some proximity and sensitivity results in quadratic integer programming we show that for any optimal solution bar z for a given separable quadratic integer programming problem there exist an optimal solution bar x for its continuous relaxation such that parallel bar z bar x parallel infty leqslant n delta a wheren is the number of variables and a is the largest absolute subdeterminant of the integer constraint matrixa also for any feasible solutionz which is not optimal for the separable quadratic integer programming problem there exists a feasible solution bar z having greater objective function value and with parallel bar z z parallel infty leqslant n delta a we further prove under some additional assumptions that the distance between a pair of optimal solutions to an integer quadratic programming problem with right hand side vectorsb andb respectively depends linearly on b b 1 finally the validity of all the results for nonseparable mixed integer quadratic programs is established the proximity results obtained in this paper are extensions of some of the results described in cook et al 1986 for linear integer programming

the c 3 theorem and a d 2 algorithm for large scale stochastic mixed integer programming set convexification this paper considers the two stage stochastic integer programming problem with an emphasis on instances in which integer variables appear in the second stage drawing heavily on the theory of disjunctive programming we characterize convexifications of the second stage problem and develop a decomposition based algorithm for the solution of such problems in particular we verify that problems with fixed recourse are characterized by scenario dependent second stage convexifications that have a great deal in common we refer to this characterization as the c3 common cut coefficients theorem based on the c3 theorem we develop a decomposition algorithm which we refer to as disjunctive decomposition d2 in this new class of algorithms we work with master and subproblems that result from convexifications of two coupled disjunctive programs we show that when the second stage consists of 0 1 milp problems we can obtain accurate second stage objective function estimates after finitely many steps this result implies the convergence of the d2 algorithm

the sample average approximation method for stochastic discrete optimization in this paper we study a monte carlo simulation based approach to stochastic discrete optimization problems the basic idea of such methods is that a random sample is generated and the expected value function is approximated by the corresponding sample average function the obtained sample average optimization problem is solved and the procedure is repeated several times until a stopping criterion is satisfied we discuss convergence rates stopping rules and computational complexity of this procedure and present a numerical example for the stochastic knapsack problem

layer solutions in a half space for boundary reactions r n 1 are denoted by y y1 yn 1 our main goal is to study bounded solutions of 1 1 that are monotone increasing say from 1 to 1 in one of the y variables we call them layer solutions of 1 1 and we study their existence uniqueness symmetry and variational properties as well as their asymptotic behavior the interest in such increasing solutions comes from some models of boundary phase transitions when the nonlinearity f is given by f u sin cu for some constant c problem 1 1 in a half plane is called the peierls nabarro problem and it appears as a model of dislocations in crystals see 21 36 the peierls nabarro problem is also central to the analysis of boundary vortices in the paper 28 which studies a model for soft thin films in micromagnetism recently derived by kohn and slastikov 26 see also 27 our main result theorem 1 2 characterizes the nonlinearities f for which there exists a layer solution of 1 1 in dimension n 2 we prove that the necessary and sufficient condition is that the potential g defined by g f has two and only two absolute minima in the interval 1 1 located at 1 under the additional hypothesis g 1 0 we also establish the uniqueness of a layer solution up to translations in the y variable the proofs of both the necessity and the sufficiency of the condition on g for existence use new ingredients which we develop in this article a first one is a nonlocal estimate as well as a conserved or hamiltonian quantity satisfied by every layer solution in dimension 2 see theorem 1 3 the estimate can be seen as

a finite branch and bound algorithm for two stage stochastic integer programs this paper addresses a general class of two stage stochastic programs with integer recourse and discrete distributions we exploit the structure of the value function of the second stage integer problem to develop a novel global optimization algorithm the proposed scheme departs from those in the current literature in that it avoids explicit enumeration of the search space while guaranteeing finite termination computational experiments on standard test problems indicate superior performance of the proposed algorithm in comparison to those in the existing literature

singular perturbation of domains and the semilinear elliptic equation ii on considere un domaine a perturbation singuliere d 1 d 2 q avec un petit parametre 0 et on etudie le comportement des solutions et leur structure du probleme aux valeurs limites elliptique semilineaire pour quand 0 est petit v f v 0 dans v v 0 sur ou f est une fonction lisse a valeur reelle non lineaire

the stochastic location model with risk pooling abstract in this paper we present a stochastic version of the location model with risk pooling lmrp that optimizes location inventory and allocation decisions under random parameters described by discrete scenarios the goal of our model called the stochastic lmrp or slmrp is to find solutions that minimize the expected total cost including location transportation and inventory costs of the system across all scenarios the location model explicitly handles the economies of scale and risk pooling effects that result from consolidating inventory sites the slmrp framework can also be used to solve multi commodity and multi period problems we present a lagrangian relaxation based exact algorithm for the slmrp the lagrangian subproblem is a non linear integer program but it can be solved by a low order polynomial algorithm we discuss simple variable fixing routines that can drastically reduce the size of the problem we present quantitative and qualitative computational results on problems with up to 150 nodes and 9 scenarios describing both algorithm performance and solution behavior as key parameters change

jacobi matrices and transversality the paper deals with smooth nonlinear ode systems in n f x such that the derivative f x has a matrix representation of jacobi type not necessarily symmetric with positive off diagonal entries a discrete functional is introduced and is discovered to be nonincreasing along the solutions of the associated linear variational system f x t y two families of transversal cones invariant under the flow of that linear system allow us to prove transversality between the stable and unstable manifolds of any two hyperbolic critical points of the given nonlinear system it is also proved that the nonwandering points are critical points a new class of morse smale systems in n is then explicitly constructed

mixed integer bilinear programming problems this paper addresses a class of problems called mixed integer bilinear programming problems these problems are identical to the well known bilinear programming problems with the exception that one set of variables is restricted to be binary valued and they arise in various production location allocation and distribution application contexts we first identify some special cases of this problem which are relatively more readily solvable even though their continuous relaxations are still nonconvex for the more general case we employ a linearization technique and design a composite lagrangian relaxation implicit enumeration cutting plane algorithm extensive computational experience is provided to test the efficacy of various algorithmic strategies and the effects of problem data on the computational effort of the proposed algorithm

a simulation based approach to two stage stochastic programming with recourse in this paper we consider stochastic programming problems where the objective function is given as an expected value function we discuss monte carlo simulation based approaches to a numerical solution of such problems in particular we discuss in detail and present numerical results for two stage stochastic programming with recourse where the random data have a continuous multivariate normal distribution we think that the novelty of the numerical approach developed in this paper is twofold first various variance reduction techniques are applied in order to enhance the rate of convergence successful application of those techniques is what makes the whole approach numerically feasible second a statistical inference is developed and applied to estimation of the error validation of optimality of a calculated solution and statistically based stopping criteria for an iterative alogrithm 1998 the mathematical programming society inc published by elsevier science b v

disjunctive decomposition for two stage stochastic mixed binary programs with random recourse this paper introduces disjunctive decomposition for two stage mixed 0 1 stochastic integer programs sips with random recourse disjunctive decomposition allows for cutting planes based on disjunctive programming to be generated for each scenario subproblem under a temporal decomposition setting of the sip problem a new class of valid inequalities for mixed 0 1 sip with random recourse is presented in particular we derive valid inequalities that allow for scenario subproblems for sip with random recourse but deterministic technology matrix and right hand side vector to share cut coefficients the valid inequalities are used to derive a disjunctive decomposition method whose derivation has been motivated by real life stochastic server location problems with random recourse which find many applications in operations research computational results with large scale instances to demonstrate the potential of the method are reported

the geometric interpretation of correspondence analysis abstract correspondence analysis is an exploratory multivariate technique that converts a data matrix into a particular type of graphical display in which the rows and columns are depicted as points the method has a long and varied history and has appeared in different forms in the psychometric and ecological literature among others in this article we review the geometry of correspondence analysis and its geometric interpretation we also discuss various extensions of correspondence analysis to multivariate categorical data multiple correspondence analysis and a variety of other data types

improved algorithm for mixed integer quadratic programs and a computational study in a recent paper 6 we suggested an algorithm for solving complicated mixed integer quadratic programs based on an equivalent formulation that employs a nonsingular transformation of variables the objectives of the present paper are two first to present an improved version of this algorithm which reduces substantially its computational requirements second to report on the results of a computational study with the revised algorithm

sensitivity theorems in integer linear programming we consider integer linear programming problems with a fixed coefficient matrix and varying objective function and right hand side vector among our results we show that for any optimal solution to a linear program max wx ax b the distance to the nearest optimal solution to the corresponding integer program is at most the dimension of the problem multiplied by the largest subdeterminant of the integral matrixa using this we strengthen several integer programming proximity results of blair and jeroslow graver and wolsey we also show that the chvatal rank of a polyhedron x ax b can be bounded above by a function of the matrixa independent of the vectorb a result which as blair observed is equivalent to blair and jeroslow s theorem that each integer programming value function is a gomory function

on structure and stability in stochastic programs with random technology matrix and complete integer recourse for two stage stochastic programs with integrality constraints in the second stage we study continuity properties of the expected recourse as a function both of the first stage policy and the integrating probability measure

a nonlinear parabolic equation with varying domain abstract this document considers a boundary value problem where the basic problem is to understand how new equilibrium solutions may appear for a given nonlinear function f as the domain omega is changed these new solutions being bifurcations from known equilibrium solutions for a given cubic function f an intuitive discussion of this phenomenon was given by hale indicating how stable non constant equilibria appear as secondary rather than primary bifurcations additional keywords continuity perturbations reprints

stochastic programming with integer variables including integer variables into traditional stochastic linear programs has considerable implications for structural analysis and algorithm design starting from mean risk approaches with different risk measures we identify corresponding two and multi stage stochastic integer programs that are large scale block structured mixed integer linear programs if the underlying probability distributions are discrete we highlight the role of mixed integer value functions for structure and stability of stochastic integer programs when applied to the block structures in stochastic integer programming well known algorithmic principles such as branch and bound lagrangian relaxation or cutting plane methods open up new directions of research we review existing results in the field and indicate departure points for their extension

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

the nonlinear resource allocation problem in this paper we study the nonlinear resource allocation problem defined as the minimization of a convex function over one convex constraint and bounded integer variables this problem is encountered in a variety of applications including capacity planning in manufacturing and computer networks production planning capital budgeting and stratified sampling despite its importance to these and other applications the nonlinear resource allocation problem has received little attention in the literature therefore we develop a branch and bound algorithm to solve this class of problems first we present a general framework for solving the continuous variable problem then we use this framework as the basis for our branch and bound method we also develop reoptimization procedures and a heuristic that significantly improve the performance of the branch and bound algorithm in addition we show how the algorithm can be modified to solve nonconvex problems so that a concave objective function can be handled

decomposition with branch and cut approaches for two stage stochastic mixed integer programming decomposition has proved to be one of the more effective tools for the solution of large scale problems especially those arising in stochastic programming a decomposition method with wide applicability is benders decomposition which has been applied to both stochastic programming as well as integer programming problems however this method of decomposition relies on convexity of the value function of linear programming subproblems this paper is devoted to a class of problems in which the second stage subproblem s may impose integer restrictions on some variables the value function of such integer subproblem s is not convex and new approaches must be designed in this paper we discuss alternative decomposition methods in which the second stage integer subproblems are solved using branch and cut methods one of the main advantages of our decomposition scheme is that stochastic mixed integer programming smip problems can be solved by dividing a large problem into smaller mip subproblems that can be solved in parallel this paper lays the foundation for such decomposition methods for two stage stochastic mixed integer programs

correspondence analysis of multivariate categorical data by weighted least squares summary a generalization of correspondence analysis to multivariate categorical data is proposed where all two way contingency tables of a set of categorical variables are simultaneously fitted by weighted least squares an alternating least squares algorithm is developed to perform the fitting this technique has a number of advantages over the usual generalization known as multiple correspondence analysis it is also an analogue of least squares factor analysis for categorical data

l shaped decomposition of two stage stochastic programs with integer recourse we consider two stage stochastic programming problems with integer recourse the l shaped method of stochastic linear programming is generalized to these problems by using generalized benders decomposition nonlinear feasibility and optimality cuts are determined via general duality theory and can be generated when the second stage problem is solved by standard techniques finite convergence of the method is established when gomory s fractional cutting plane algorithm or a branch and bound algorithm is applied

bifurcations caused by perturbing the domain in an elliptic equation we consider the neumann problem u u up on a continuous family of bounded domains which approach as 0 a set 0 with two connected components and analyze it as a bifurcation problem with two parameters and the bifurcation diagrams and the qualitative properties of the bifurcation sets for p odd and p even are obtained and the relations between them are studied by considering the problem u u au2 u3 for different values of a

the singularly perturbed domain and the characterization for the eigenfunctions with neumann boundary condition on considere un domaine mobile 0 parametre qui degenere partiellement quand 0 et on donne une caracterisation elaboree des comportements des fonctions propres du laplacien avec la condition limite de neumann

an analysis and synthesis of multiple correspondence analysis optimal scaling dual scaling homogeneity analysis and other methods for quantifying categorical multivariate data we discuss a variety of methods for quantifying categorical multivariate data these methods have been proposed in many different countries by many different authors under many different names in the first major section of the paper we analyze the many different methods and show that they all lead to the same equations for analyzing the same data in the second major section of the paper we introduce the notion of a duality diagram and use this diagram to synthesize the many superficially different methods into a single method

solving stochastic programs with integer recourse by enumeration a framework using gro bner basis reductions in this paper we present a framework for solving stochastic programs with complete integer recourse and discretely distributed right hand side vector using grobner basis methods from computational algebra to solve the numerous second stage integer programs using structural properties of the expected integer recourse function we prove that under mild conditions an optimal solution is contained in a finite set furthermore we present a basic scheme to enumerate this set and suggest improvements to reduce the number of function evaluations needed

jointly constrained biconvex programming this paper presents a branch and bound algorithm for minimizing the sum of a convex function in x a convex function in y and a bilinear term in x and y over a closed set such an objective function is called biconvex with biconcave functions similarly defined the feasible region of this model permits joint constraints in x and y to be expressed the bilinear programming problem becomes a special case of the problem addressed in this paper we prove that the minimum of a biconcave function over a nonempty compact set occurs at a boundary point of the set and not necessarily an extreme point the algorithm is proven to converge to a global solution of the nonconvex program we discuss extensions of the general model and computational experience in solving jointly constrained bilinear programs for which the algorithm has been implemented

a survey for the quadratic assignment problem the quadratic assignment problem qap one of the most difficult problems in the np hard class models many real life problems in several areas such as facilities location parallel and distributed computing and combinatorial data analysis combinatorial optimization problems such as the traveling salesman problem maximal clique and graph partitioning can be formulated as a qap in this paper we present some of the most important qap formulations and classify them according to their mathematical sources we also present a discussion on the theoretical resources used to define lower bounds for exact and heuristic algorithms we then give a detailed discussion of the progress made in both exact and heuristic solution methods including those formulated according to metaheuristic strategies finally we analyze the contributions brought about by the study of different approaches

reformulating nonlinear combinatorial optimization problems for higher computational efficiency abstract it is important to develop computationally efficient algorithms to solve combinatorial optimization problems equally important if not more so is to find a way to reformulate the original combinatorial optimization problem provided this is possible so that the resulting equivalent problem can be solved easily and efficiently using commercially available codes this paper proposes an equivalent formulation to deal basically with the linearization of nonlinear expressions in 0 1 variables the equivalent formulation technique proposed in this paper linearizes a binary quadratic integer problem of n variables by introducing only n new linear constraints whereas the most economical method in the literature requires the addition of 2n of such constraints the same technique is extended to linearize the binary cubic expression of n variables this time requiring the addition of at most 3n auxiliary linear constraints and 3n continuous variables also discussed in the paper is the computational superiority of the proposed method over the existing ones in the literature through a series of randomly generated test problems

an algorithm for indefinite integer quadratic programming abstract we present an algorithm for finding the global minimum of an indefinite quadratic function over the integers contained in a compact convex set to find this minimum the algorithm first transforms the problem into an equivalent problem with a separable objective function it then uses a branch and bound search on the values of the constraints rather than the variables of the transformed problem

a linearization procedure for quadratic and cubic mixed integer problems several techniques of linearization have appeared in the literature the technique of f glover which seems to be the most efficient linearizes a binary quadratic integer problem of n variables by introducing n new continuous variables and 4n auxiliary linear constraints the new technique proposed in this paper is not only useful in linearizing binary quadratic and cubic integer problems but also applicable to the case of quadratic and to a certain class of cubic mixed integer problems it is shown that the new technique further reduces the number of auxiliary linear constraints from 4n to n while keeping the number of new continuous variables at n for the binary quadratic integer problem of n variables and it requires in the case of a certain class of cubic mixed integer problems having 2n of 0 1 variables only 3n auxiliary linear constraints and the same number of new continuous variables the analytical superiority of the new linearization technique has also been observed in terms of the nu

two stage integer programs with stochastic right hand sides a superadditive dual approach we consider two stage pure integer programs with discretely distributed stochastic right hand sides we present an equivalent superadditive dual formulation that uses the value functions in both stages we give two algorithms for finding the value functions to solve the reformulation after obtaining the value functions we develop a global branch and bound approach and a level set approach to find an optimal tender we show that our method can solve randomly generated instances whose extensive forms are several orders of magnitude larger than the extensive forms of those instances found in the literature

the value function of an integer program we consider integer programs in which the objective function and constraint matrix are fixed while the right hand side varies the value function gives for each feasible right hand side the criterion value of the optimal solution we provide a precise characterization of the closed form expression for any value function

patterns in parabolic problems with nonlinear boundary conditions we obtain existence of asymptotically stable nonconstant equilibrium solutions for semilinear parabolic equations with nonlinear boundary conditions on small domains connected by thin channels we prove the convergence of eigenvalues and eigenfunctions of the laplace operator in such domains this information is used to show that the asymptotic dynamics of the heat equation in this domain is equivalent to the asymptotic dynamics of a system of two ordinary differential equations diffusively weakly coupled the main tools employed are the invariant manifold theory and a uniform trace theorem

large time dynamics of a nonlinear spring mass damper model abstract in this paper we consider a nonlinear strongly damped wave equation as a model for a controlled spring mass damper system and give some results concerning its large time behaviour it can be seen that the infinite dimensional system admits a two dimensional attracting manifold where the equation is well represented by a classical nonlinear oscillations ode which can be exhibited explicitly in contrast to other papers this one applies invariant manifold theory to a problem whose linear part is not self adjoint

on optimal allocation of indivisibles under uncertainty the optimal allocation of indivisible resources is formalized as a stochastic optimization problem involving discrete decision variables a general stochastic search procedure is proposed which develops the concept of the branch and bound method the main idea is to process large collections of possible solutions and to devote more attention to the most promising groups by gathering more information to reduce the uncertainty and by narrowing the search area the optimal solution can be found with probability one special techniques for calculating stochastic lower and upper bounds are discussed the results are illustrated by a computational experiment

two stage quadratic integer programs with stochastic right hand sides we consider two stage quadratic integer programs with stochastic right hand sides and present an equivalent reformulation using value functions we propose a two phase solution approach the first phase constructs value functions of quadratic integer programs in both stages the second phase solves the reformulation using a global branch and bound algorithm or a level set approach we derive some basic properties of value functions of quadratic integer programs and utilize them in our algorithms we show that our approach can solve instances whose extensive forms are hundreds of orders of magnitude larger than the largest quadratic integer programming instances solved in the literature

mixed integer quadratic programming this paper considers mixed integer quadratic programs in which the objective function is quadratic in the integer and in the continuous variables and the constraints are linear in the variables of both types the generalized benders decomposition is a suitable approach for solving such programs however the program does not become more tractable if this method is used since benders cuts are quadratic in the integer variables a new equivalent formulation that renders the program tractable is developed under which the dual objective function is linear in the integer variables and the dual constraint set is independent of these variables benders cuts that are derived from the new formulation are linear in the integer variables and the original problem is decomposed into a series of integer linear master problems and standard quadratic subproblems the new formulation does not introduce new primary variables or new constraints into the computational steps of the decomposition algorithm

on a level set characterization of the value function of an integer program and its application to stochastic programming we propose a level set approach to characterize the value function of a pure linear integer program with inequality constraints we study theoretical properties of our characterization and show how they can be exploited to optimize a class of stochastic integer programs through a value function reformulation specifically we develop algorithmic approaches that solve two stage multidimensional knapsack problems with random budgets yielding encouraging computational results

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

integer programming duality price functions and sensitivity analysis recently a duality theory for integer programming has been developed here we examine some of the economic implications of this theory in particular the necessity of using price functions in place of prices and the possibility of carrying out sensitivity analysis of optimal solutions in addition we consider the form of price functions that are generated by known algorithms for integer programming