subproblems each having one of the two matrices of the original problem as its constraints there is one lagrangean multiplier per variable we prove that the optimal value of this new lagrangean dual dominates the optimal value of the lagrangean dual obtained by relaxing one set of constraints and give a necessary condition for a strict improvement we show on an example that the resulting bound improvement can be substantial we show on a complex practical problem how lagrangean decomposition may help uncover hidden special structures and thus yield better solution methodology

classification of minimal actions of a compact kac algebra with amenable dual on injective factors of type iii abstract we classify a certain class of minimal actions of a compact kac algebra with amenable dual on injective factors of type iii the structural analysis of type iii factors and the canonical extension of endomorphisms introduced by izumi are our main technical tools

the characteristic square of a factor and the cocycle conjugacy of discrete group actions on factors this paper has three main purposes rst to associate to any separable factor m a canonical commuting exact square of groups called the characteristic square ofm secondly to formulate at least in a preliminary way a methodology for approaching non smooth classi cation problems in analysis and thirdly to use this approach in conjunction with the characteristic square to complete the cocycle conjugacy classi cation of actions of discrete amenable groups on injective factors of type iii1 this result and previous works on the cocyle conjugacy classi cation problem combine to produce a cocycle conjugacy classi cation theorem which applies simultaneously to all countable amenable groups and all injective separable factors and in which the characteristic square and our new methodology play a crucial role the cocycle conjugacy classi cation problem has by now a long and distinguished history beginning with connes ground breaking outer conjugacy classi cation of single automorphisms of the approximately nite dimensional or afd factor r0 of type ii1 c3 in the intervening 20 years enormous progress has been made the rst step was taken by v jones j who identi ed the true nature of the cocycle conjugacy invariants and completed the classi cation of actions of nite groups on r0 ocneanu o re ned this work and introduced new analytic techniques enabling him to extend the classi cation to countable amenable groups acting on the semi nite afd factor r0 1 and to some actions on afd factors of type iii also sutherland and takesaki using groupoid and duality techniques originating in jt extended the classi cation results to invent math 132 331 380 1998

an exact method based on lagrangian decomposition for the 0 1 quadratic knapsack problem abstract the 0 1 quadratic knapsack problem qkp consists in maximizing a quadratic pseudo boolean function with positive coefficients subject to a linear capacity constraint in this paper we present an exact method to solve this problem this method makes use of the computation of an upper bound for qkp which is derived from lagrangian decomposition it allows us to find the optimum of instances with up to 150 variables whatever their density and with up to 300 variables for medium and low density

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

solving the 0 1 quadratic knapsack problem with a competitive quantum inspired evolutionary algorithm quadratic knapsack problem qkp extends the canonical simple knapsack problem where the value obtained by selecting a subset of objects is a function dependent not only on the value corresponding to individual objects selected but also on their pair wise selection qkp is np hard in stronger sense i e no pseudo polynomial time algorithm is known to exist which can solve qkp instances qkp has been studied intensively due to its simple structure yet challenging difficulty and numerous applications quantum inspired evolutionary algorithm qiea belongs to the class of evolutionary algorithms and exhibits behaviour of an estimation of distribution algorithm eda qiea provides a generic framework that has to be carefully tailored for a given problem to obtain an effective implementation thus several forms of qiea exist in the literature these have been successfully applied on many hard problems a new qiea qiea psa is proposed with improved exploration and exploitation capabilities computational experiments on these benchmarks show that qiea psa is improved significantly both in terms of the quality of solutions and speed of convergence on several benchmark qkp instances the ideas incorporated are general enough and can be utilized with advantage on other similar and not so similar problems

parametric convex quadratic relaxation of the quadratic knapsack problem abstract we consider a parametric convex quadratic programming cqp relaxation for the quadratic knapsack problem qkp this relaxation maintains partial quadratic information from the original qkp by perturbing the objective function to obtain a concave quadratic term the nonconcave part generated by the perturbation is then linearized by a standard approach that lifts the problem to matrix space we present a primal dual interior point method to optimize the perturbation of the quadratic function in a search for the tightest upper bound for the qkp we prove that the same perturbation approach when applied in the context of semidefinite programming sdp relaxations of the qkp cannot improve the upper bound given by the corresponding linear sdp relaxation the result also applies to more general integer quadratic problems finally we propose new valid inequalities on the lifted matrix variable derived from cover and knapsack inequalities for the qkp and present separation problems to generate cuts for the current solution of the cqp relaxation our best bounds are obtained alternating between optimizing the parametric quadratic relaxation over the perturbation and applying cutting planes generated by the valid inequalities proposed

asymptotic behavior of the quadratic knapsack problem abstract we study the quadratic knapsack problem on instances where the profits are independent random variables defined on the interval 0 1 and the knapsack capacity is proportional to the number of items we assume that the weights are arbitrary numbers from the interval 0 1 we show asymptotically that the objective value of a very easy heuristic is not far away from the optimal solution more specifically we show that the ratio of the optimal solution and the objective value of this heuristic almost surely tends to 1 as the size of the knapsack instance tends to infinity as a consequence using randomly generated instances following this scheme seems to be inappropriate for studying the performance of heuristics and to some extend exact methods however such instances are frequently used in the literature for this purpose additionally we introduce a class of test instances based on hidden cliques for which finding a good solution is much harder we support this by theoretical observations as well as by performing computational experiments

classification of minimal actions of a compact kac algebra with amenable dual we show the uniqueness of minimal actions of a compact kac algebra with amenable dual on the afd factor of type ii1 this particularly implies the uniqueness of minimal actions of a compact group our main tools are a rohlin type theorem the 2 cohomology vanishing theorem and the evans kishimoto type intertwining argument

compact abelian group actions on injective factors we classify compact abelian group actions on injective type iii factors up to conjugacy which completes the final step of classification of compact abelian group actions on injective factors

lagrangean methods for the 0 1 quadratic knapsack problem abstract it is well known that the lagrangean decomposition provides better bounds than the lagrangean relaxation does nevertheless the lagrangean decomposition bound is harder to compute than the lagrangean relaxation bound thus one might wonder what is the best lagrangean method to use in a branch and bound algorithm in this paper we give an answer to such a question for the 0 1 quadratic knapsack problem we first study the lagrangean decomposition for this problem and give new necessary optimality conditions for the dual problem which allow us to elaborate a heuristic method for solving the lagrangean decomposition dual problem we then introduce this method in chaillou hansen mahieu s branch and bound algorithm where upper bounds were computed by lagrangean relaxation

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

compact abelian prime actions on von neumann algebras we classify the compact abelian actions on semifinite injective von neumann algebras with factor fixed point algebra the method uses that the nonzero eigenspaces of such an action contain unitaries which give rise to a classifying invariant

on the regularity of solutions of optimal transportation problems we give a necessary and sufficient condition on the cost function so that the map solution of monge s optimal transportation problem is continuous for arbitrary smooth positive data this condition was first introduced by ma trudinger and wang 24 30 for a priori estimates of the corresponding monge ampere equation it is expressed by a socalled cost sectional curvature being non negative we show that when the cost function is the squared distance of a riemannian manifold the cost sectional curvature yields the sectional curvature as a consequence if the manifold does not have non negative sectional curvature everywhere the optimal transport map cannot be continuous for arbitrary smooth positive data the non negativity of the cost sectional curvature is shown to be equivalent to the connectedness of the contact set between any cost convex function the proper generalization of a convex function and any of its supporting functions when the cost sectional curvature is uniformly positive we obtain that optimal maps are continuous or holder continuous under quite weak assumptions on the data compared to what is needed in the euclidean case this case includes the quadratic cost on the round sphere

a new upper bound for the 0 1 quadratic knapsack problem the 0 1 quadratic knapsack problem qkp consists in maximizing a positive quadratic pseudo boolean function subject to a linear capacity constraint we present in this paper a new method based on lagrangian decomposition for computing an upper bound of qkp we report computational experiments which demonstrate the sharpness of the bound relative error very often less than 1 for large size instances up to 500 variables

refractor surfaces determined by near field data abstract in this paper we study the near field refractor problem with point source at the origin and prescribed target on the given receiver surface this nonvariational problem can be studied in the framework of prescribed jacobian equations we construct the corresponding generating function and show that the aleksandrov and the brenier type solutions are equivalent our main result establishes local smoothness of aleksandrov s solutions when the data is smooth and when the medium containing the source has smaller refractive index than the medium containing the target this is done by deriving the monge ampere type equation that smooth solutions satisfy and establishing the validity of the mtw condition for a large class of receiver surfaces which in turn implies the local c 2 regularity of the refractor

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

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

the non commutative flow of weights on a von neumann algebra abstract the flow of weights of connes and takesaki is a canonical functor from the category of separable factors to the category of ergodic flows the non commutative flow of weights is another canonical functor from the category of separable factors to the category of covariant systems of semi finite von neumann algebras equipped with trace scaling one parameter automorphism groups with conjugations as morphisms the constructions of these two functors are very similar the flow of weights functor is obtained by looking at all semi finite normal weights on a factor with the murray von neumann equivalence relation the non commutative flow of weights functor is obtained by relating an arbitrary pair of faithful semi finite normal weights by the connes cocycle not only does this construction put a period to the search for a canonical construction of the core of a factor of type iii but it also allows us to put the characteristic square of a factor obtained by katayama sutherland and takesaki in a new perspective the power of this new approach is seen in an ultimate solution to a long standing question of extending the extended modular automorphism of a dominant weight to an arbitrary weight which has been left open ever since the introduction of extended modular automorphisms by connes and takesaki over 20 years ago the construction of the functor ties together the theory of l p spaces of haagerup kosaki hilsum terp and izumi to the structure theory of a factor of type iii in fact the non commutative flow of weights is obtained by the analytic continuation of l p spaces to a pure imaginary value of p

dual mean field search for large scale linear and quadratic knapsack problems abstract an implementation of mean field annealing to deal with large scale linear and non linear binary optimization problems is given mean field annealing is based on the analogy between combinatorial optimization and interacting physical systems at thermal equilibrium specifically a mean field approximation of the boltzmann distribution given by a lagrangian that encompass the objective function and the constraints is calculated the original discrete task is in this way transformed into a continuous variational problem in our version of mean field annealing no temperature parameter is used but a good starting point in the dual space is given by a thermodynamic limit argument the method is tested in linear and quadratic knapsack problems with sizes that are considerably larger than those used in previous studies of mean field annealing dual mean field annealing is capable to find high quality solutions in running times that are orders of magnitude shorter than state of the art algorithms moreover as may be expected for a mean field theory the solutions tend to be more accurate as the number of variables grow

constrained 0 1 quadratic programming basic approaches and extensions abstract we describe the simplest technique to tackle 0 1 quadratic programs with linear constraints among those that turn out to be successful in practice this method is due to and familiar to the quadratic assignment experts even if it took some time to realize that most approaches to the problem could be interpreted in these terms whereas it does not appear to be widely known outside this community since the technique is completely general and is by far the most successful one in several other cases such as quadratic knapsack we illustrate it in its full generality pointing out its relationship to lagrangian and linear programming relaxations and discussing further extensions we believe that this method should be in the background of every practitioner in combinatorial optimization

the structure of the automorphism group of an injective factor and the cocycle conjugacy of discrete abelian group actions we complete classification of discrete abelian or finite group actions on injective type iii1 factors up to cocycle conjugacy we also give a proof for connes characterization of the ker mod and cnt m for an injective factor m of type iii 0 introduction the purpose of this paper is to give a proof of connes announcement on approximately inner automorphisms and centrally trivial automorphisms of an injective 1980 mathematics subject classification 1985 revision 46l40

a simplified binary artificial fish swarm algorithm for 0 1 quadratic knapsack problems this paper proposes a simplified binary version of the artificial fish swarm algorithm s bafsa for solving 0 1 quadratic knapsack problems this is a combinatorial optimization problem which arises in many fields of optimization in s bafsa trial points are created by using crossover and mutation in order to make the points feasible a random heuristic drop item procedure is used the heuristic add item is also implemented to improve the quality of the solutions and a cyclic reinitialization of the population is carried out to avoid convergence to non optimal solutions to enhance the accuracy of the solution a swap move heuristic search is applied on a predefined number of points the method is tested on a set of benchmark 0 1 knapsack problems

on the supermodular knapsack problem in this paper we introduce binary knapsack problems where the objective function is nonlinear and investigate their lagrangean and continuous relaxations some of our results generalize previously known theorems concerning linear and quadratic knapsack problems we investigate in particular the case in which the objective function is supermodular under this hypothesis although the problem remains np hard we show that its lagrangean dual and its continuous relaxation can be solved in polynomial time we also comment on the complexity of recognizing supermodular functions the particular case in which the knapsack constraint is of the cardinality type is also addressed and some properties of its optimal value as a function of the right hand side are derived

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

using modifications to grover s search algorithm for quantum global optimization we study the problem of finding a global optimal solution to discrete optimization problems using a heuristic based on quantum computing methods knowledge of quantum computing ideas is not necessary to read this paper we focus on a successful quantum computing method introduced by baritompa bulger and wood that we refer to as the bbw algorithm and develop two modifications first we modify the bbw algorithm to achieve a dramatic speedup that lets us extend the known bbw static schedule from 33 to 43 points thereby increasing its applicability we further modify it by converting it from a static method to a dynamic one experimental results show the value of this latter modification

on the reflector shape design in this paper we study the problem of recovering the reflecting surface in a reflector system which consists of a point light source a reflecting surface and an object to be illuminated this problem involves a fully nonlinear partial differential equation of monge

on strict convexity and continuous differentiability of potential functions in optimal transportation this note concerns the relationship between conditions on cost functions and domains the convexity properties of potentials in optimal transportation and the continuity of the associated optimal mappings in particular we prove that if the cost function satisfies the condition a3 introduced in our previous work with xinan ma the densities and their reciprocals are bounded and the target domain is convex with respect to the cost function then the potential is continuously differentiable and its dual potential strictly concave with respect to the cost function our results extend by different and more direct proof similar results of loeper proved by approximation from our earlier work on regularity

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

on the second boundary value problem for monge ampere type equations and geometric optics in this paper we prove the existence of classical solutions to second boundary value problems for generated prescribed jacobian equations as recently developed by the second author thereby obtaining extensions of classical solvability of optimal transportation problems to problems arising in near field geometric optics our results depend in particular on a priori second derivative estimates recently established by the authors under weak co dimension one convexity hypotheses on the associated matrix functions with respect to the gradient variables a3w we also avoid domain deformations by using the convexity theory of generating functions to construct unique initial solutions for our homotopy family thereby enabling application of the degree theory for nonlinear oblique boundary value problems

the nonlinear knapsack problem algorithms and applications abstract we present a survey of algorithms and applications for the nonlinear knapsack problem or the nonlinear resource allocation problem in its most general form the nonlinear knapsack problem will be defined as a nonlinear optimization problem with just one constraint bounds on the variables and in some cases a set of specially structured constraints such as generalized upper bounds gubs this problem is encountered either directly or as a subproblem in a variety of applications including production planning financial modeling stratified sampling and capacity planning in manufacturing health care and computer networks by taking advantage of the special structure of the problem efficient solution methods can be developed problem classes addressed here include continuous and integer problems convex and nonconvex problems separable and nonseparable problems and problems with additional specially structured constraints

a tight linearization and an algorithm for zero one quadratic programming problems this paper is concerned with the solution of linearly constrained zero one quadratic programming problems problems of this kind arise in numerous economic location decision and strategic planning situations including capital budgeting facility location quadratic assignment media selection and dynamic set covering a new linearization technique is presented for this problem which is demonstrated to yield a tighter continuous or linear programming relaxation than is available through other methods an implicit enumeration algorithm which uses lagrangian relaxation benders cutting planes and local explorations is designed to exploit the strength of this linearization computational experience is provided to demonstrate the usefulness of the proposed linearization and algorithm

a fast parametric maximum flow algorithm and applications the classical maximum flow problem sometimes occurs in settings in which the arc capacities are not fixed but are functions of a single parameter and the goal is to find the value of the parameter such that the corresponding maximum flow or minimum cut satisfies some side condition finding the desired parameter value requires solving a sequence of related maximum flow problems in this paper it is shown that the recent maximum flow algorithm of goldberg and tarjan can be extended to solve an important class of such parametric maximum flow problems at the cost of only a constant factor in its worst case time bound faster algorithms for a variety of combinatorial optimization problems follow from the result

c1 α estimates for the parallel refractor abstract we consider the parallel refractor problem when the planar radiating source lies in a medium having higher refractive index than the medium in which the target is located we prove local c 1 estimates for parallel refractors under suitable geometric assumptions on the source and target and under local regularity hypotheses on the target set we also discuss existence of refractors under energy conservation assumptions