In column generation schemes, particularly those proposed for set partitioning type problems, dynamic programming algorithms are applied to solve the respective pricing subproblem. In addition to traditional dominance criteria for state space reduction, we develop a simple generic lower bounding criterion based on the dual optimal solution of the restricted master problem. Key words: Dynamic programming; Column generation; Linear programming Column generation is a prominent---and often the solely applicable---method to cope with linear programming problems with a colossal number of variables. In recent years we have been witnessing the optimal solution of truly large problems, but still the need for faster algorithms remains, especially in the various practical application areas. The pricing subproblem usually constitutes the crucial part of a column generation scheme, not uncommonly for the reason of being a hard combinatorial optimization problem by itself. Although a dynamic program..

Marco E. Lübbecke

In this note, we aim at reducing the state space of dynamic programming algorithms used as column generators in solving the linear programming relaxation of set partitioning problems arising from practical applications. We propose a simple generic lower bounding criterion based on the respective dual optimal solution of the restricted master program

Lübbecke, Marco E.

DepositOnce

FAKULTÄT IIMATHEMATIK UNDNATURWISSENSCHAFTENInstitut für MathematikDUAL VARIABLE BASED FATHOMING INDYNAMIC PROGRAMS FOR COLUMNGENERATIONbyMARCO E. LÜBBECKENo. 2003/42Dual Variable Based Fathoming in Dynamic Programs forColumn GenerationMarco E. LübbeckeTechnische Universität BerlinInstitut für Mathematik, Sekr. MA 6-1Straße des 17. Juni 136, D-10623 Berlin, GermanyE-mail: m.luebbecke@math.tu-berlin.deAugust 19, 2003AbstractIn this note, we aim at reducing the state space of dynamic programming algorithms used as col-umn generators in solving the linear programming relaxation of set partitioning problems arisingfrom practical applications. We propose a simple generic lower bounding criterion based on therespective dual optimal solution of the restricted master program.Key words: Dynamic programming, column generation, state space reduction1 MotivationColumn generation is a well-established method to solve large-scale (integer) linear programs, seee.g., [3, 5, 8]. In recent years we have seen the optimal solution of problems with millions of variablesin various practical application areas. Often, a natural modeling approach is a formulation as a setpartitioning style program. Classical examples are airline crew pairing [2] and vehicle routing [5].Here, the pricing problem, i.e., the task to compute a favorable column to enter the basis or to provethat none such exists, commonly constitutes an N P -complete combinatorial optimization problem;dynamic programming algorithms have been proposed for exact solutions, see e.g., [2, 4, 5, 6, 9].This note provides a simple means to reduce the state space of these algorithms, rediscovering thefathoming technique in dynamic programming [1, 10, 12].2 Dynamic Programs as Column GeneratorsStressing their practical relevance [3], we confine ourselves to linear programming (LP) relaxationsof set partitioning problems. That is, we minimize cT x subject to Ax = 1, x ≥ 0, where A ∈ {0,1}m×n,and n is typically very large. We assume c ≥ 0. The columns of A encode a set S of admissible subsetsof an m-set S via ai j = 1 iff i ∈ j ∈ S ⊆ 2S , i = 1, . . . ,m. Usually n = |S|  2m, and set membershipin S is defined by a set C of problem specific constraints. An example: A set S of m customershas to be visited, each exactly once, by a fleet of vehicles. Rules C ensure e.g., that time windows,vehicle capacities, and precedence relations among customers are respected. This results in the set Sof (incidence vectors of) feasible vehicle routes, and associated costs c j , j ∈ S.1The generic column generation scheme starts off with a small subset of S, the associated variablesconstitute the initial restricted master program (RMP). Adjoining negative reduced cost columns,and re-optimization of the RMP alternately proceed. Modern implementations usually use problemspecific heuristics for solving the pricing problem, cf. [3, 4], but this is not followed up here. Instead,we concentrate on an exact algorithm which has to be executed at some point in time, at least whenone wishes to prove optimality of the RMP.Let x and u denote the primal and dual optimal solutions, respectively, associated with the currentRMP. Then, the pricing problem amounts to determiningz := min{c j −∑i∈ j ui | j ∈ S}. (1)If z ≥ 0, no reduced cost coefficient is negative, and x, embedded in Rn, optimally solves the originalLP as well. Otherwise, when the minimum in (1) is attained for index j ′ ∈ S, column a j′ is adjoined tothe RMP. Implicit enumeration of S can be accomplished by a dynamic program, which starts fromR = /0, and successively appends elements from S , the most recent of which is denoted by k. Thecurrent stage of the algorithm is represented by state (R,k) with R ⊆ S , k ∈ R. A transition from(R,k) to (R′,k′) is feasible iff (R′,k′) is compliant to a set of constraints C , which usually depend on(R,k). In the simplest case, each state is associated with its reduced cost c(R,k). Note the assumptionof additive costs along transitions. For every R generated during the process there exists a supersetR ⊇ R with R ∈ S. These so-called final states—which represent the feasible solutions for the pricingproblem—are finally considered by the dynamic program, at least implicitly. The minimum reducedcost feasible solution among these determines z in (1).3 Dual Variable Based FathomingBesides classical dominance relations among states, lower bounds can be exploited to reduce the statespace of dynamic programs [1, 10, 12]. Consider a particular state (R,k) during the assumed dynamicprogramming pricing algorithm. Denote by z the cost of a currently cheapest final state—referred toas the incumbent, initialized with a known upper bound, possibly infinity. Candidates for extension ofR come from (but because of C need not be identical to) S \R. Now only final states having negativecost are interesting. Therefore, if LB(R,k) ≥ min{0,z} holds for a lower bound LB(R,k) on the bestpossible reduced cost coefficient obtainable by (subsequent) transitions of state (R,k), we prune thesearch, which we call fathoming of the current state. In the following we present such a simple bound.The idea is to relax constraints in C imposed on the extension of R. An immediate choice wouldbe a total relaxation, i.e., disregarding C altogether. Denote by S+ = {i ∈ S | ui > 0}. No state derivedfrom (R,k) can have cost smaller thanLB(R,k) = c(R,k)−∑i∈S+\R ui , (2)which—although straightforward—has not been generally stated, to the best of our knowledge. Ingeneral, this lower bound is possibly weak, but can be refined as follows. Not relaxing all C , andutilizing some structural information on elements in S one might be able to restrict S+ to a moremeaningful set to be substituted in (2). For instance, upper bounds on max j∈S | j| can be applied tostrengthen LB(R,k). In practical applications, such information is often available, e.g., the maximalnumber of customers on a vehicle tour. Another likely situation is to have i1, i2 ∈ S+ such that thereis no j′ ∈ S with R∪ {i1, i2} ⊆ j′, that is, two incompatible elements, implied by C , e.g., due to2conflicting time windows. We would eliminate from S+ one which attains min{ui1 ,ui2}. This can begeneralized to more than two elements [7], still, the tradeoff between such efforts and computationalgains should be kept in mind. If known, the minimal cost incurred by transitions from the current to afinal state can be added to LB(R,k), e.g., the cost for a vehicle to return to a depot.Suppose we fathom a state (R,k) when γ · LB(R,k) ≥ min{0,z} with 0 ≤ γ ≤ 1. We alreadydiscussed the case γ = 1. For γ = 0, the inequality becomes redundant; all states would be eliminated.Assume that LB(R,k) ≤ 0, for otherwise the multiplication by γ would have no effect on the criterion.Then, for 0 < γ < 1, we have γ ·LB(R,k) > LB(R,k); more states than before are fathomed. In otherwords, an incumbent possibly with z > z will be considered optimal. Note, that (z− z)/z < 1 alwaysholds for any incumbent with 0 > z ≥ z. That is, as long as some negative reduced cost column iscomputed, we have a trivial upper bound on the relative error incurred for any pricing heuristic. Mostnotably, in this case the heuristic just described allows for a better approximation guarantee.Lemma 1 Let z ≤ z < 0. When LB(R,k) is replaced by γ ·LB(R,k) with 0 < γ ≤ 1, then (z− z)/z ≤1− γ.Proof. The modified lower bound is more effective only if LB(R,k) < z ≤ γ ·LB(R,k) for some state(R,k). When such an additional elimination takes place, we obtain for the incumbentz− z ≤ γ ·LB(R,k)− z ⇐⇒z− zz≥ γ ·LB(R,k)z−1 ≥ γ ·1−1 .The last inequality follows from LB(R,k) ≤ z < 0, which holds by definition of the lower bound.Constraining z to be non-positive immediately yields the claim. The case z ≥ 0 is uninteresting; the assumed dynamic program truly returns that no negativereduced cost columns exist.Let us finally remark that a variant of (2) is suited for preprocessing the data prior to solving apricing problem. Again, we assume non-negative cost coefficients. If u p < 0 and ∑i∈S+ ui + up ≤ 0,then the element in S corresponding to p cannot be promisingly incorporated in any feasible solutionto the pricing problem, and therefore can be discarded throughout the calculation.4 Concluding RemarksOur fathoming criterion also applies in the common presence of convexity constraints in the LP. Whenthe cost structure is such that j1 ⊆ j2 ⊆ S ⇒ c j1 ≤ c j2 , considering the set covering relaxation Ax ≥ 1is no loss of optimality. The assumption A ∈ {0,1}m×n may be temporarily relaxed for computationalease. The non-binary columns encode the repeated appearance of elements in a set S. In this case,(2) is no longer a lower bound. Independently of our work a similar bound has been proposed [11],however, in a pricing algorithm of branch-and-bound style, where the use of bounds is essential to themethod.Traditionally used dominance rules in dynamic programming require an efficiently manageableglobal overview of states, and only allow for discarding states which are already generated. Opposedto that, the benefit and advantage of our criterion is its ability to be checked locally in the sense thatonly knowledge about the current state is necessary. Future state transitions may be avoided in thefirst place. Computational experiments with (2), conducted in the context of a locomotive schedulingproblem at in-plant railroads [9], indicate considerable state space reductions of more than 90%.3AcknowledgementThis research was funded by the German Federal Ministry of Education and Research (BMBF) undergrants no. 03-ZI7BR2-1 and 03-ZIM2BS. It was done while the author was with the Department ofMathematical Optimization, Braunschweig University of Technology. I acknowledge the commentsof two anonymous referees which lead to a more readable note.References[1] O.G. Alekseev and I.F. Volodos’. Combined use of dynamic programming and branch-and-bound methods in discrete-programming problems. Automat. Remote Control, 37:557–565,1976.[2] R. Anbil, J.J. Forrest, and W.R. Pulleyblank. Column generation and the airline crew pairingproblem. In Proceedings of the International Congress of Mathematicians Berlin, Extra VolumeICM 1998 of Doc. Math. J. DMV, pages III 677–686, August 1998.[3] C. Barnhart, E.L. Johnson, G.L. Nemhauser, M.W.P. Savelsbergh, and P.H. Vance. Branch-and-price: Column generation for solving huge integer programs. Oper. Res., 46(3):316–329, 1998.[4] G. Desaulniers, J. Desrosiers, and M.M. Solomon. Accelerating strategies in column generationmethods for vehicle routing and crew scheduling problems. In C.C. Ribeiro and P. Hansen,editors, Essays and Surveys in Metaheuristics, pages 309–324, Boston, 2001. Kluwer.[5] J. Desrosiers, Y. Dumas, M.M. Solomon, and F. Soumis. Time constrained routing and schedul-ing. In M.O. Ball, T.L. Magnanti, C.L. Monma, and G.L. Nemhauser, editors, Network Rout-ing, volume 8 of Handbooks in Operations Research and Management Science, pages 35–139.North-Holland, Amsterdam, 1995.[6] J. Desrosiers, P. Pelletier, and F. Soumis. Plus court chemin avec contraintes d’horaires. RAIRORech. Opér., 17(4):357–377, 1983. In French.[7] M.E. Lübbecke. Engine Scheduling by Column Generation. PhD thesis, Braunschweig Univer-sity of Technology, Cuvillier Verlag, Göttingen, 2001.[8] M.E. Lübbecke and J. Desrosiers. Selected topics in column generation. Les Cahiers du GERADG-2002-64, HEC Montréal, Canada, 2002. Submitted.[9] M.E. Lübbecke and U.T. Zimmermann. Engine routing and scheduling at industrial in-plantrailroads. Transportation Sci., 37(2):183–197, 2003.[10] R.E. Marsten and T.L. Morin. A hybrid approach to discrete mathematical programming. Math.Programming, 14:21–40, 1978.[11] A. Mehrotra and M.A. Trick. Cliques and clustering: A combinatorial approach. Oper. Res.Lett., 22(1):1–12, 1998.[12] T.L. Morin and R.E. Marsten. Branch-and-bound strategies for dynamic programming. Oper.Res., 24(4):611–627, 1976.4Reports from the group“Combinatorial Optimization and Graph Algorithms”of the Department of Mathematics, TU Berlin2003/43 Michael R. Bussieck, Thomas Lindner, and Marco E. Lübbecke: A Fast Algorithm for Near CostOptimal Line Plans2003/42 Marco E. Lübbecke: Dual Variable Based Fathoming in Dynamic Programs for Column Generation2003/37 Sándor P. Fekete, Marco E. Lübbecke, and Henk Meijer: Minimizing the Stabbing Number of Match-ings, Trees, and Triangulations2003/25 Daniel Villeneuve, Jacques Desrosiers, Marco E. Lübbecke, and François Soumis: On Compact For-mulations for Integer Programs Solved by Column Generation2003/24 Alex Hall, Katharina Langkau, and Martin Skutella: An FPTAS for Quickest Multicommodity Flowswith Inflow-Dependent Transit Times2003/23 Sven O. Krumke, Nicole Megow, and Tjark Vredeveld: How to Whack Moles2003/22 Nicole Megow and Andreas S. Schulz: Scheduling to Minimize Average Completion Time Revisited:Deterministic On-Line Algorithms2003/16 Christian Liebchen: Symmetry for Periodic Railway Timetables2003/12 Christian Liebchen: Finding Short Integral Cycle Bases for Cyclic Timetabling762/2002 Ekkehard Köhler and Katharina Langkau and Martin Skutella: Time-Expanded Graphs for Flow-Dependent Transit Times761/2002 Christian Liebchen and Leon Peeters: On Cyclic Timetabling and Cycles in Graphs752/2002 Ekkehard Köhler and Rolf H. Möhring and Martin Skutella: Traffic Networks and Flows Over Time739/2002 Georg Baier and Ekkehard Köhler and Martin Skutella: On the k-splittable Flow Problem736/2002 Christian Liebchen and Rolf H. Möhring: A Case Study in Periodic Timetabling723/2001 Berit Johannes: Scheduling Parallel Jobs to Minimize Makespan716/2001 Christian Liebchen: The Periodic Assignment Problem (PAP) May Be Solved Greedily711/2001 Esther M. Arkin, Michael A. Bender, Sándor P. Fekete, Joseph S. B. Mitchell, and Martin Skutella:The Freeze-Tag Problem: How to Wake Up a Swarm of Robots710/2001 Esther M. Arkin, Sándor P. Fekete, and Joseph S. B. Mitchell: Algorithms for Manufacturing Paper-clips and Sheet Metal Structures705/2000 Ekkehard Köhler: Recognizing Graphs without Asteroidal Triples704/2000 Ekkehard Köhler: AT-free, coAT-free Graphs and AT-free Posets702/2000 Frederik Stork: Branch-and-Bound Algorithms for Stochastic Resource-Constrained Project Schedul-ing700/2000 Rolf H. Möhring: Scheduling under uncertainty: Bounding the makespan distribution698/2000 Sándor P. Fekete, Ekkehard Köhler, and Jürgen Teich: More-dimensional packing with order con-straints697/2000 Sándor P. Fekete, Ekkehard Köhler, and Jürgen Teich: Extending partial suborders and implicationclasses696/2000 Sándor P. Fekete, Ekkehard Köhler, and Jürgen Teich: Optimal FPGA module placement with tem-poral precedence constraints695/2000 Sándor P. Fekete, Henk Meijer, André Rohe, and Walter Tietze: Solving a “hard” problem to approx-imate an “easy” one: heuristics for maximum matchings and maximum Traveling Salesman Problems694/2000 Esther M. Arkin, Sándor P. Fekete, Ferran Hurtado, Joseph S. B. Mitchell, Marc Noy, Vera Sacristánand Saurabh Sethia: On the reflexivity of point sets693/2000 Frederik Stork and Marc Uetz: On the representation of resource constraints in project scheduling691/2000 Martin Skutella and Marc Uetz: Scheduling precedence constrained jobs with stochastic processingtimes on parallel machines689/2000 Rolf H. Möhring, Martin Skutella, and Frederik Stork: Scheduling with AND/OR precedence con-straints685/2000 Martin Skutella: Approximating the single source unsplittable min-cost flow problem684/2000 Han Hoogeveen, Martin Skutella, and Gerhard J. Woeginger: Preemptive scheduling with rejection683/2000 Martin Skutella: Convex quadratic and semidefinite programming relaxations in Scheduling682/2000 Rolf H. Möhring and Marc Uetz: Scheduling scarce resources in chemical engineering681/2000 Rolf H. Möhring: Scheduling under uncertainty: optimizing against a randomizing adversary680/2000 Rolf H. Möhring, Andreas S. Schulz, Frederik Stork, and Marc Uetz: Solving project schedulingproblems by minimum cut computations (Journal version for the previous Reports 620 and 661)674/2000 Esther M. Arkin, Michael A. Bender, Erik D. Demaine, Sándor P. Fekete, Joseph S. B. Mitchell, andSaurabh Sethia: Optimal covering tours with turn costs669/2000 Michael Naatz: A note on a question of C. D. Savage667/2000 Sándor P. Fekete and Henk Meijer: On geometric maximum weight cliques666/2000 Sándor P. Fekete, Joseph S. B. Mitchell, and Karin Weinbrecht: On the continuous Weber and k-median problems664/2000 Rolf H. Möhring, Andreas S. Schulz, Frederik Stork, and Marc Uetz: On project scheduling withirregular starting time costs661/2000 Frederik Stork and Marc Uetz: Resource-constrained project scheduling: from a Lagrangian relax-ation to competitive solutionsReports may be requested from: Sekretariat MA 6–1Fakultt II – Institut fr MathematikTU BerlinStraße des 17. Juni 136D-10623 Berlin – Germanye-mail: klink@math.TU-Berlin.DEReports are also available in various formats fromhttp://www.math.tu-berlin.de/coga/publications/techreports/and via anonymous ftp asftp://ftp.math.tu-berlin.de/pub/Preprints/combi/Report-number-year.ps

Dual Variable Based Fathoming in Dynamic Programs for Column Generation

CiteSeerX

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.45.2272

Dual Variable Based Fathoming in Dynamic Programs for Column Generation

Abstract

Similar works

Full text

Available Versions

DepositOnce

CiteSeerX