a note on node aggregation and benders decomposition node aggregation methods have been previously studied as a means for approximating large scale transportation problems in this paper we show how the approximations inherent in benders decomposition method can be combined with node aggregation in optimizing large scale capacitated plant location problems

a production transportation problem with stochastic demand and concave production costs the problem of transporting goods from a set of supply points factories to a set of demand points customers so as to minimize linear transportation costs is well known and very efficient solution methods exist in the well known facility location problem one also includes fixed charges for the supply points on the other hand in ordinary transportation problems stochastic demand has been introduced and modelled by convex costs yielding the stochastic transportation problem stp solved by for example the methods in cooper and leblanc 1977 and holmberg and jornsten 1984 however the simultaneous use of these generalizations has recieved little attention until now only a few suggestions for solution methods can be found leblanc 1977 franca and luna 1982 in this paper we consider generalizing this problem even further by introducing general concave costs at the supply points as well as convex costs at the demand points fixed charges are not the only cost structures that occur when producing goods at a factory economies of scale very often yield other concave cost functions

the lagrangian relaxation method for solving integer programming problems this article originally appeared in management science january 1981 volume 27 number 1 pp 1 18 published by the institute of management sciences one of the most computationally useful ideas of the 1970s is the observation that many hard integer programming problems can be viewed as easy problems complicated by a relatively small set of side constraints dualizing the side constraints produces a lagrangian problem that is easy to solve and whose optimal value is a lower bound for minimization problems on the optimal value of the original problem the lagrangian problem can thus be used in place of a linear programming relaxation to provide bounds in a branch and bound algorithm this approach has led to dramatically improved algorithms for a number of important problems in the areas of routing location scheduling assignment and set covering this paper is a review of lagrangian relaxation based on what has been learned in the last decade

solving mixed integer nonlinear programs by outer approximation a wide range of optimization problems arising from engineering applications can be formulated as mixed integer nonlinear programming problems minlps duran and grossmann 1986 suggest an outer approximation scheme for solving a class of minlps that are linear in the integer variables by a finite sequence of relaxed milp master programs and nlp subproblems

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

convex constrained programmes with unattained infima abstract we consider a problem of minimization of convex function f x over the convex region r where the objective function and the feasible region have a common direction of recession in cases when one of these directions is not in the constancy space of the objective function then the minimal solution is not achieved even if the function f x is bounded below over the region r many algorithms if applied to this class of programmes do not guarantee convergence to the global infimum our approach to this problem leads to derivation of the equation of the feasible parametrized curve c t such that the infimum of the logarithmic penalty function along this curve is equal to the global infimum of the objective function over the region r we show that if all functions defining the program are analytic then c t is also an analytic function the equation of the curve can be successfully used to determine the global infimum in particular unboundedness of the convex constrained programmes in cases when the application of classical methods such as the steepest descent method fails to converge to the global infimum

a cross decomposition algorithm for capacitated facility location we present an implementation of the cross decomposition method to solve the capacitated facility location problem the method unifies benders decomposition and lagrangean relaxation into a single framework that involves successive solutions to a transportation problem and a simple plant location problem computational tests on problems with as many as 100 candidate facility locations and 200 customers provide evidence that the approach is superior to several other methods the specialized cross decomposition program is on a common set of test problems about ten times faster than several recently developed programs we also present a simple algorithm to strengthen benders cuts

ergodic primal convergence in dual subgradient schemes for convex programming lagrangean dualization and subgradient optimization techniques are frequently used within the field of computational optimization for finding approximate solutions to large structured optimization prob lems the dual subgradient scheme does not automatically produce primal feasible solutions there is an abundance of techniques for computing such solutions via penalty functions tangential approximation schemes or the solution of auxiliary primal programs all of which require a fair amount of computational effort we consider a subgradient optimization scheme applied to a lagrangean dual formulation of a convex program and construct at minor cost an ergodic sequence of subproblem solutions which converges to the primal solution set numerical experiments performed on a traffic equilibrium assignment problem under road pricing show that the computation of the ergodic sequence results in a considerable improvement in the quality of the primal solutions obtained compared to those generated in the basic subgradient scheme

a cross decomposition scheme with integrated primal dual multi cuts for two stage stochastic programming investment planning problems we describe a cross decomposition algorithm that combines benders and scenario based lagrangean decomposition for two stage stochastic programming investment planning problems with complete recourse where the first stage variables are mixed integer and the second stage variables are continuous the algorithm is a novel cross decomposition scheme and fully integrates primal and dual information in terms of primal dual multi cuts added to the benders and the lagrangean master problems for each scenario the potential benefits of the cross decomposition scheme are demonstrated with numerical experiments on a number of instances of a facility location problem under disruptions in the original formulation where the underlying lp relaxation is weak the cross decomposition method outperforms multi cut benders decomposition if the formulation is improved with the addition of tightening constraints the performance of both decomposition methods improves but cross decomposition clearly remains the best method for large scale problems

on the treatment of thermal and quantum annealing by stochastic variational means we consider the control of a diffusion process represented by a stochastic differential equation of the ito type the diffusion process is one of a random walk on the configuration space of a multimodal function and the goal of the control is to determine the optimum variation of the temperature in order to anneal into the global minimum of the function such optimal controls are derived using stochastic variational means by the viscosity solution of a second order hamilton jacobi bellman equation or by minimizing a feynman kac wiener path integral the extension to quantized annealing and the possibilities of a differential game theoretic formulation for extra diffusive annealing are also considered

a reduced gradient method for quadratic programs with quadratic constraints and lp constrained lp approximation problems abstract a dual algorithm is developed for solving a general class of nonlinear programs that properly contains all convex quadratic programs with quadratic constraints and l p constrained l p approximation problems the general dual program to be solved has essentially linear constraints but the objective function is nondifferentiable when certain variables equal zero modifications to the reduced gradient method for linearly constrained problems are presented that overcome the numerical difficulties associated with the nondifferentiable objective function these modifications permit blocks of variables to move to and away from zero on certain iterations even though the objective function is nondifferentiable at points having a block of variables equal to zero

cross decomposition applied to the stochastic transportation problem abstract in this paper we give a solution method for the stochastic transportation problem based on cross decomposition developed by van roy 1980 solution methods to the derived sub and master problems are discussed and computational results are given for a number of large scale test problems we also compare the efficiency of the method with other methods suggested for the stochastic transportation problem the frank wolfe algorithm and separable programming

constraint partitioning in penalty formulations for solving temporal planning problems in this paper we study the partitioning of constraints in temporal planning problems formulated as mixed integer nonlinear programming minlp problems constraint partitioning is attractive because it leads to much easier subproblems where each is a significant relaxation of the original problem moreover each subproblem is very similar to the original problem and can be solved by any existing solver with little or no modification constraint partitioning however introduces global constraints that may be violated when subproblems are evaluated independently to reduce the overhead in resolving such global constraints we develop in this paper new conditions and algorithms for limiting the search space to be backtracked in each subproblem using a penalty formulation of a minlp where the constraint functions of the minlp are transformed into non negative functions we present a necessary and sufficient extended saddle point condition espc for constrained local minimization when the penalties are larger than some thresholds our theory shows a one to one correspondence between a constrained local minimum of the minlp and an extended saddle point of the penalty function hence one way to find a constrained local minimum is to increase gradually the penalties of those violated constraints and to look for a local minimum of the penalty function using any existing algorithm until a solution to the constrained model is found next we extend the espc to constraint partitioned minlps and propose a partition and resolve strategy for resolving violated global constraints across subproblems using the discrete space aspen and the mixed space mips planners to solve subproblems we show significant improvements on some planning benchmarks both in terms of the quality of the plans generated and the execution times to find them

controlled perturbations for quadratically constrained quadratic programs consider a minimization problem of a convex quadratic function of several variables over a set of inequality constraints of the same type of function the duel program is a maximization problem with a concave objective function and a set of constrains that are essentially linear however the objective function is not differentiable over the constraint region in this paper we study a general theory of dual perturbations and derive a fundamental relationship between a perturbed dual program and the original problem based on this relationship we establish a perturbation theory to display that a well controlled perturbation on the dual program can overcome the nondifferentiability issue and generate an optimal dual solution for an arbitrarily small number a simple linear program is then constructed to make an easy conversion from the dual solution to a corresponding optimal primal solution moreover a numerical example is included to illustrate the potential of this controlled perturbation scheme

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

cross decomposition for mixed integer programming many methods for solving mixed integer programming problems are based either on primal or on dual decomposition which yield respectively a benders decomposition algorithm and an implicit enumeration algorithm with bounds computed via lagrangean relaxation these methods exploit either the primal or the dual structure of the problem we propose a new approach cross decomposition which allows exploiting simultaneously both structures the development of the cross decomposition method captures profound relationships between primal and dual decomposition it is shown that the more constraints can be included in the langrangean relaxation provided the duality gap remains zero the fewer the benders cuts one may expect to need if the linear programming relaxation has no duality gap only one benders cut is needed to verify optimality

a lagrangian based heuristic for the capacitated lot sizing problem in parallel machines this paper addresses the capacitated lot sizing problem involving the production of multiple items on unrelated parallel machines a production plan should be determined in order to meet the forecast demand for the items without exceeding the capacity of the machines and minimize the sum of production setup and inventory costs a heuristic based on the lagrangian relaxation of the capacity constraints and subgradient optimization is proposed initially the heuristic is tested on instances of the single machine problem and results are compared with heuristics from the literature for parallel machines and small problems the heuristic performance is tested against optimal solutions and for larger problems it is compared with the lower bound provided by the lagrangian relaxation

on the resemblance between the kornai liptak and cross decomposition techniques for block angular linear programs in this paper the resemblance is demonstrated between the master and subproblems generated by the kornai liptak algorithm and the subproblems obtained by using the cross decomposition method on li

cone of recession and unboundedness of convex functions abstract we consider the problem of determining whether or not a convex function f x is bounded below over r n our focus is on investigating the properties of the vectors in the cone of recession 0 f of f x which are related to the unboundedness of the function

on using approximations of the benders master problem abstract in this paper we study some approximate ways of solving the benders master problem in benders decomposition the applications of lagrange duality and surrogate duality to the benders master problem are found to be equivalent we also find that the lagrangean dual of the benders master problem cannot yield better bounds than the partial lagrangean dual of the original problem compared to solving the benders master problem exactly the approximations often yields worse bounds and a lack of controllability which might prohibit the generation of necessary benders cuts for a milp problem this might yield worse bounds than its lp relaxation a convergence test originating from cross decomposition can be used to check the progress of a method using such approximations

a branch and cut algorithm for nonconvex quadratically constrained quadratic programming we present a branch and cut algorithm that yields in finite time a globally e optimal solution with respect to feasibility and optimality of the nonconvex quadratically constrained quadratic programming problem the idea is to estimate all quadratic terms by successive linearizations within a branching tree using reformulation linearization techniques rlt to do so four classes of linearizations cuts depending on one to three parameters are detailed for each class we show how to select the best member with respect to a precise criterion the cuts introduced at any node of the tree are valid in the whole tree and not only within the subtree rooted at that node in order to enhance the computational speed the structure created at any node of the tree is flexible enough to be used at other nodes computational results are reported that include standard test problems taken from the literature some of these problems are solved for the first time with a proof of global optimality

selected topics in column generation dantzig wolfe decomposition and column generation devised for linear programs is a success story in large scale integer programming we outline and relate the approaches and survey mainly recent contributions not yet found in textbooks we emphasize the growing understanding of the dual point of view which has brought considerable progress to the column generation theory and practice it stimulated careful initializations sophisticated solution techniques for the restricted master problem and subproblem as well as better overall performance thus the dual perspective is an ever recurring concept in our selected topics

the benders decomposition algorithm a literature review the benders decomposition algorithm has been successfully applied to a wide range of difficult optimization problems this paper presents a state of the art survey of this algorithm emphasizing its use in combinatorial optimization we discuss the classical algorithm the impact of the problem formulation on its convergence and the relationship to other decomposition methods we introduce a taxonomy of algorithmic enhancements and acceleration strategies based on the main components of the algorithm the taxonomy provides the framework to synthesize the literature and to identify shortcomings trends and potential research directions we also discuss the use of the benders decomposition to develop efficient meta heuristics describe the limitations of the classical algorithm and present extensions enabling its application to a broader range of problems

valid inequalities and facets of the capacitated plant location problem recently several successful applications of strong cutting plane methods to combinatorial optimization problems have renewed interest in cutting plane methods and polyhedral characterizations of integer programming problems in this paper we investigate the polyhedral structure of the capacitated plant location problem our purpose is to identify facets and valid inequalities for a wide range of capacitated fixed charge problems that contain this prototype problem as a substructure

extensions to a lagrangean relaxation approach for the capacitated warehouse location problem abstract in this paper we present a lower bound for the capacitated warehouse location problem based upon the lagrangean relaxation of a mixed integer formulation of the problem where we use subgradient optimisation in an attempt to maximise this lower bound problem reduction tests based upon this lower bound and the original problem are given incorporating this bound and the reduction tests into a tree search procedure enables us to solve problems involving up to 50 warehouses and 150 customers

an elementary survey of general duality theory in mathematical programming we survey some recent developments in duality theory with the idea of explaining and unifying certain basic duality results in both nonlinear and integer programming the idea of replacing dual variables prices by price functions suggested by everett and developed by gould is coupled with an appropriate dual problem with the consequence that many of the results resemble those used in linear programming the dual problem adopted has a traditional economic interpretation and dual feasibility then provides a simple alternative to concepts such as conjugate functions or subdifferentials used in the study of optimality in addition we attempt to make precise the relationship between primal dual and saddlepoint results in both the traditional lagrangean and the more general duality theories and to see the implications of passing from prices to price functions finally and perhaps surprisingly it appears that all the standard algorithms terminate by constructing primal and dual feasible solutions of equal value i e by satisfying generalised optimality conditions

convex functions with unbounded level sets and applications to duality theory a class of convex functions with unbounded level sets but good behavior at infinity analyse non lineaire gauthier villars paris 1989 pp 101 122 is investigated characterizations and properties are given the results are then applied to studying sequential approximation schemes for optimization problems and to duality theory when the involved functions have unbounded level sets in particular the convergence properties of stationary sequences for the dual of a convex program are studied and methods for associating with it a primal sequence converging to a solution of the primal problem are demonstrated

a new cross decomposition method for stochastic mixed integer linear programming abstract two stage stochastic mixed integer linear programming milp problems can arise naturally from a variety of process design and operation problems these problems with a scenario based formulation lead to large scale milps that are well structured when first stage variables are mixed integer and second stage variables are continuous these milps can be solved efficiently by classical decomposition methods such as dantzig wolfe decomposition dwd lagrangian decomposition and benders decomposition bd or a cross decomposition strategy that combines some of the classical decomposition methods this paper proposes a new cross decomposition method where bd and dwd are combined in a unified framework to improve the solution of scenario based two stage stochastic milps this method alternates between dwd iterations and bd iterations where dwd restricted master problems and bd primal problems yield a sequence of upper bounds and bd relaxed master problems yield a sequence of lower bounds the method terminates finitely to an optimal solution or an indication of the infeasibility of the original problem case study of two different supply chain systems a bioproduct supply chain and an industrial chemical supply chain show that the proposed cross decomposition method has significant computational advantage over bd and the monolith approach when the number of scenarios is large

new bounding and decomposition approaches for milp investment problems multi area transmission and generation planning under policy constraints abstract we propose a novel two phase bounding and decomposition approach to compute optimal and near optimal solutions to large scale mixed integer investment planning problems that have to consider a large number of operating subproblems each of which is a convex optimization our motivating application is the planning of power transmission and generation in which policy constraints are designed to incentivize high amounts of intermittent generation in electric power systems the bounding phase exploits jensen s inequality to define a lower bound which we extend to stochastic programs that use expected value constraints to enforce policy objectives the decomposition phase in which the bounds are tightened improves upon the standard benders algorithm by accelerating the convergence of the bounds the lower bound is tightened by using a jensen s inequality based approach to introduce an auxiliary lower bound into the benders master problem upper bounds for both phases are computed using a sub sampling approach executed on a parallel computer system numerical results show that only the bounding phase is necessary if loose optimality gaps are acceptable however the decomposition phase is required to attain optimality gaps use of both phases performs better in terms of convergence speed than attempting to solve the problem using just the bounding phase or regular benders decomposition separately

laplace operator and random walk on one dimensional nonhomogeneous lattice a classical result of probability theory states that under suitable space and time renormalization a random walk converges to brownian motion we prove an analogous result in the case of nonhomogeneous random walk on onedimensional lattice under suitable conditions on the nonhomogeneous medium we prove convergence to brownian motion and explicitly compute the diffusion coefficient the proofs are based on the study of the spectrum of random matrices of increasing dimension

a multi product loading problem a model and solution method abstract an important operational problem arises during the transportation and delivery of several products which cannot be mixed in the same vehicle at regular intervals the vehicle has compartments to keep the products separately therefore a scheme of allocation of compartments which we call vehicle loading problem to maximize the efficiency of the system while the demands for the products at the destination s are satisfied a mixed binary model is developed for this multi product loading problem the solution method is based on simultaneously exploring the primal and dual structures derived from the lagrangian relaxation subset sum problems are obtained as subproblems to the partial lagrangian an algorithm is developed and its convergence is proved the efficiency of the method is demonstrated by running randomly chosen test problems an initial solution finding method is also developed

behavior of general one dimensional diffusion processes we develop simple rigorous techniques to estimate the behavior of general one dimensional diffusion processes any one dimensional diffusion process with drift can be mapped onto a symmetric diffusion through an explicit change of variable for such processes we can estimate explicitly the diffusion exponent the recurrence properties and the large fluctuations in a second part we apply these results to different models including the sinai random walk diffusion in a random drift and we show how the main features of the diffusion can be readily handled

solving the staircase cost facility location problem with decomposition and piecewise linearization abstract facility location problems with staircase costs i e fixed costs that appear at several levels of production are large structured mixed integer programming problems which often are quite hard to solve in this paper solution methods based on convex piecewise linearization and benders decomposition are investigated using convex piecewise linearization only the integer variables that are needed to improve the approximation are introduced and computational tests show that in average only a few are needed benders decomposition can be used to solve the resulting problems and we show how to recalculate benders cuts when new variables are introduced so they still can be used when the approximation is improved

lagrangian relaxation based solution procedures for a multiproduct capacitated facility location problem with choice of facility type this paper presents exact and heuristic solution procedures for a multiproduct capacitated facility location mpcfl problem in which the demand for a number of different product families must be supplied from a set of facility sites and each site offers a choice of facility types exhibiting different capacities mpcfl generalizes both the uncapacitated or simple facility location ufl problem and the pure integer capacitated facility location problem we define a branch and bound algorithm for mpcfl that utilizes bounds formed by a lagrangian relaxation of mpcfl which decomposes the problem into ufl subproblems and easily solvable 0 1 knapsack subproblems the ufl subproblems are solved by the dual based procedure of erlenkotter we also present a subgradient optimization lagrangian relaxation based heuristic for mpcfl computational experience with the algorithm and heuristic are reported the mpcfl heuristic is seen to be extremely effective generating solutions to the test problems that are on average within 2 of optimality and the branch and bound algorithm is successful in solving all of the test problems to optimality

solving the uncapacitated network design problem by a lagrangean heuristic and branch and bound the network design problem is a multicommodity minimal cost network flow problem with fixed costs on the arcs i e a structured linear mixed integer programming problem it has various applications such as construction of new links in transportation networks topological design of computer communication networks and planning of empty freight car transportation on railways we present a lagrangean heuristic within a branch and bound framework as a method for finding the exact optimal solution of the uncapacitated network design problem with single origins and destinations for each commodity the simplest problem in this class but still np hard the lagrangean heuristic uses a lagrangean relaxation as subproblem solving the lagrange dual with subgradient optimization combined with a primal heuristic the benders subproblem yielding primal feasible solutions computational tests on problems of various sizes up to 1000 arcs 70 nodes and 138 commodities or 40 nodes and 600 commodities and of several different structures lead to the conclusion that the method is quite powerful outperforming for example a state of the art mixed integer code both with respect to problem size and solution time

inexact cuts in benders decomposition benders decomposition is a well known technique for solving large linear programs with a special structure in particular it is a popular technique for solving multistage stochastic linear programming problems early termination in the subproblems generated during benders decomposition assuming dual feasibility produces valid cuts that are inexact in the sense that they are not as constraining as cuts derived from an exact solution we describe an inexact cut algorithm prove its convergence under easily verifiable assumptions and discuss a corresponding dantzig wolfe decomposition algorithm the paper is concluded with some computational results from applying the algorithm to a class of stochastic programming problems that arise in hydroelectric scheduling

a note on an implementation of a method for quadratic semi infinite programming for convex quadratic semi infinite programming problems afortran package is described a first coarse grid is successively refined in such a way that the solution on the foregoing grids can be used on the one hand as starting points for the subsequent grids and on the other hand to considerably reduce the number of constraints which have to be considered in the subsequent problems this enables an efficient treatment of large problems with moderate storage requirements powell s 1983 numerically stable convex quadratic programming implementation is used to solve the qp subproblems

on the convergence of cross decomposition cross decomposition is a recent method for mixed integer programming problems exploiting simultaneously both the primal and the dual structure of the problem thus combining the advantages of dantzig wolfe decomposition and benders decomposition finite convergence of the algorithm equipped with some simple convergence tests has been proved stronger convergence tests have been proposed but not shown to yield finite convergence in this paper cross decomposition is generalized and applied to linear programming problems mixed integer programming problems and nonlinear programming problems with and without linear parts using the stronger convergence tests finite exact convergence is shown in the first cases unbounded cases are discussed and also included in the convergence tests the behaviour of the algorithm when parts of the constraint matrix are zero is also discussed the cross decomposition procedure is generalized by using generalized benders decomposition in order to enable the solution of nonlinear programming problems

minimal representation of quadratically constrained convex feasible regions in this paper we are concerned with characterizing minimal representations of feasible regions defined by both linear and convex quadratic constraints we say that representation is minimal if every other representation has either more quadratic constraints or has the same number of quadratic constraints and at least as many linear constraints we will prove that a representation is minimal if and only if it contains no redundant constraints no pseudo quadratic constraints and no implicit equality constraints we define a pseudo quadratic constraint as a quadratic constraint that can be replaced by a finite number of linear constraints in order to prove the minimal representation theorem we also prove that if the surfaces of two quadratic constraints match on a ball then they match everywhere in this paper we also provide algorithms that can be used to detect implicit equalities and pseudoquadratic constraints the redundant constraints can be identified using the hypersphere directions hd method

linear mean value cross decomposition a generalization of the kornai liptak method abstract we propose a modification of cross decomposition called mean value cross decomposition for linear programming problems the method is a generalization of the kornai liptak method and eliminates the need for using master problems the base for the method is the subproblem phase in cross decomposition where we iterate between the dual subproblem and the primal subproblem however instead of using the last solution of one subproblem as input to the other and vice versa we use the average mean value of all previously obtained solutions we show that this is equivalent to the brown robinson method for a matrix game and use this fact to prove convergence of the procedure finally we present some computational results comparing this method to other primal dual methods and to using a standard lp code

decomposition in general mathematical programming in this paper a unifying framework is presented for the generalization of the decomposition methods originally developed by benders 1962 and dantzig and wolfe 1960 these generalizations calledvariable decomposition andconstraint decomposition respectively are based on the general duality theory developed by tind and wolsey the framework presented is of a general nature since there are no restrictive conditions imposed on problem structure moreover inaccuracies and duality gaps that are encountered during computations are accounted for the two decomposition methods are proven not to cycle if certain fairly general conditions are met furthermore finite convergence can be ensured under the traditional finiteness conditions and asymptotic convergence can be guaranteed once certain continuity conditions are met the obvious symmetry between both types of decomposition methods is explained by establishing a duality relation between the two which extends a similar result in linear programming a remaining asymmetry in the asymptotic convergence results is argued to be a direct consequence of a fundamental asymmetry that resides in the tind wolsey duality theory it can be shown that in case the latter asymmetry disappears the former does too other decomposition techniques such as lagrangean decomposition and cross decomposition turn out to be captured by the general framework presented here as well

an algorithm to determine boundedness of quadratically constrained convex quadratic programmes abstract in this paper we present an algorithm which can be used to determine whether or not a convex quadratic objective function is bounded from below over a feasible region defined by convex quadratic constraints if there are m constraints and n variables the algorithm terminates after at most min m 1 n 1 iterations each iteration requires the identification of implicit equality constraints in a system of homogeneous linear inequality and equality constraints and the identification can be completed by the solution of linear programmes in addition the algorithm has the advantage of providing a mechanism to reduce both the number of constraints and the dimension of the problem

a constrained least squares regularization method for nonlinear ill posed problems this paper deals with a method for solving ill posed nonlinear hilbert space operator equations f x y regularization is obtained by solving a constrained least squares regularization problem min f x y 2 quad text subject to j x leqq beta 2 beta serves as a regularization parameter and j x is a quadratic penalty functional to robustly and efficiently solve this regularization problem we apply a trust region method at each iteration the quadratic penalty constraint is retained a gauss newton approximation to the objective functional is taken and we add a quadratic trust region constraint the resulting quadratic subproblem is then reformulated as a nonlinear complementarily problem and solved using newton s method this paper applies methods to find approximate solutions to a severely ill posed nonlinear first kind integral equation arising in geophysics the method of generalized cross validation gcv is used to pick the regularization parameter when random er

a one dimensional diffusion process in a wiener medium we consider a one dimensional diffusion process in a random medium which is itself generated by a wiener process the asymptotic behaviour is investigated using the self similarity of the medium process the results are analogous to the lattice case

mean value cross decomposition applied to integer programming problems abstract the mean value cross decomposition method for linear programming problems is a modification of ordinary cross decomposition that eliminates the need for using the benders or dantzig wolfe master problems as input to the dual subproblem the average of a part of all known dual solutions of the primal subproblem is used and as input to the primal subproblem the average of a part of all known primal solutions of the dual subproblem in this paper we study the lower bounds on the optimal objective function value of linear pure integer programming problems obtainable by the application of mean value cross decomposition and find that this approach can be used to get lower bounds ranging from the bound obtained by the lp relaxation to the bound obtained by the lagrangean dual we examplify by applying the technique to the clustering problem and give some preliminary computational results

a method of analytic centers for quadratically constrained convex quadratic programs an interior point method is developed for maximizing a concave quadratic function under convex quadratic constraints the algorithm constructs a sequence of nested convex sets and finds their approximate centers using a partial newton step given the first convex set and its approximate center the total arithmetic operations required to converge to an approximate solution are of order o sqrt m m n n 2 ln varepsilon where m is the number of constraints n is the number of variables and varepsilon is determined by the desired tolerance of the optimal value and the size of the first convex set a method to initialize the algorithm is also proposed so that the algorithm can start from an arbitrary perhaps infeasible point

on the choice of step size in subgradient optimization abstract this paper recommends some procedures for the selection of step sizes in the context of subgradient optimization the first of these procedures is developed in detail in this study and is a theoretically convergent scheme this method has two phases the first phase is designed to accelerate the solution procedure towards an optimal solution while the second phase helps to close in on an optimal solution a second technique recommended is a simple minded scheme which although not theoretically convergent seems to be computationally very efficient these two methods are shown to compare favorably with held wolfe and crowder s scheme for prescribing step sizes we also suggest some modifications of the latter scheme to make it computationally more efficient

computational aspects of two segment separable programming recursive separable programming algorithms based on local two segment approximations are described for the solution of separable convex programs details are also given for the computation of lower bounds on the optimal value by both a primal and a dual approach and these approaches are compared computational comparisons of the methods are provided for a variety of test problems including a water supply application with more than 600 constraints and more than 900 variables and an econometric modelling problem with more than 200 variables

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

unboundedness of a convex quadratic function subject to concave and convex quadratic constraints abstract we present conditions for the existence of upper and lower bounds on convex quadratic objective functions subject to concave and convex quadratic constraints we also present techniques for determining whether or not the conditions are satisfied

reoptimization with the primal dual interior point method reoptimization techniques for an interior point method applied to solving a sequence of linear programming problems are discussed conditions are given for problem perturbations that can be absorbed in merely one newton step the analysis is performed for both short step and long step feasible path following methods a practical procedure is then derived for an infeasible path following method it is applied in the context of crash start for several large scale structured linear programs numerical results with oops a new object oriented parallel solver demonstrate the efficiency of the approach for large structured linear programs crash start leads to about 40 reduction in the number of iterations and translates into a 25 reduction of the solution time the crash procedure parallelizes well and speed ups between 3 1 3 8 on four processors are achieved

cross decomposition applied to integer programming problems duality gaps and convexification in parts we study the lower bounds on the optimal objective function value of linear pure integer programming problems obtainable by the convexification in parts that results from using benders or cross decomposition and the best lower bounds obtainable by the convexification resulting from lagrangian relaxation together with subgradient optimization or dantzig wolfe decomposition a comparison shows that generalized benders and generalized cross decomposition yield the best of these bounds while ordinary benders decomposition yields bounds that are sometimes better and sometimes worse than those of lagrangian relaxation however cross decomposition can be used to automatically get the best of the two bounds the conclusion of this paper is that cross decomposition can be useful for getting good lower bounds for pure integer programming problems bounds that can be made better than those of the frequently used lagrangian relaxation

generalized benders decomposition for one class of minlps with vector conic constraint in this paper we mainly study one class of mixed integer nonlinear programming problems minlps with vector conic constraint in banach spaces duality theory of convex vector optimization problems applied to this class of minlps is deeply investigated with the help of duality we use the generalized benders decomposition method to establish an algorithm for solving this minlp several convergence theorems on the algorithm are also presented the convergence theorems generalize and extend the existing results on minlps in finite dimensional spaces

extensions to the generalised assignment heuristic for vehicle routing abstract although tabu search has generally given the best results for benchmark vehicle routing problems it has been observed that such heuristics require substantial computing times and several parameter settings thus when good quality solutions are required quickly there is still a need for relatively fast and effective heuristics the generalised assignment heuristic for the basic vehicle routing problem vrp can be extended by adjusting seed positions to reduce the optimal objective value for the generalised assignment problem gap combined with a simple lagrangean heuristic this gives a relatively straightforward method for obtaining solutions within a few percent of the best known values for benchmark problems

exact and heuristic solution approaches for the mixed integer setup knapsack problem we consider a class of knapsack problems that include setup costs for families of items an individual item can be loaded into the knapsack only if a setup cost is incurred for the family to which it belongs a mixed integer programming formulation for the problem is provided along with exact and heuristic solution methods the exact algorithm uses cross decomposition the proposed heuristic gives fast and tight bounds in addition a benders decomposition algorithm is presented to solve the continuous relaxation of the problem this method for solving the continuous relaxation can be used to improve the performance of a branch and bound algorithm for solving the integer problem computational performance of the algorithms are reported and compared to cplex