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

an efficient trajectory method for probabilistic production inventory distribution problems we consider a supply chain operating in an uncertain environment the customers demand is characterized by a discrete probability distribution a probabilistic programming approach is adopted for constructing an inventory production distribution plan over a multiperiod planning horizon the plan does not allow the backlogging of the unsatisfied demand and minimizes the costs of the supply chain while enabling it to reach a prescribed nonstockout service level it is a strategic plan that hedges against undesirable outcomes and that can be adjusted to account for possible favorable realizations of uncertain quantities a modular integrated and computationally tractable method is proposed for the solution of the associated stochastic mixed integer optimization problems containing joint probabilistic constraints with dependent right hand side variables the concept of p efficiency is used to construct a finite number of demand trajectories which in turn are employed to solve problems with joint probabilistic constraints we complement this idea by designing a preordered set based preprocessing algorithm that selects a subset of promising p efficient demand trajectories finally to solve the resulting disjunctive mixed integer programming problem we implement a special column generation algorithm that limits the risk of congestion in the resources of the supply chain the methodology is validated on an industrial problem faced by a large chemical supply chain and turns out to be very efficient it finds a solution with a minimal integrality gap and provides substantial cost savings

generalized drinfel d sokolov hierarchies a general approach is adopted to the construction of integrable hierarchies of partial differential equations a series of hierarchies associated to untwisted kac moody algebras and conjugacy classes of the weyl group of the underlying finite lie algebra is obtained the generalized kdv hierarchies of v g drinfel d and v v sokolov are obtained as the special case for the coxeter element various examples of the general formalism are treated in some detail including the fractional kdv hierarchies

algorithms for a multi level network optimization problem in this paper we work on a multi level network optimization problem that integrates into the same model important aspects of i discrete facility location ii topological network design and iii network dimensioning potential applications for the model are discussed stressing its growing importance the multi level network optimization problem treated is defined and a mathematical programming formulation is presented we make use of a branch and bound algorithm based on lagrangean relaxation lower bounds to introduce some new powerful auxiliary algorithms to exactly solve the problem we conduct a set of computational experiments that indicate the quality of the proposed approach

matrix formulation of hamiltonian structures of constrained kp hierarchy we give a matrix formulation of the hamiltonian structures of constrained kp hierarchy first we derive from the matrix formulation the hamiltonian structure of the one constraint kp hierarchy which was originally obtained by oevel and strampp we then generalize the derivation to the multiconstraint case and show that the resulting bracket is actually the second gelfand dickey bracket associated with the corresponding lax operator the matrix formulation of the hamiltonian structure of the one constraint kp hierarchy in the form introduced in the study of matrix model is also discussed

generalized drinfeld sokolov reductions and kdv type hierarchies generalized drinfeld sokolov ds hierarchies are constructed through local reductions of hamiltonian flows generated by monodromy invariants on the dual of a loop algebra following earlier work of de groot et al reductions based upon graded regular elements of arbitrary heisenberg subalgebras are considered we show that in the case of the nontwisted loop algebra l gl n graded regular elements exist only in those heisenberg subalgebras which correspond either to the partitions ofn into the sum of equal numbersn pr or to equal numbers plus onen pr 1 we prove that the reduction belonging to the grade 1 regular elements in the casen pr yields thep p matrix version of the gelfand dickeyr kdv hierarchy generalizing the scalar casep 1 considered by ds the methods of ds are utilized throughout the analysis but formulating the reduction entirely within the hamiltonian framework provided by the classical r matrix approach leads to some simplifications even forp 1

facets and algorithms for capacitated lot sizing the dynamic economic lot sizing model which lies at the core of numerous production planning applications is one of the most highly studied models in all of operations research and yet capacitated multi item versions of this problem remain computationally elusive we study the polyhedral structure of an integer programming formulation of a single item capacitated version of this problem and use these results to develop solution methods for multi item applications in particular we introduce a set of valid inequalities for the problem and show that they define facets of the underlying integer programming polyhedron computational results on several single and multiple product examples show that these inequalities can be used quite effectively to develop an efficient cutting plane branch and bound procedure moreover our results show that in many instances adding certain of these inequalities a priori to the problem formulation and avoiding the generation of cutting planes can be equally effective

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

a lagrangian heuristic for the capacitated plant location problem with single source constraints abstract in this paper we look at the lagrangian relaxations of the capacitated plant location problem with single source constraints in terms of the bounds and solution techniques we propose a heuristic based on a lagrangian relaxation of the problem we also show how this heuristic can be used to solve other combinatorial problems that are related to this problem such as the concentrator location problem and the bin packing problem

generating alternative mixed integer programming models using variable redefinition dropping the complicating constraints in a mixed integer linear program often yields a special structure subproblem that can be reformulated using a different set of decision variables once the new variables have been identified the entire problem can be reformulated in terms of the new variables we develop a theory of variable redefinition based on relating the two sets of decision variables by a linear transformation and describe methods for reformulating the special structure problem the reformulated models have a more accurate linear relaxation than the problems from which they were derived an important property within the context of linear programming based branch and bound modeling approaches

probabilistic properties of the dual structure of the multidimensional knapsack problem and fast statistically efficient algorithms properties of several dual characteristics of the multidimensional knapsack problem such as the probability of existence of optimal and optimal feasible lagrange function generalized saddle points magnitude of relative duality gap etc are investigated for different probabilistic models sufficient conditions of good asymptotic behavior of the dual characteristics are given a fast statistically efficient approximate algorithm with linear running time complexity for problems with random coefficients is presented

a constrained nonlinear 0 1 program for data allocation abstract this paper analyzes the problem of allocating copies of relations from a global database to the sites of a geographically distributed communication network the objective of the allocation is to minimize the total cost due to transmissions generated by queries from the various sites including queries that access multiple relations this allocation problem is modeled as a constrained nonlinear 0 1 subproblems generated during subgradient optimization are solved as optimization some of the unconstrained quadratic 0 1 subproblems generated during subgradient optimization are solved as maximum flow problems while the others require implicit enumeration depending on the nature of the objective function coefficients of the subproblems our solution approach is tested extensively on data allocation problems with as many as 100 sites and 20 relations on a set of randomly generated test problems our approach was close to two orders of magnitude faster than the general purpose integer programming code osl

stronger k tree relaxations for the vehicle routing problem abstract a lagrangian based exact solution algorithm for the vehicle routing problem vrp defined on an undirected graph is introduced in this paper lower bounds are obtained by allowing exponentially many inequalities as candidates to lagrangian dualization three different families of strong valid inequalities each one with exponentially many elements are used within vrp formulations for each of them separation procedures are proposed for points that define incidence vectors of k trees violated inequalities identified in this way are then dualized in a relax and cut framework upper bounds are generated through a lagrangian clarke and wright heuristic a variable fixation test based on approximating linear programming reduced costs is also implemented computational results are presented for the proposed algorithm

a lagrangean relaxation approach for very large scale capacitated lot sizing in this paper we develop a lagrangean relaxation based heuristic procedure to generate near optimal solutions to very large scale capacitated lot sizing problems clsp with setup times and limited overtime our computational results show that large problems involving several thousand products and several thousand 0 1 integer variables can be solved in a reasonable amount of computer time to within one percent of their optimal solution the proposed procedure is general enough to be applied directly or with slight modification to real life production problems

capacitated multi item lot sizing problems with time windows this research concerns a new family of capacitated multi item lot sizing problems namely lot sizing problems with time windows two classes of the problem are analyzed and solved using different lagrangian heuristics capacity constraints and a subset of time window constraints are relaxed resulting in particular single item time window problems that are solved in polynomial time other relaxations leading to the classical wagner whitin problem are also tested several smoothing heuristics are implemented and tested and their results are compared the gaps between lower and upper bounds for most problems are very small less than 1 moreover the proposed algorithms are robust and do not seem to be too affected when different parameters of the problem are varied

a heuristic method for the set covering problem we present a lagrangian based heuristic for the well known set covering problem scp the algorithm was initially designed for solving very large scale scp instances involving up to 5 000 rows an

short term scheduling of thermal electric generators using lagrangian relaxation this paper presents an expanded formulation of the unit commitment problem in which hundreds of thermal electric generators must be scheduled on an hourly basis for up to 7 days at a time the underlying model incorporates the full set of costs and constraints including setup production ramping and operational status and takes the form of a mixed integer nonlinear control problem lagrangian relaxation is used to disaggregate the model by generator into separate subproblems which are then solved with a nested dynamic program the strength of the methodology lies partially in its ability to construct good feasible solutions from information provided by the dual thus the need for branch and bound is eliminated in addition the inclusion of the ramping constraint provides new insight into the limitations of current techniques computational experience with the proposed algorithm indicates that problems containing up to 100 units and 48 time periods can be readily solved in reasonable times duality gaps of less than 1 were achieved in all cases

heuristics for multilevel lot sizing with a bottleneck in this paper we present a heuristic method based on lagrangian relaxation for multilevel lot sizing when there is a single bottleneck facility a series of lagrangian relaxations one for each item in the product structure is imbedded in a branch and bound procedure the objective is to find a production schedule that fits within available capacity at minimum cost the method has two solution phases dual and primal in the dual phase of the procedure implied costs of setups and production are determined based on a tentative schedule the primal phase is repeated with these new prices and we iterate to reach a good solution the solution procedure is first tested on two special cases uncapacitated multilevel lot sizing and the capacitated single level multi item lot sizing problem the results show that the solution procedure can provide better solutions than some heuristics designed especially for those problems test results on the bottleneck problem indicate that good feasible solutions are found for problems too difficult to solve with exact methods

hydrodynamic reductions of the dispersionless harry dym hierarchy we investigate the reductions of the dispersionless harry dym hierarchy to systems of finitely many partial differential equations these equations must satisfy the compatibility condition and they are diagonalizable and semi hamiltonian by imposing a further constraint the compatibility is reduced to a system of algebraic equations whose solutions are described

m idempotent and self dual morphological filters in this paper we present a comprehensive analysis of self dual and m idempotent operators we refer to an operator as m idempotent if it converges after m iterations we focus on an important special case of the general theory of lattice morphology spatially variant morphology which captures the geometrical interpretation of spatially variant structuring elements we demonstrate that every increasing self dual morphological operator can be viewed as a morphological center necessary and sufficient conditions for the idempotence of morphological operators are characterized in terms of their kernel representation we further extend our results to the representation of the kernel of m idempotent morphological operators we then rely on the conditions on the kernel representation derived and establish methods for the construction of m idempotent and self dual morphological operators finally we illustrate the importance of the self duality and m idempotence properties by an application to speckle noise removal in radar images

theoretical aspects of gray level morphology after a brief discussion of the extension of mathematical morphology to complete lattices the space of gray level functions is considered and the concept of a threshold set is introduced it is shown how one can use binary morphological operators and thresholding techniques to build a large class of gray level morphological operators particular attention is given to the class of so called flat operators i e operators which commute with thresholding it is also shown how to define dilations and erosions with nonflat structuring elements if the gray level set is finite it is reported that mere truncation yields wrong results

the integrable hierarchy constructed from a pair of kdv type hierarchies and its associated w algebra for any two arbitrary positive integers n and m using them th kdv hierarchy and the n m th kdv hierarchy as building blocks we are able to construct another integrable hierarchy referred to as the n m th kdv hierarchy thew algebra associated to the second hamiltonian structure of the n m th kdv hierarchy calledw n m algebra is isomorphic via a miura map to the direct sum of aw m algebra aw n m algebra and an additionalu 1 current algebra in turn from the latter we can always construct a representation of aw algebra

design of a degree constrained minimal spanning tree with unreliable links and node outage costs abstract the degree constrained minimal spanning tree dcmst problem with unreliable links and node outage costs consists of finding links in a network to connect a set of terminal nodes to a central node while minimizing the expected annual expenditure the number of ports available on each terminal node limits the number of incident links the degree constraint each terminal node in the network has an associated node outage cost which is the economic cost incurred by the network user whenever that node is disabled due to failure of a link we formulate this problem as an integer programming problem and present a lagrangian relaxation method which for each choice of lagrangian multipliers provides a lower bound for the optimal objective function value a subgradient optimization method is used to search for multipliers which yield good lower bounds a branch exchange heuristic procedure makes modifications to each infeasible solution of the lagrangian relaxation in order to find good feasible solutions the quality of these heuristic solutions is estimated using the best obtained lower bounds experimental results over a wide range of problem structures show that the branch exchange heuristic method yields verifiably good solutions to this problem

a branch and bound algorithm for the total weighted tardiness problem this paper presents a new branch and bound algorithm for the single machine total weighted tardiness problem it obtains lower bounds using a lagrangian relaxation approach with subproblems that are total weighted completion time problems the well known subgradient optimization technique is replaced by a multiplier adjustment method that leads to an extremely fast bound calculation the method incorporates various devices for checking dynamic programming dominance in the search tree extensive computational results for problems with up to 50 jobs show the superiority of the algorithm over existing methods

scheduling of manufacturing systems using the lagrangian relaxation technique the practical solutions for three manufacturing scheduling problems are examined as each problem is formulated constraints are added or modified to reflect increasing real world complexity the first problem considers scheduling single operation jobs on identical machines the second problem is concerned with scheduling multiple operation jobs with simple fork join precedence constraints on identical machines the third problem is the job shop problem in which multiple operation jobs with general precedence constraints are scheduled on multiple machine types langrangian relaxation is used to decompose each of the scheduling problems into job or operation level subproblems the subproblems are easier to solve than the original problem and have intuitive appeal this technique results in algorithms which generate near optimal schedules efficiently while giving a lower bound on the optimal cost in resolving the scheduling problem from one time instant to the next the lagrange multipliers from the last schedule can be used to initialize the multipliers further reducing the computation time

exact and approximation algorithms for the operational fixed interval scheduling problem the operational fixed interval scheduling problem ofisp is characterized as the problem of scheduling a number of jobs each with a fixed starting time a fixed finishing time a priority index and a job class the objective is to find an assignment of jobs to machines with maximal total priority the problem is complicated by the restrictions that i each machine can handle only one job at a time ii each machine can handle only jobs from a prespecified subset of all possible job classes and iii preemption is not allowed it follows from the above that ofisp has both the character of a job scheduling problem and the character of an assignment problem in this paper we discuss the occurrence of the problem in practice and we present newly developed exact and approximation algorithms for solving ofisp finally some computational results are shown

integrable coupled kdv systems we give the conditions for a system of n coupled korteweg de vries kdv type of equations to be integrable we find the recursion operators of each subclass and give all examples for n 2

bi hamiltonian structures of the coupled akns hierarchy and the coupled yajima oikawa hierarchy the hamiltonian theory for the two component akns hierarchy and yajima oikawa hierarchy is considered from the viewpoint of reduction we show that the second hamiltonian structures of the former is a dirac reduction of the sl 3 current algebra while the latter is related to the classical w 3 4 algebra

a matrix generation approach for eigenvalue optimization we study the extension of a column generation technique to nonpolyhedral models in particular we study the problem of minimizing the maximum eigenvalue of an affine combination of symmetric matrices at each step of the algorithm a restricted master problem in the primal space corresponding to the relaxed dual original problem is formed a query point is obtained as an approximate analytic center of a bounded set that contains the optimal solution of the dual problem the original objective function is evaluated at the query point and depending on its differentiability a column or a matrix is added to the restricted master problem we discuss the issues of recovering feasibility after the restricted master problem is updated by a column or a matrix the computational experience of implementing the algorithm on randomly generated problems are reported and the cpu time of the matrix generation algorithm is compared with that of the primal dual interior point methods on dense and sparse problems using the software sdpt3 our numerical results illustrate that the matrix generation algorithm outperforms primal dual interior point methods on dense problems with no structure and also on a class of sparse problems

an efficient algorithm for the lagrangean dual of nonlinear knapsack problems with additional nested constraints abstract this paper presents an efficient algorithm for solving the lagrangean dual of nonlinear knapsack problems with additional nested constraints the dual solution provides a feasible primal solution if it exists and associated lower and upper bounds on the optimal objective function value of the primal problem computational experience is cited indicating computation time number of dual iterations and tightness of the bounds

hierarchical design of an integrated production and 2 echelon distribution system abstract this study uses the method of lagrangean relaxation in the hierarchical design of an integrated model of production distribution functions in a 2 echelon system a mixed integer mathematical model is developed with a centralized planning perspective to address production and distribution decisions simultaneously in order to solve the resulting large scale problem the lagrangean relaxation is used to decouple the imbedded distribution and production subproblems and subgradient optimization is implemented to coordinate the information flow between these in a hierarchical manner this corresponds to a decentralized organizational design where a central agent coordinates the information exchange between the distribution and production organizational units a forward heuristic designed to solve the distribution subproblem is shown to provide good solutions hierarchical interdependency is incorporated into the lagrangean heuristic such that distribution decisions are placed in the top level to restrict the solution of the production subproblem in the lower level

a method for the cutting stock problem with different qualities abstract an operations research project at a swedish company is reported the project deals with the problem of allocating wood products with different quality requirements to be cut from wooden boards which in turn consists of a number of quality zones a major difficulty is the existence of distinct time limits since the cross cutting of the boards is done in a real time process in this paper we derive a mathematical model which includes a set packing problem that accurately describes the real world problem this problem is then solved using a combination of lagrangean relaxation and subgradient optimization numerical results for practical problems are reported the results indicate a valid model and satisfactory efficiency of the proposed solution procedure

using lagrangean techniques to solve hierarchical production planning problems this paper proposes and tests a procedure for decomposing a large scale production planning problem modeled as a mixed integer linear program we interpret this decomposition in the context of hax and meal s hierarchical framework for production planning the procedure decomposes the production planning problem into two subproblems which correspond to the aggregate planning subproblem and a disaggregation subproblem in the hax meal framework the linking mechanism for these two subproblems is an inventory consistency relationship which is priced out by a set of lagrange multipliers the best values for the multipliers are found by an iterative procedure which may be interpreted as a feedback mechanism in the hax meal framework at each iteration the procedure finds both a lower bound on the optimal value to the production planning problem and a feasible solution from which an upper bound is obtained our computational tests show that the best feasible solution found from this procedure is very close to o

facilities location in a competitive environment a promethee based multiple criteria analysis abstract in this paper a methodology for locating facilities in a competitive environment is introduced in this methodology the multi criteria approach and the notion of efficiency are appropriately integrated the classic distance criterion is combined with a second one a quality criterion to produce a more realistic representation of a customer s decision process

an exact method for the two echelon single source capacitated facility location problem facility location problems form an important class of integer programming problems with applications in the telecommunication distribution and transportation industries in this paper we are conc

a benders decomposition based heuristic for the hierarchical production planning problem abstract in this paper a heuristic procedure for determining good feasible solutions to a multi item dynamic production planning problem is described and evaluated the problem which is modelled as a mixed integer linear program has a hierarchical structure where production items are aggregated into families and families into product types a decomposition of the original problem creates a trivial subproblem providing feasible solutions on the type level and a master problem which separates into one uncapacitated lot sizing problem for each family the latter problems can be solved by dynamic programming with very little computational effort besides the nice structure of both the sub and master problem the dual information from the subproblem can be given interesting economic interpretations as distinguished from hax and meal s hierarchical framework for production planning and also they hybrid approach suggested by graves in 1982 there exists a direct information flow from the family level problem to the type level problem in form of a production plan the heuristic is essentially an application of benders decomposition the only difference being that the benders cuts are relaxed using lagrangean multipliers a subgradient procedure is used to update the lagrangean multipliers one iteration of this heuristic requires less computational effort compared to grave s algorithm this is due to the fact that the benders subproblem can be solved by inspection while graves algorithm involves solving an lp problem on the type level for thirtysix medium size test problems the average deviation from optimum is 2 34 and the deviation ranges between 0 00 and 5 95

using confidence limits for the global optimum in combinatorial optimization let z be the unknown optimal value of a combinatorial optimization problem in minimization form and let z be a constant satisfying probz z 1 for a fixed 0 1 then z is called a statistical or probabilistic lower bound at significance level 1 we discuss how statistical lower bounds can be used to measure the effectiveness of an approximate solution and to fathom subproblems in a branch and bound approach after reviewing some methods for obtaining statistical lower bounds we report on extensive computational experience for two notoriously difficult combinatorial optimization problems the traveling salesman problem and the quadratic assignment problem from our experience we conclude that statistical bounds are highly useful for quadratic assignment problems in this problem context the quality of simple analytic formulas is comparable to the quality of the maximum likelihood formula in general using statistics to aid in the solution of combinatorial optimization problems should be most useful when such problems are large

mathematics of dispersive water waves a commuting hierarchy of dispersive water wave equations makes a three hamiltonian system which belongs to a general class of nonstandard integrable systems whose theory is developed the modified water wave hierarchy is a bi hamiltonian system its modification bifurcates the water wave hierarchy and the hierarchies of the korteweg de vries and the modified korteweg de vries equations as well as the classical miura map are given new representations through various specializations of nonstandard systems

lagrangian relaxation versus genetic algorithm based matheuristic for a large partitioning problem abstract this paper is concerned with a partitioning problem one of the applications and the motivation for this research is the problem of class formation for training and retraining sessions at large electricity distributors two different approaches are developed one is based on the quadratic multiple knapsack formulation and lagrangian relaxation the other is a matheuristic developed as an amalgamation of genetic algorithms and integer programming the approaches are tested by means of computational experiments both heuristics outperformed the direct application of quadratic programming with the lagrangian relaxation based approach performing the best on average and the genetic algorithm based approach performing the best on the larger test cases

optimization based production planning with hybrid dynamics and constraints optimizing production planning has tremendous economic impact for many industrial production systems in this paper the planning problem of a class of production systems with hybrid dynamics and constraints is considered with practical background of power generation planning and other applications the problem is solved within the lagrangian relaxation framework with the system wide demand and resource limit constraints relaxed by lagrange multipliers a new method is developed in this paper to obtain the exact optimal solutions to the subproblems with hybrid dynamics and constraints efficiently without discretizing the continuous production levels or introducing intermediate levels of relaxation a novel definition of the discrete state associated with a consecutive time span is introduced so that solving each subproblem is converted into solving a number of continuous optimization problems and a discrete optimization problem separately an efficient double dynamic programming dp method is developed to solve these subproblems and the principle of optimality is guaranteed for both the continuous and discrete problem the production levels in a consecutive running span with non convex piecewise linear cost functions are determined in a dp forward sweep without discretization the dp method is then applied to determine the optimal discrete operating states across time efficiently numerical testing results demonstrate that the new method is efficient and effective for optimization based production planning with the complex hybrid dynamics and constraints

wronskian type solutions for the vector k constrained kp hierarchy motivated by a relation of the 1 constrained kadomtsev petviashvili kp hierarchy with the 2 component kp hierarchy the tau functions of the vector k constrained kp hierarchy are constructed by using an analogue of the baker akhiezer m 1 point function these tau functions are expressed in terms of wronskian type determinants

capacitated lot sizing and scheduling by lagrangean relaxation abstract development of new models and solution procedures for production planning has been of research interest for several decades implementation of these models has resulted in lower production costs by reducing inventories number of setups and labor costs in this paper we develop several optimal near optimal procedures for the capacitated lot sizing and scheduling problem clsp with setup times limited regular time and limited overtime we formulate a mixed integer linear programming model of the problem and solve it by lagrangean relaxation we experiment with alternative lagrangean relaxations and develop new procedures to solve these relaxations overall the capacity constraints relaxation seems to be superior to the demand constraints relaxation our results show that large problems can be solved in reasonable computer times and within one percent accuracy of the optimal solutions we solved 99 8 i e 99 items and 8 periods 50 12 and 50 8 problems in 30 61 36 25 and 12 65 seconds of cdc cyber 730 computer time respectively our procedures are general enough to be applied directly or with slight modifications in real life production settings

dynamic aggregation of set partitioning constraints in column generation column generation is often used to solve problems involving set partitioning constraints such as vehicle routing and crew scheduling problems when these constraints are in large numbers and the columns have on average more than 8 12 nonzero elements column generation often becomes inefficient because solving the master problem requires very long solution times at each iteration due to high degeneracy to overcome this difficulty we introduce a dynamic constraint aggregation method that reduces the number of set partitioning constraints in the master problem by aggregating some of them according to an equivalence relation to guarantee optimality this equivalence relation is updated dynamically throughout the solution process tests on the linear relaxation of the simultaneous vehicle and crew scheduling problem in urban mass transit show that this method significantly reduces the size of the master problem degeneracy and solution times especially for larger problems in fact for an instance involving 1 600 set partitioning constraints the master problem solution time is reduced by a factor of 8

a distributed computation algorithm for solving portfolio problems with integer variables a portfolio problem with integer variables can facilitate the use of complex models including models containing discrete asset values transaction costs and logical constraints this study proposes a distributed algorithm for solving a portfolio program to obtain a global optimum for a portfolio problem with n integer variables the objective function first is converted into an ellipse function containing n separated quadratic terms next the problem is decomposed into m equal size separable programming problems solvable by a distributed computation system composed of m personal computers linked via the internet the numerical examples illustrate that the proposed method can obtain the global optimum effectively for large scale portfolio problems involving integral variables

conditional subgradient optimization theory and applications abstract we generalize the subgradient optimization method for nondifferentiable convex programming to utilize conditional subgradients firstly we derive the new method and establish its convergence by generalizing convergence results for traditional subgradient optimization secondly we consider a particular choice of conditional subgradients obtained by projections which leads to an easily implementable modification of traditional subgradient optimization schemes to evaluate the subgradient projection method we consider its use in three applications uncapacitated facility location two person zero sum matrix games and multicommodity network flows computational experiments show that the subgradient projection method performs better than traditional subgradient optimization in some cases the difference is considerable these results suggest that our simply modification may improve subgradient optimization schemes significantly this finding is important as such schemes are very popular especially in the context of lagrangean relaxation

a multiperiod two echelon multicommodity capacitated plant location problem abstract in this paper we deal with a facility location problem where one desires to establish facilities at two different distribution levels by selecting the time periods our model intends to minimize the total cost for meeting demands for all the products specified over the planning horizon at various customer locations while satisfying the capacity requirements of the production plants and intermediate warehouses we address this problem by means of a formulation as a mixed integer programming problem a lagrangean relaxation is proposed to solve the problem together with a heuristic procedure that constructs feasible solutions of the original problem from the solutions at the lower bounds obtained by the relaxed problems computational tests are provided showing the good performance of this approach for a wide range of problems

descent direction algorithm with multicommodity flow problem for signal optimization and traffic assignment jointly network managers wish to optimize control parameters such as signal setting which are very related to the traffic assignment models on the other hand traffic assignment patterns as an important instrument for predicting the amount of flow on network links are dependent to control decisions according to the significance of this concept some important papers about this mutually relation are reviewed in this paper then we implement a nonlinear algorithm on a minimal cost multicommodity flow mcmf problem to optimize some control policies subject to optimal flows although we take signal times into account but this approach has a far more reaching application in urban network control and design we employ a hybrid intelligent algorithm integrating decent direction algorithm and an interior point algorithm in a mutually consistent scheme for obtaining optimal signals and equilibrium flows an example is given to illustrate the effectiveness of our scheme

on hamiltonian perturbations of hyperbolic systems of conservation laws i quasi triviality of bi hamiltonian perturbations we study the general structure of formal perturbative solutions to the hamiltonian perturbations of spatially one dimensional systems of hyperbolic pdes vt v x 0 under certain genericity assumptions it is proved that any bi hamiltonian perturbation can be eliminated in all orders of the perturbative expansion by a change of coordinates on the infinite jet space depending rationally on the derivatives the main tool is in constructing the so called quasimiura transformation of jet coordinates eliminating an arbitrary deformation of a semisimple bi hamiltonian structure of hydrodynamic type the quasi triviality theorem we also describe following 35 the invariants ofsuchbi hamiltonian structures with respect to the group of miura type transformations depending polynomially on the derivatives c

a mathematical programming model and solution for scheduling production orders in shanghai baoshan iron and steel complex in this paper we investigate the production order scheduling problem derived from the production of steel sheets in shanghai baoshan iron and steel complex baosteel a deterministic mixed integer programming mip model for scheduling production orders on some critical and bottleneck operations in baosteel is presented in which practical technological constraints have been considered the objective is to determine the starting and ending times of production orders on corresponding operations under capacity constraints for minimizing the sum of weighted completion times of all orders due to large numbers of variables and constraints in the model a decomposition solution methodology based on a synergistic combination of lagrangian relaxation linear programming and heuristics is developed unlike the commonly used method of relaxing capacity constraints this methodology alternatively relaxes constraints coupling integer variables with continuous variables which are introduced to the objective function by lagrangian multipliers the lagrangian relaxed problem can be decomposed into two sub problems by separating continuous variables from integer ones the sub problem that relates to continuous variables is a linear programming problem which can be solved using standard software package osl while the other sub problem is an integer programming problem which can be solved optimally by further decomposition the subgradient optimization method is used to update lagrangian multipliers a production order scheduling simulation system for baosteel is developed by embedding the above lagrangian heuristics computational results for problems with up to 100 orders show that the proposed lagrangian relaxation method is stable and can find good solutions within a reasonable time

finding discrete global minima with a filled function for integer programming the filled function method is an approach to find the global minimum of multidimensional functions this paper proposes a new definition of the filled function for integer programming problem a filled function which satisfies this definition is presented furthermore we discuss the properties of the filled function and design a new filled function algorithm numerical experiments on several test problems with up to 50 integer variables have demonstrated the applicability and efficiency of the proposed method

solving mixed integer programming problems using automatic reformulation in this paper we describe computational experience in solving mixed 0 1 programming problems using strong valid inequalities as cutting planes in particular we report on the solution to optimality of 18 medium to large size problems including production planning problems with setup costs and capacity constraints multilevel distribution planning problems drainage and heating system design problems and electricity generator scheduling problems the solution approach uses the theory of strong valid inequalities that we developed in a series of earlier papers here we report specifically on the implementation of an experimental system mpsarx which consists of the sciconic mathematical programming system and an automatic reformulation executor arx that use this theory and on the results obtained with this system

dual applications of proximal bundle methods including lagrangian relaxation of nonconvex problems we exhibit useful properties of proximal bundle methods for finding min sf where f and s are convex we show that they asymptotically find objective subgradients and constraint multipliers involved in optimality conditions multipliers of objective pieces for max type functions and primal and dual solutions in lagrangian decomposition of convex programs when applied to lagrangian relaxation of nonconvex programs they find solutions to relaxed convexified versions of such programs numerical results are presented for unit commitment in power production scheduling

reliable facility location design under the risk of disruptions reliable facility location models consider unexpected failures with site dependent probabilities as well as possible customer reassignment this paper proposes a compact mixed integer program mip formulation and a continuum approximation ca model to study the reliable uncapacitated fixed charge location problem rufl which seeks to minimize initial setup costs and expected transportation costs in normal and failure scenarios the mip determines the optimal facility locations as well as the optimal customer assignments and is solved using a custom designed lagrangian relaxation lr algorithm the ca model predicts the total system cost without details about facility locations and customer assignments and it provides a fast heuristic to find near optimum solutions our computational results show that the lr algorithm is efficient for mid sized rufl problems and that the ca solutions are close to optimal in most of the test instances for large scale problems the ca method is a good alternative to the lr algorithm that avoids prohibitively long running times

optimal release times in a single server an optimal control perspective the paper is concerned with the basic structure of optimal control of discrete event dynamic processes defined over a max plus algebra only a simple system is considered namely a single server processing a given sequence of jobs but the structural conditions that are discovered may lead to extensions to more general systems the problem in question is how to optimally control the completion output times of the jobs by assigning their release input times so as to minimize a measure of the discrepancy between the completion times and given desired due dates the concept of the costate is applied to the discrete dynamics to identify structural optimality conditions and in the case of quadratic cost measures the optimal control is shown to be computable by a state feedback law that is linear in the max plus algebra

modeling and solving the two facility capacitated network loading problem this paper studies a topical and economically significant capacitated network design problem that arises in the telecommunications industry in this problem given point to point communication demand in a network must be met by installing loading capacitated facilities on the arcs loading a facility incurs an arc specific and facility dependent cost this paper develops modeling and solution approaches for loading facilities to satisfy the given demand at minimum cost we consider two approaches for solving the underlying mixed integer program a lagrangian relaxation strategy and a cutting plane approach that uses three classes of valid inequalities that we identify for the problem we show that a linear programming formulation that includes these inequalities always approximates the value of the mixed integer program at least as well as the lagrangian relaxation bound our computational results on a set of prototypical telecommunication data show that including these inequalities considerably improves the gap between the integer programming formulation and its linear programming relaxation from an average of 25 to an average of 8 these results show that strong cutting planes can be an effective modeling and algorithmic tool for solving problems of the size that arise in the telecommunications industry

a critical review of discrete filled function methods in solving nonlinear discrete optimization problems many real life problems can be modeled as nonlinear discrete optimization problems such problems often have multiple local minima and thus require global optimization methods due to high complexity of these problems heuristic based global optimization techniques are usually required when solving large scale discrete optimization or mixed discrete optimization problems one of the more recent global optimization tools is known as the discrete filled function method nine variations of the discrete filled function method in literature are identified and a review on theoretical properties of each method is given some of the most promising filled functions are tested on various benchmark problems numerical results are given for comparison

inverse optimization an application to the capacitated plant location problem lagrangean techniques have had wide application to the optimization of discrete optimization problems inverse optimization refers to the fact that each time a lagrangean calculation is made for a specific problem with a given resources vector an optimal solution is obtained for a related problem with a suitably adjusted resources vector this property is studied in depth for the capacitated plant location problem and new parametric methods for that problem are suggested computational experience is reported

solving the fixed charge problem with lagrangian relaxation and cost allocation heuristics abstract a new heuristic algorithm is presented for solving large general fixed charge problems based on langrangian relaxation and cost allocation heuristics limited computational results indicate the new algorithm provides good solutions to large problems with reasonable computational requirements it demonstrates significant improvement over current methods for solving large general fixed charge problems

constraints of the kadomtsev petviashvili hierarchy for the kadomtsev petviashvili kp hierarchy constructed in terms of the famous sato theory a k constraint is proposed that leads the hierarchy to the nonlinear system involving a finite number of dynamical coordinates the eigenvalue problem of the k constrained system is naturally obtained from the linear system of the kp hierarchy which takes the form of kth order polynomial coupled with a first order one thus we are able to derive the correspondent lax pair recursion operator bi hamiltonian structures and conserved quantities the constraints for the bkp hierarchy are also sketched

the constrained modified kp hierarchy and the generalized miura transformations in this letter we consider the second hamiltonian structure of the constrained modified kp hierarchy after mapping the lax operator to a pure differential operator the second structure becomes the sum of the second and the third gelfand dickey brackets defined by this differential operator we simplify this hamiltonian structure by factorizing the lax operator into linear terms

classical w algebras we reconsider the relation between classicalw algebras and deformations of differential operators emphasizing the consistency with diffeomorphisms generators of thew algebra that arek differentials are constructed by a systematic procedure the method extends following drinfeld and sokolov tow algebras based on arbitrary simple lie algebras

progressive interval heuristics for multi item capacitated lot sizing problems we consider a family of n items that are produced in or obtained from the same production facility demands are deterministic for each item and each period within a given horizon of t periods if in a given period an order is placed setup costs are incurred the aggregate order size is constrained by a capacity limit the objective is to find a lot sizing strategy that satisfies the demands for all items over the entire horizon without backlogging and that minimizes the sum of inventory carrying costs fixed order costs and variable order costs all demands cost parameters and capacity limits may be time dependent in the basic joint setup cost js model the setup cost of an order does not depend on the composition of the order the joint and item dependent setup cost jis model allows for item dependent setup costs in addition to the joint setup costs we develop and analyze a class of so called progressive interval heuristics a progessive interval heuristic solves a js or jis problem over a progressively larger time interval always starting with period 1 but fixing the setup variables of a progressively larger number of periods at their optimal values in earlier iterations different variants in this class of heuristics allow for different degrees of flexibility in adjusting continuous variables determined in earlier iterations of the algorithm for the js model and the two basic implementations of the progressive interval heuristics we show under some mild parameter conditions that the heuristics can be designed to be e optimal for any desired value of e 0 with a running time that is polynomially bounded in the size of the problem they can also be designed to be simultaneously asymptotically optimal and polynomially bounded a numerical study covering both the js and jis models shows that a progressive interval heuristic generates close to optimal solutions with modest computational effort and that it can be effectively used to solve large scale problems

a branch and bound algorithm for a production scheduling problem in an assembly system under due date constraints abstract this paper focuses on a production scheduling problem in a tree structured assembly system operating on a make to order basis due dates are considered as constraints in the problem that is tardiness is not allowed the objective of the problem is to minimize holding costs of final product inventory as well as work in process inventory a mixed integer linear programming model is given and a branch and bound b b algorithm based on a lagrangian relaxation method is developed in the algorithm a subgradient method is used for obtaining good lower bounds of subproblems or partial solutions while a lagrangian heuristic is used for upper bounds results of computational experiments on randomly generated test problems showed that the b b algorithm found optimal solutions of problems of moderate sizes in a reasonable amount of computation time

a new lagrangian relaxation approach to the generalized assignment problem abstract in this paper we present a new lagrangian relaxation approach to the generalized assignment problem gap the new approach is based on a reformulation of gap into an equivalent problem which is then relaxed by traditional lagrangian relaxation techniques the reformulation is created by introducing a set of auxiliary variables and a number of coupling constraints by relaxing the coupling constraints we get subproblems where both types of constraint structures present in the gap are active the lower bound from the lagrangian dual of this approach is shown to be at least as strong as that from the best of the traditional lagrangian relaxation approaches an important feature is also the possibility of generating upper bounds quite easily the new approach has been tested on 10 problems of size m 4 n 25 i e 100 0 1 variables and the results have been compared with the traditional approaches

a branch and bound algorithm for the singly constrained assignment problem the singly constrained assignment problem scap is a linear assignment problem lap with one extra side constraint e g due to a time restriction the scap is in contrast to the lap difficult to solve a branch and bound algorithm is presented to solve the scap to optimality lower bounds are obtained by lagrangean relaxation computational results show that the algorithm is able to solve different types of scap instances up to size n 1000 within short running times on a standard personal computer

scheduling of design projects with uncertain number of iterations abstract a short product design cycle is critical to the success of companies in the era of time based competition the underlying design activities however are often interlinked and quite uncertain for example some activities may have to be iterated several times to meet the design criteria furthermore time critical projects suffer the risk of failure if they cannot meet established target dates generating good and robust schedules is thus critical especially under the concurrent engineering paradigm where the delay of a single task may have a domino effect on subsequent tasks and on other projects sharing designers and or resources this paper studies the scheduling of design projects with uncertain number of iterations while managing design risks a separable problem formulation that balances modeling accuracy and computation complexity is created with the goal to minimize project tardiness and risk penalties an optimization based methodology that combines lagrangian relaxation stochastic dynamic programming and ordinal optimization is developed numerical results supported by simulation demonstrate that near optimal solutions are obtained and uncertainties are effectively managed for problems of practical sizes

generating fenchel cutting planes for knapsack polyhedra the author recently proposed a class of cutting planes for integer programs called fenchel cuts which distinguish themselves from more conventional cuts in that they are generated by directly seeking to solve the separation problem rather than by using explicit knowledge of the polyhedral structure of the integer program an algorithm for generating fenchel cuts is presented and described in detail for the separation problem associated with knapsack polyhedra computational results are presented for a collection of real world integer programs to demonstrate the effectiveness of the cutting planes

an adaptation of sh heuristic to the location set covering problem abstract in a recent paper a new surrogate heuristic sh has been proposed for the set covering problem here we present an adaptation of it in order to solve more efficiently the location set covering problem we will show that our new version not only outperforms algorithm sh but that it is more accurate than the pair cma fmc its power is experimentally tested over a set of 65 randomly generated problems

annular filters for binary images a binary annular filter removes isolated points in the foreground and the background of an image here the adjective isolated refers to an underlying adjacency relation between pixels which may be different for foreground and background pixels in this paper annular filters are represented in terms of switch pairs a switch pair consists of two operators which govern the removal of points from foreground and background respectively in the case of annular filters switch pairs are completely determined by foreground and background adjacency it is shown that a specific triangular condition in terms of both adjacencies is required to establish idempotence of the resulting annular filter in the case of translation invariant operators an annular filter takes the form x spl rarr x spl oplus a spl cap x spl cup x spl ominus b where a and b are structuring elements satisfying some further conditions when a spl cap b spl cap a spl oplus b spl ne o it is an idempotent morphological filter when a spl cup b spl sub a spl oplus b it is a strong filter and in this case it can be obtained by composing in either order the annular opening x spl rarr x spl oplus a spl cap x and the annular closing x spl rarr spl cup x spl oplus b

convergent lagrangian and contour cut method for nonlinear integer programming with a quadratic objective function in this paper we present an efficient exact solution method for solving nonlinear separable integer programming problems with a quadratic objective function the proposed method combines the lagrangian dual method with a duality reduction scheme using contour cut at each iteration of the algorithm lower and upper bounds of the problem are determined by the lagrangian dual search to eliminate the duality gap a novel cut and partition scheme is derived by exploring the special structure of the quadratic contour the method finds an exact solution of the problem in a finite number of iterations computational results are reported for problems with up to 2000 integer variables comparison results with other methods are also presented

single facility resource allocation under capacity based economies and diseconomics of scope we consider the optimal allocation of a resource in a single facility production environment in the presence of capacity based economies and diseconomies of scope this setting generalizes the usual approach to single facility resource allocation by allowing for the effective capacity of a facility to be a nonlinear function of the number of different items produced or the services delivered by the facility economies or diseconomies of scope are attributable to factors such as production changeover time overall process management requirements and complementary production requirements that vary with the product or service mix we consider the problem setting in which the effective capacity depends on the number of tasks assigned to the facility the resulting model scope generalizes the well known 0 1 knapsack problem we also consider the more general problem gencap in which capacity consumption depends on the specific set of tasks assigned to the facility we define tabu search heuristics as well as exact branch and bound algorithms for scope and gencap on the basis of extensive computational experience the solution procedures are seen to be extremely effective in particular the heuristics consistently obtain high quality solutions to the test problems furthermore the tractability of solving problems to optimality is demonstrated through the solution of scope problems having as many as 500 tasks and gencap problems involving as many as 50 tasks and more than 16 500 nonlinear capacity interactions

the tv break packing problem abstract instead of selling advertisement spots one by one some french satellite channels decided in 2002 to modify their commercial offer in order to sell packages of spots these new general conditions of sale lead to an interesting optimization problem that we named the tv b reak p acking p roblem tvbp we establish its np hardness and study various resolutions approaches including linear programming lp lagrangian relaxation lr constraint programming cp and local search ls finally we propose a generic cp ls hybridization scheme branch and move whose application to the tvbp obtained the best results in our experiments dual upper bounds of the maximal revenue are also computed

a multiplier adjustment method for the generalized assignment problem we describe a branch and bound algorithm for the generalized assignment problem in which bounds are obtained from a lagrangian relaxation with the multipliers set by a heuristic adjustment method the algorithm was tested on a large sample of small random problems and a number of large problems derived from a vehicle routing application computation times were reasonable in all cases and the branch and bound trees generated had nearly two orders of magnitude fewer nodes than for competing algorithms although comparison of running times on different machines is difficult the multiplier adjustment method appears to be about one order of magnitude faster than the best previously existing algorithms for this problem

single machine scheduling with symmetric earliness and tardiness penalties abstract this research focuses on scheduling jobs with varying processing times and distinct due dates on a single machine subject to earliness and tardiness penalties hence this work will find application in a just in time jit production environment the scheduling problem is formulated as a 0 1 linear integer program with three sets of constraints where the objective is to minimize the sum of the absolute deviations between job completion times and their respective due dates the first two sets of constraints are equivalent to the supply and demand constraints of an assignment problem the third set which represents the process time non overlap constraints is relaxed to form the lagrangian dual problem the dual problem is then solved using the subgradient algorithm efficient heuristics have also been developed in this work to yield initial primal feasible solutions and to convert primal infeasible solutions to feasibility the computational results show that the relative deviation from optimality obtained by the subgradient algorithm is less than 3 for problem sizes varying from 10 to 100 jobs