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

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

balanced line bundles on fano varieties a conjecture of batyrev and manin relates arithmetic properties of varieties with ample anticanonical class to geometric invariants in particular counting functions defined by metrized ample line bundles and the corresponding asymptotics of rational points of bounded height are interpreted in terms of cones of effective divisors and certain thresholds with respect to these cones this framework leads to the notion of balanced line bundles whose counting functions conjecturally capture generic distributions of rational points we investigate balanced line bundles in the context of the minimal model program with special regard to the classification of fano threefolds

proof theory for linear lattices abstract we make a study of lattices representable by commuting equivalence relations which we call linear lattices we develop a proof theory for implications valid in linear lattices which differs from classical proof theories in that its deductions transform representative graphs rather than well formed formulas using graphtheoretic arguments we establish a duality theorem and a normal form theorem for this proof theory

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

a perturbation based heuristic for the capacitated multisource weber problem this paper proposes a perturbation based heuristic for the capacitated multisource weber problem this procedure is based on an effective use of borderline customers several implementations are considered and the two most appropriate are then computationally enhanced by using a reduced neighbourhood when solving the transportation problem computational results are presented using data sets from the literature originally used for the uncapacitated case with encouraging results

an algorithm for solving large capacitated warehouse location problems abstract in this paper we present a lower bound for the capacitated warehouse location problem based upon lagrangean relaxation of a mixed integer formulation of the problem feasible solution exclusion constraints are used together with problem reduction tests derived from both the original problem and the lagrangean relaxation by incorporating the lower bound and the reduction tests into a tree search procedure we are able to solve problems involving up to 500 potential warehouse locations and 1000 customers

vector bundles of rank 2 and linear systems on algebraic surfaces let s be a smooth complex algebraic surface and let l be a divisor on s it is well known that the linear system il ksi where ks is the canonical divisor plays an important role in understanding the geometry of s several authors see for example 8 11 studied various properties of this system our paper treats the same subject but the point of view is different from the works mentioned above the philosophy underlying our approach is almost classical points on s more generally effective 0 cycles in special position with respect to il ksi see definition below contain information about the geometry of s this point of view was recently revived in 5 also 10 in fact this paper contains the technique of going from the effective 0 cycle z in special position with respect to il ksi to the geometry of s for the sake of completeness we give a definition and state a theorem which can be found in 5 10

benders decomposition without separability a computational study for capacitated facility location problems abstract benders is one of the most famous decomposition tools for mathematical programming and it is the method of choice e g in mixed integer stochastic programming its hallmark is the capability of decomposing certain types of models into smaller subproblems each of which can be solved individually to produce local information notably cutting planes to be exploited by a centralized master problem as its name suggests the power of the technique comes essentially from the decomposition effect i e the separability of the problem into a master problem and several smaller subproblems in this paper we address the question of whether the benders approach can be useful even without separability of the subproblem i e when its application yields a single subproblem of the same size as the original problem in particular we focus on the capacitated facility location problem in two variants the classical linear case and a congested case where the objective function contains convex but non separable quadratic terms we show how to embed the benders approach within a modern branch and cut mixed integer programming solver addressing explicitly all the ingredients that are instrumental for its success in particular we discuss some computational aspects that are related to the negative effects derived from the lack of separability extensive computational results on various classes of instances from the literature are reported with a comparison with the state of the art exact and heuristic algorithms the outcome is that a clever but simple implementation of the benders approach can be very effective even without separability as its performance is comparable and sometimes even better than that of the most effective and sophisticated algorithms proposed in the previous literature

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

arguesian identities in the congruence variety of abelian groups abstract a class of identities in the grassmann cayley algebra was found by m j hawrylycz 1994 geometric identities in invariant theory ph d thesis massachusetts institute of technology which yields a large number of geometric theorems on the incidence of subspaces of projective spaces in a previous paper we established a link between such identities in the grassmann cayley algebra and a family of inequalities in the class of linear lattices i e the lattices of commuting equivalence relations we proved that a subclass of identities found by hawrylycz namely the arguesian identities of order 2 can be systematically translated into inequalities holding in linear lattices however it is not known whether the arguesian identities of higher orders have such latticial extensions in this paper we give an affirmative answer to the above question in the congruence variety of abelian groups we prove that every arguesian identity regardless of the order can be systematically translated into a lattice inequality holding in the congruence variety of abelian groups in particular such a lattice inequality holds in the lattices of subspaces of vector spaces which are characteristic free and independent of dimensions as a consequence many classical theorems of projective geometry including desargues bricard fontene and their higher dimensional generalizations can be extended to lattice inequalities in the general projective spaces with the variables representing subspaces of arbitrary dimensions

facility location with increasing production costs abstract facility location models determine the set of locations on a network that minimize the sum of the costs of investment production and distribution to meet a known set of demands in this paper we introduce a new type of facility location model which combines aspects of the well studied simple uncapacitated and capacitated facility location problems its distinctive feature is that unit production costs are modeled as increasing with scale of output such cost functions have practical value in handling cases in which capacity can be stretched by incurring some additional cost e g by adding workers indeed it is shown that average total costs are minimized at a point where average production costs are rising four different formulations for this problem are proposed using linear programming plus branch and bound as a solution method the four formulations are tested and compared on a set of 216 problems with randomly generated data

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

arguesian identities in invariant theory abstract having been motivated by an example of doubilet rota and stein stud appl math 56 1976 185 216 we present a technique for constructing geometric identities in a grassmann cayley algebra each identity represents a projective invariant closely related to the theorem of desargues in the plane and its generalizations to higher dimensional projective space the construction employs certain combinatorial properties of matchings in bipartite graphs we also prove a dimension independence result for arguesian identities thereby connecting the identities with lattice theory

on the exact solution of large scale simple plant location problems abstract the simple plant location problem deals with the selection of facility sites from a set of possible locations with the intention of serving a given set of customers from the chosen locations the objective is to minimize the sum of the fixed charges for establishing the facilities and the transportation costs corresponding to the supply of the demands we show how to modify a primal dual version of erlenkotter s exact algorithm to get an improved procedure computational experience with large scale problems indicates that compared with erlenkotter s code an implementation of the modified algorithm is faster by more than one order of magnitude for real life problems with 1500 potential facility sites the increase of computational speed may even exceed the factor of 100

capacitated facility location network design problems abstract we introduce a combined facility location network design problem in which facilities have constraining capacities on the amount of demand they can serve this model has a number of applications in regional planning distribution telecommunications energy management and other areas our model includes the classical capacitated facility location problem cflp on a network as a special case we present a mixed integer programming formulation of the problem and several classes of valid inequalities are derived to strengthen its lp relaxation computational experience with problems with up to 40 nodes and 160 candidate links is reported and a sensitivity analysis provides insight into the behavior of the model in response to changes in key problem parameters

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

arguesian identities in linear lattices abstract a class of identities in the grassmann cayley algebra which yields a large number of geometric theorems on the incidence of subspaces of projective spaces was found by hawrylycz geometric identities in invariant theory ph d thesis massachusetts institute of technology 1994 in this paper we establish a link between such identities in the grassmann cayley algebra and a class of inequalities in the class of linear lattices i e the lattices of commuting equivalence relations we prove that a subclass of identities found by hawrylycz namely the arguesian identities of order 2 can be systematically translated into inequalities holding in linear lattices as a consequence we obtain a family of geometric theorems on the incidence of subspaces that are characteristic free and independent of dimensions

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

lagrangean heuristics for location problems abstract in this paper we present a framework for developing lagrangean heuristics heuristics based upon lagrangean relaxation and subgradient optimisation with respect to location problems computational results are given for four different location problems p median uncapacitated warehouse location capacitated warehouse location and capacitated warehouse location with single source constraints these results indicate that the framework presented in this paper is robust i e it gives good quality solutions for each of these different location problems

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

on the geometry of thin exceptional sets in manin s conjecture manin s conjecture predicts the rate of growth of rational points of a bounded height after removing those lying on an exceptional set we study whether the exceptional set in manin s conjecture is a thin set using the minimal model program and boundedness of log fano varieties

an algorithm for solving dynamic capacitated plant location problems with discrete expansion sizes in the dynamic capacitated plant location problem dcplp the task is to find a time schedule and sizes for installing facilities at plant locations to minimize the discounted cost of capital expenditures over the planning horizon the costs include setup costs for establishing facilities volume dependent operational costs and transportation costs for distributing demand from facilities to customers we consider a class of the dcplp in which the available facilities have finite capacities and the number of facility types is relatively small so that the expansion sizes cannot be modeled by continuous variables the dcplp is formulated as a combinatorial optimization problem that allows consideration of more than one fatality type and finds the optimum mix of facilities in each location we describe an optimization algorithm for solving the dcplp based on the lagrangian relaxation technique algorithms for converting infeasible optimal solutions of the lagrangian to a feasible solution of the dcplp are presented the procedure has been tested on both randomly generated and real life based problems computational results indicate that the algorithm produces solutions within 3 of the lower bounds for a wide range of input data

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

lagrangean dual ascent algorithms for computing bounds in capacitated plant location problems abstract given a mixed integer programming problem if one dualizes all linking constraints and the constraints containing only integer variables the optimum of the lagrangean dual is equal to the optimum of the lp relaxation if one strengthens this dual by adding valid inequalities such that the problem still decomposes into a continuous and a pure integer problem and is relatively easy to solve due to some special structure one may be able to devise dual ascent steps for the lagrangean dual this is investigated for the capacitated plant location problem whose strengthened lagrangean relaxed problem decomposes into a transportation problem and a knapsack problem computational experiments are reported based on some very simple ascent steps

a branch and price algorithm for the capacitated facility location problem the capacitated facility location problem cflp is a well known combinatorial optimization problem with applications in distribution and production planning it consists in selecting plant sites from a finite set of potential sites and in allocating customer demands in such a way as to minimize operating and transportation costs a number of solution approaches based on lagrangean relaxation and subgradient optimization has been proposed for this problem subgradient optimization does not provide a primal fractional optimal solution to the corresponding master problem however in order to compute optimal solutions to large or difficult problem instances by means of a branch and bound procedure information about such a primal fractional solution can be advantageous in this paper a stabilized column generation method is therefore employed in order to solve a corresponding master problem exactly the column generation procedure is then employed within a branch and price algorithm for computing optimal solutions to the cflp computational results are reported for a set of larger and difficult problem instances

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

the capacitated plant location problem this paper provides a review of the various solution methods for the capacitated plant location problem heuristic and exact procedures that have appeared in the literature are covered the review also examines two innovative concepts the lagrangian heuristic and variable splitting as proposed for the capacitated plant location problem

the coordinatization of arguesian lattices we show that the auxiliary planar ternary ring of an n frame in an arguesian lattice n 3 is indeed an associative ring with unit the addition of two weak necessary conditions allows us to coordinatize a hyperplane of this n frame this generalizes the classical work of von neumann baer inaba j6nsson and j6nsson monk

a lagrangean relax and cut approach for the two stage capacitated facility location problem abstract the two stage capacitated facility location problem is to find the optimal locations of depots to serve customers with a given demand the optimal assignment of customers to depots and the optimal product flow from plants to depots in order to solve this problem a lagrangean heuristic is proposed which is based on the relaxation of the capacity constraints the resulting lagrangean subproblem is an uncapacitated facility location problem with an aggregate capacity constraint and can be solved efficiently by branch and bound methods the best bound available from this relaxation is computed by means of a weighted dantzig wolfe decomposition approach feasible solutions are constructed from those of the lagrangean subproblems by applying simple reassignment procedures furthermore the relaxation is strengthened by adding and dualizing valid inequalities which are violated by the fractional primal solution of the dual master program computational results and comparisons with some other bounds and heuristics for this problem are presented

rational points of bounded height on fano varieties a prime pe7z let v be an algebraic variety defined over f and li a metrized line bundle on v i e a system l consisting of a line bundle l and a family of banach v adic metrics on l f for all v s satisfying well known conditions cf 1 chap vi then we can define a height function hl x f i r by the formula h x is x 1 where s is a local section of l non vanishing

approximate solutions to large scale capacitated facility location problems this article presents a heuristic algorithm for solving large scale capacitated facility location problems the algorithm consists of solving a single source facility location problem and then solving a minimum cost network problem the single source problem is solved by applying an iterative procedure in which lower bounds on the optimal objective value are generated through lagrangian relaxation techniques upper bounds on the optimal objective value are obtained by modifying the lagrangian solution to restore feasibility the iterative procedure is terminated when the difference between the two bounds is within a prespecified tolerance further improvement of the objective value is attained by subjecting the single source solution to a network optimization code the heuristic algorithm has been tested on over 300 problems derived in one form or another from the kuehn hamburger test problems the testing has also been undertaken on the problems of van roy which were generated using netgen computational results indicate that our approach can be effective in producing near optimal solutions for problems with as many as 1000 customers and 100 potential facility locations most problems converged within 3 percent of the optimal value requiring only a modest amount of computation on an ibm 3090

neron severi groups under specialization andre used hodge theoretic methods to show that in a smooth proper family x b of varieties over an algebraically closed field k of characteristic 0 there exists a closed fiber having the same picard number as the geometric generic fiber even if k is countable we give a completely different approach to andre s theorem which also proves the following refinement in a family of varieties with good reduction at p the locus on the base where the picard number jumps is p adically nowhere dense our proof uses the p adic lefschetz 1 1 theorem of berthelot and ogus combined with an analysis of p adic power series we prove analogous statements for cycles of higher codimension assuming a p adic analogue of the variational hodge conjecture and prove that this analogue implies the usual variational hodge conjecture applications are given to abelian schemes and to proper families of projective varieties

accelerating benders decomposition algorithmic enhancement and model selection criteria this paper proposes methodology for improving the performance of benders decomposition when applied to mixed integer programs it introduces a new technique for accelerating the convergence of the algorithm and theory for distinguishing good model formulations of a problem that has distinct but equivalent mixed integer programming representations the acceleration technique is based upon selecting judiciously from the alternate optima of the benders subproblem to generate strong or pareto optimal cuts this methodology also applies to a much broader class of optimization algorithms that includes dantzig wolfe decomposition for linear and nonlinear programs and related cutting plane type algorithms that arise in resource directive and price decomposition when specialized to network location problems this cut generation technique leads to very efficient algorithms that exploit the underlying structure of these models in discussing the proper formulation of mixed integer programs we suggest criteri

computational results from a new lagrangean relaxation algorithm for the capacitated plant location problem abstract in this paper we present an algorithm for the capacitated plant location problem based on a reformulation obtained by the introduction of auxiliary variables by relaxation of the coupling constraints two separate subproblems are obtained this is the basis for the so called variable splitting approach we present a complete algorithmic procedure including solution procedures for subproblems heuristics for the generation of feasible solutions and suggestion for branching rules we also present numerical results for capacitated plant location problems of the size x n where m 10 20 n 20 50

modelling a fertiliser distribution system abstract the fertiliser production distribution problem is formulated as a mixed zero one integer linear programming problem the distinguishing features of the problem are described benders decomposition is used as a solution procedure two lagrangean relaxation procedures are developed to solve the primal problem one of them converging at a much faster rate the computational performance and the implementation details of the solution procedure are discussed

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

a comparison of heuristics and relaxations for the capacitated plant location problem abstract approaches proposed in the literature for the capacitated plant location problem are compared the comparison is based on new theoretical and computational results the main emphasis is on relaxations in particular dominance relations among the various relaxations found in the literature are identified in the computational study the relaxations are compared as a function of various characteristics of the test problems several of these relaxations can be used to generate heuristic feasible solutions that are better than the classical greedy or interchange heuristics both in computing time and in the quality of the solutions found

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

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