a generalization of polyak s convergence result for subgradient optimization this paper generalizes a practical convergence result first presented by polyak this new result presents a theoretical justification for the step size which has been successfully used in several specialized algorithms which incorporate the subgradient optimization approach

uniqueness and existence of positive solutions for degenerate logistic type elliptic equations on rn abstract in this paper we consider the existence and uniqueness of positive solutions for the degenerate logistic type equation u a x u b x u q x r n n 2 we show that under rather general conditions on a x and b x there exists a unique positive solution for large x our results are an improvement on the corresponding ones in w dong y du unbounded principal eigenfunctions and the logistic equation on r n bull austral math soc 67 2003 413 427 and y du l ma logistic type equations on r n by a squeezing method involving boundary blow up solutions j london math soc 64 2 2001 107 124

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

a cycle based evolutionary algorithm for the fixed charge capacitated multi commodity network design problem this paper presents an evolutionary algorithm for the fixed charge multicommodity network design problem mcndp which concerns routing multiple commodities from origins to destinations by designing a network through selecting arcs with an objective of minimizing the fixed costs of the selected arcs plus the variable costs of the flows on each arc the proposed algorithm evolves a pool of solutions using principles of scatter search interlinked with an iterated local search as an improvement method new cycle based neighborhood operators are presented which enable complete or partial re routing of multiple commodities an efficient perturbation strategy inspired by ejection chains is introduced to perform local compound cycle based moves to explore different parts of the solution space the algorithm also allows infeasible solutions violating arc capacities while performing the ejection cycles and subsequently restores feasibility by systematically applying correction moves computational experiments on benchmark mcndp instances show that the proposed solution method consistently produces high quality solutions in reasonable computational times

resource constrained management of heterogeneous assets with stochastic deterioration we consider a collection of heterogeneous assets which exhibit independent stochastic behavior but are economically interdependent due to resource constraints on management decisions we represent the collection of assets as a network of nonhomogeneous markov decision processes linked by side constraints to facilitate a procedure for obtaining nonrandomized decision policies we express the resource limitations as integrated chance constraints and relax them into the objective function within a lagrangian penalty term we utilize subgradient optimization to establish upper bounds and a greedy randomized lagrangian repair heuristic to obtain feasible solutions we empirically validate the tightness of these a posteriori upper and lower bounds with computational experiments on a pair of applications for pavement maintenance we examine the effect that the reward structure of each constituent mdp has on the maintenance policy in the presence of budget constraints for equipment replacement we consider constraints on two resource types

lagrangian based heuristics for the multicommodity network flow problem with fixed costs on paths we study the multicommodity network flow problem with fixed costs on paths with specific application to the empty freight car distribution process of a rail operator the classification costs for sending a group of cars do not depend on the number of cars in the group as long as the group is kept together as one unit arcs correspond to trains so we have capacity restrictions on arcs but fixed costs on the paths corresponding to routes for groups of cars as solution method we propose a lagrangian based heuristic using dual subgradient search and primal heuristics based on path information of the lagrangian subproblem solutions the method illustrates several ways of exploiting the specific structures of the problem computational tests indicate that the method is able to generate fairly good primal feasible solutions and lower bounds on the optimal objective function value

on the obstructed lagrangian floer theory abstract lagrangian floer homology in a general case has been constructed by fukaya oh ohta and ono where they construct an a algebra or an a bimodule from lagrangian submanifolds they developed obstruction and deformation theories of the lagrangian floer homology theory but for obstructed lagrangian submanifolds the standard lagrangian floer homology cannot be defined we explore several well known homology theories on these a objects which are hochschild and cyclic homology for an a objects and chevalley eilenberg or cyclic chevalley eilenberg homology for their underlying l objects we show that these homology theories are well defined and invariant even in the obstructed cases due to the existence of m 0 the standard homological algebra does not work and we develop analogous homological algebra over novikov fields we provide computations of these homology theories in some cases we show that for an obstructed a algebra with a non trivial primary obstruction chevalley eilenberg floer homology vanishes whose proof is inspired by the comparison with cluster homology theory of lagrangian submanifolds by cornea and lalonde in contrast we also provide an example of an obstructed case whose cyclic floer homology is non vanishing

an effective linear approximation method for separable programming problems among the numerous applications of piecewise linearization methods include data fitting network analysis logistics and statistics in the early 1950s a concave function was found to be able to be linearized by introducing 0 1 variables most textbooks in operations research offer such methods for expressing linear approximations various methods of linearization have also been developed in recent literature nevertheless the transformed linear scheme has a severe shortcoming most standard procedures for linearizing typically involve a large increase in the number of binary variables consequently the gains to be derived from dealing with linear functions are quite likely to be nullified by the increase in the size of the problem conventional methods for linearizing a concave function with m break points require m 1 binary variables however when m becomes large the computation will be very time consuming and may cause a heavy computational burden this study proposes an effective approach in which only log 2 m 1 binary variables are used the proposed method has the following features i it offers more convenient and efficient means of expressing a piecewise linear function ii fewer 0 1 variables are used iii the computational results show that the proposed method is much more efficient and faster than the conventional one especially when the number of break points becomes large

multicommodity network flows the impact of formulation on decomposition this paper investigates the impact of problem formulation on dantzig wolfe decomposition for the multicommodity network flow problem these problems are formulated in three ways origin destination specific destination specific and product specific the path based origin destination specific formulation is equivalent to the tree based destination specific formulation by a simple transformation supersupply and superdemand nodes are appended to the tree based product specific formulation to create an equivalent path based product specific formulation we show that solving the path based problem formulations by decomposition results in substantially fewer master problem iterations and lower cpu times than by using decomposition on the equivalent tree based formulations computational results on a series of multicommodity network flow problems are presented

elements of the lagrangian whitney trick we investigate the possibility of a lagrangian whitney trick a process to remove a pair of intersection points of a self transverse lagrangian immersion by a homotopy through lagrangian immersions there is a model for which a lagrangian whitney trick with compact support works assuming the model satisfies an area capacity condition reduction of more general cases to the model not necessarily fulfilling the area capacity requirement is possible if the given pair of double points admits a suitable symplectic disc and a certain maslov viterbo index is 1 we look into an example to see the actualities of the maslov viterbo index and the area capacity conditions

the unregularized gradient flow of the symplectic action the symplectic action can be defined on the space of smooth paths in a symplectic manifold p which join two lagrangian submanifolds of p to pursue a new approach to the variational theory of this function we define on a subset of the path space the flow whose trajectories are given by the solutions of the cauchy riemann equation with respect to a suitable almost complex structure on p in particular we prove compactness and transversality results for the set of bounded trajectories

the weighted uncapacitated planned maintenance problem complexity and polyhedral properties abstract this paper introduces the weighted uncapacitated planned maintenance problem wupmp based on guaranteed maximum service intervals the wupmp pursues the finding of a maintenance schedule that minimizes the resulting total fixed and variable costs one finding is that significant polyhedral attributes of its solution space are derived among them quasi integrality that allows for applying an integral simplex algorithm is proven moreover we prove strong np hardness and propose an exact solution procedure that is polynomial if the number of considered maintenance activities or the number of periods is constant since at least one restriction applies to most real world applications the algorithm provides practical decision support furthermore the complexity status of various polynomial special cases of the wupmp is resolved

a dual ascent procedure for large scale uncapacitated network design the fixed charge network design problem arises in a variety of problem contexts including transportation communication and production scheduling we develop a family of dual ascent algorithms for this problem this approach generalizes known ascent procedures for solving shortest path plant location steiner network and directed spanning tree problems our computational results for several classes of test problems with up to 500 integer and 1 98 million continuous variables and constraints show that the dual ascent procedure and an associated drop add heuristic generate solutions that in almost all cases are guaranteed to be within 1 to 4 of optimality moreover the procedure requires no more than 150 seconds on an ibm 3083 computer the test problems correspond to dense and sparse networks including some models that arise in freight transport

a new obstruction to embedding lagrangian tori n let 2n be endowed with the canonical symplectic form a i 1 dxi dy i and consider a lagrangian embedding of a compact manifold j l r 2 a a well known result due to g r o m o v g states that the cohomology class of a in h z e 2 l is non zero we identify l with its image in r2 an equivalent assertion is that the liouville form 2 y 7 1 xldy i pulls back to a closed form j 2 on l whose cohomology class does not vanish g r o m o v s proof uses subtle properties of holomorphic maps into c he shows that there is always a holomorphic disk a in c with boundary in l on such a disk a is positive hence sa a soa 2 0 in this paper we shall only consider the case l t and give an elementary proof of gromov s result elementary should be understood in the sense of number theory that is without using holomorphic functions in fact our result is somewhat more precise and can be stated as follows

lagrangian singularities of invariant tori of hamiltonian systems with two degrees of freedom on considere des projections lagrangiennes du 2 tore invariant sous un flot hamiltonien g t

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

positive solutions of yamabe type equations on complete manifolds and applications we study the semilinear equation u a x u b x u 1 on a complete riemannian manifold we determine conditions on the coefficients that guarantee existence and nonexistence of positive solutions a very general uniqueness result is also established our main results are valid without explicit curvature assumptions and appear to be new even in rm

a priori estimates uniqueness and existence of positive solutions of yamabe type equations on complete manifolds abstract we study the asymptotic behaviour of non negative solutions of yamabe type equations on a complete riemannian manifold then we provide a comparison result based on a form of the weak maximum principle at infinity which together with the a priori estimates previously obtained yields uniqueness under very general ricci assumptions the paper ends with an existence result and an application to the non compact yamabe problem

a capacity improvement lower bound for fixed charge network design problems network design problems concern flows over networks in which a fixed charge must be incurred before an arc becomes available for use the uncapacitated multicommodity network design problem is modeled with aggregate and disaggregate forcing constraints forcing constraints ensure logical relationships between the fixed charge related and the flow related decision variables a new lower bound for this problem referred to as the capacity improvement ci bound is presented and an efficient implementation scheme using shortest path and linearized knapsack programs is described a key feature of the ci lower bound is that it is based on the lp relaxation of the aggregate version of the problem a numerical example illustrates that the ci lower bound can converge to the optimal objective function value of the ip formulation

uniqueness of boundary blow up solutions on exterior domain of rn abstract in this paper we consider the existence and uniqueness of positive solutions of the degenerate logistic type elliptic equation u a x u b x u q 1 u x r n d u d where n 2 d r n is a bounded domain with smooth boundary and a x b x are continuous functions on r n with b x 0 b x 0 we show that under rather general conditions on a x and b x for large x there exists a unique positive solution our results improve the corresponding ones in w dong y du unbounded principal eigenfunctions and the logistic equation on r n bull austral math soc 67 2003 413 427 and y du l ma logistic type equations on r n by a squeezing method involving boundary blow up solutions j london math soc 2 64 2001 107 124

a class of convergent primal dual subgradient algorithms for decomposable convex programs in this paper we develop a primal dual subgradient algorithm for preferably decomposable generally nondifferentiable convex programming problems under usual regularity conditions the algorithm employs a lagrangian dual function along with a suitable penalty function which satisfies a specified set of properties in order to generate a sequence of primal and dual iterates for which some subsequence converges to a pair of primal dual optimal solutions several classical types of penalty functions are shown to satisfy these specified properties a geometric convergence rate is established for the algorithm under some additional assumptions this approach has three principal advantages firstly both primal and dual solutions are available which prove to be useful in several contexts secondly the choice of step sizes which plays an important role in subgradient optimization is guided more determinably in this method via primal and dual information thirdly typical subgradient algorithms suffer from the lack of an appropriate stopping criterion and so the quality of the solution obtained after a finite number of steps is usually unknown in contrast by using the primal dual gap the proposed algorithm possesses a natural stopping criterion

complete conformal metrics with negative scalar curvature in compact riemannian manifolds soit m g une variete de riemann compacte de dimension n 3 et soit une sous variete lisse fermee de dimension d alors il y a une metrique conforme complete sur m m a courbure scalaire negative constante si et seulement si d n 2 2

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

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

variable disaggregation in network flow problems with piecewise linear costs we study mixed integer programming formulations based upon variable disaggregation for generic multicommodity network flow problems with nonconvex piecewise linear costs a problem class that arises frequently in many application domains in telecommunications transportation and logistics we present several structural results for these formulations and we analyze the results of extensive experiments on a large set of instances with various characteristics in particular we show that the linear programming relaxation of an extended disaggregated model approximates the objective function by its lower convex envelope in the space of commodity flows together the theoretical and computational results allow us to suggest which formulation might be the most appropriate depending on the characteristics of the problem instances

the maslov class of the lagrange surfaces and gromov s pseudo holomorphic curves for an immersed lagrange submanifold w c t x one can define a nonnegative integer topologic invariant m w such that the image of h1 w z under the maslov class is equal to m w z in this paper the value of m w is calculated for the case of a two dimensional oriented manifold x with the universal cover homeomorphic to r2 and an embedded lagrange torus w it is proved that if x t2 and w is homologic to the zero section then m w 0 in all the other cases m w 2 the last result is true also for a wide class of oriented properly embedded lagrange surfaces in t r the proof is based on the gromov s theory of pseudo holomorphic curves some applications to the hamiltonian mechanics are mentioned

prescribing scalar curvatures on the conformal classes of complete metrics with negative curvature let m n g be a complete noncompact riemannian manifold with the curvature bounded between two negative constants given a function k on m n in terms of the behaviors of k at infinite we give a fairly complete answer to when the k can be the scalar curvature function of a complete metric g 1 which is conformal to g

railroad blocking a network design application in this study we formulate the railroad blocking problems as a network design problem with maximum degree and flow constraints on the nodes and propose a heuristic lagrangian relaxation approach to solve the problem the newapproach decomposes the complicated mixed integer programming problem into two simple subproblems so that the storage requirement and computational effort are greatly reduced a set of inequalities are added to one subproblem to tighten the lower bounds and facilitate generating feasible solutions subgradient optimization is used to solve the lagrangian dual an advanced dual feasible solution is generated to speed up the convergence of the subgradient method the model is tested on blocking problems from a major railroad and the results show that the blocking plans generated have the potential to reduce the railroad s operating costs by millions of dollars annually

punctured holomorphic curves and lagrangian embeddings we use a neck stretching argument for holomorphic curves to produce symplectic disks of small area and maslov class with boundary on lagrangian submanifolds of nonpositive curvature applications include the proof of audin s conjecture on the maslov class of lagrangian tori in linear symplectic space the construction of a new symplectic capacity obstructions to lagrangian embeddings into uniruled symplectic manifolds a quantitative version of arnold s chord conjecture and estimates on the size of weinstein neighbourhoods the main technical ingredient is transversality for the relevant moduli spaces of punctured holomorphic curves with tangency conditions

scalar curvature and conformal deformation of hyperbolic space abstract let h m gh m 3 be the m dimensional hyperbolic space with its riemannian metric gh of sectional curvature 1 and let k be a smooth function on h m in the first part of this article we establish sufficient conditions for k to be respectively not to be the scalar curvature of some complete metric u4 m 2 gh pointwise conformal to gh in the second part we prove results for the two dimensional case singularity and uniquens questions

uniqueness of boundary blow up solutions on unbounded domain of rn abstract in this paper we consider the existence of positive solutions of the degenerate logistic type elliptic equation u a x u b x f u x r n d u d where n 2 d r n is a bounded smooth domain and a x b x are continuous functions on r n with b x 0 b x 0 especially b x 0 on d and f u c 0 we show that under rather general conditions on a x and b x for large x and f u behaves like u q where q 1 without the behavior of b x near the boundary d it will be shown that there exists a unique positive solution our results improve the corresponding ones in w dong pang uniqueness of boundary blow up solutions on exterior domain of r n j math anal appl 330 2007 654 664 and y du l ma logistic type equations on r n by a squeezing method involving boundary blow up solutions j london math soc 2 64 2001 107 124

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

a lagrangian heuristic based branch and bound approach for the capacitated network design problem the capacitated network design problem is a multicommodity minimal cost network flow problem with fixed charges on the arcs and is well known to be np hard the problem type is very common in the context of transportation networks telecommunication networks etc in this paper we propose an efficient method for this problem based on a lagrangian heuristic within a branch and bound framework the lagrangian heuristic uses a lagrangian relaxation to obtain easily solved subproblems and solves the lagrangian dual by subgradient optimization it also includes techniques for finding primal feasible solutions the lagrangian heuristic is then embedded into a branch and bound scheme that yields further primal improvements special penalty tests and cutting criteria are developed the branch and bound scheme can either be an exact method that guarantees the optimal solution of the problem or be a quicker heuristic the method has been tested on networks of various structures and sizes computational comparisons between this method and a state of the art mixed integer code are presented the method is found to be capable of generating good feasible solutions to large scale problems within reasonable time and data storage limits

positive solutions of yamabe type equations with function coefficients on graphs abstract in this paper we study the yamabe type semi linear equation u a x u b x u 1 on both finite and locally finite graphs we determine certain conditions on the coefficients of the equation that guarantee existence of positive solutions in both cases some uniqueness results are also established

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

a note on a dual ascent approach to the fixed charge capacitated network design problem abstract we show by a counterexample that the dual ascent procedure proposed by herrmann ioannou minis and proth in a 1996 issue of the european journal of operational research is incorrect in the sense that it does not generate a valid lower bound to the optimal value of fixed charge capacitated network design problems we provide a correct dual ascent procedure based on the same ideas and we give an interpretation of it in terms of a simple lagrangean relaxation although correct this procedure is not effective as in general it provides a less tighter bound than the linear programming relaxation

a lagrangean heuristic for the facility location problem with staircase costs in this paper we develop and investigate heuristic solution methods for the capacitated facility location problem with staircase shaped production cost functions these problems are linear structured mixed integer programming problems with a large proportion of integer variables lagrangean relaxation and subgradient optimization is used to obtain a dual solution and as a base for an efficient primal heuristic in the lagrangean scheme we use a convex piecewise linearization of the staircase shaped cost functions to get a good initial upper bound as well as an initial dual solution based on the subproblem solution we construct a transportation problem which yields primal feasible solutions for comparison we have generalized the well known add heuristic for the capacitated plant location model to the staircase cost facility location problem to improve this heuristic we have combined it with certain priority rules computational results are given indicating that the lagrangean heuristic is quite promising

examples of symplectic structures in this paper we construct symplectic forms tilde omega k k geqq 0 on a compact manifold tilde y which have the same homotopy theoretic invariants but which are not diffeomorphic

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

bundle methods for sum functions with easy components applications to multicommodity network design we propose a version of the bundle scheme for convex nondifferentiable optimization suitable for the case of a sum function where some of the components are easy that is they are lagrangian functions of explicitly known compact convex programs this corresponds to a stabilized partial dantzig wolfe decomposition where suitably modified representations of the easy convex subproblems are inserted in the master problem as an alternative to iteratively inner approximating them by extreme points thus providing the algorithm with exact information about a part of the dual objective function the resulting master problems are potentially larger and less well structured than the standard ones ruling out the available specialized techniques and requiring the use of general purpose solvers for their solution this strongly favors piecewise linear stabilizing terms as opposed to the more usual quadratic ones which in turn may have an adverse effect on the convergence speed of the algorithm so that the overall performance may depend on appropriate tuning of all these aspects yet very good computational results are obtained in at least one relevant application the computation of tight lower bounds for fixed charge multicommodity min cost flow problems

hamiltonian collapsing of irrational lagrangian submanifolds with small first betti number among the main symplectic invariants of a closed lagrange submanifoldl of the cotangent of n is the tubular radiusr l defined as the smallest tube d r n 1 of n t n in whichl can be pushed by an hamiltonian diffeotopy of n we show here using pseudo holomorphic techniques that such a submanifold cannot collapse if the first betti number ofl is smaller than 3 and if the maslov class ofl does not vanish in other words r l is then strictly positive and one can actually give an explicit lower bound in terms of the liouville and maslov classes ofl

unknottedness of real lagrangian tori in s 2 times s 2 we prove the hamiltonian unknottedness of real lagrangian tori in the monotone s 2 times s 2 namely any real lagrangian torus in s 2 times s 2 is hamiltonian isotopic to the clifford torus the proof is based on a neck stretching argument gromov s foliation theorem and the cieliebak schwingenheuer criterion

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