Abstract
Given a mixed-integer programming problem with two matrix constraints, it is possible to define a Lagrangean relaxation such that the relaxed problem decomposes additively into two subproblems, each having one of the two matrices of the original problem as its constraints. There is one Lagrangean multiplier per variable. We prove that the optimal value of this new Lagrangean dual dominates the optimal value of the Lagrangean dual obtained by relaxing one set of constraints and give a necessary condition for a strict improvement. We show on an example that the resulting bound improvement can be substantial. We show on a complex practical problem how Lagrangean decomposition may help uncover hidden special structures and thus yield better solution methodology.
Similar content being viewed by others
References
O. Bilde and J. Krarup, “Bestemmelse af optimal beliggenhed af produktionssteder” Research Report, IMSOR, The Technical University of Denmark (1967).
O. Bilde and J. Krarup, “Sharp Lower Bounds and Efficient Algorithms for the Simple Plant Location Problem,”Annals of Discrete Mathematics 1 (1977) 79–97.
M.E. Dyer, “Calculating Surrogate Constraints,”Mathematical Programming 19 (1980) 255–278.
D. Erlenkotter, “A Dual-Based Procedure for Uncapacitated Facility Location,”Operations Research 26 (1978) 992–1009.
M.L. Fisher, R. Jaikumar and L. Van Wassenhove, “A Multiplier Adjustment Method for the Generalized Assignment Problem,”Management Science 32 (1986) 1095–1103.
A. Frank, “A Weighted Matroid Intersection Algorithm,”Journal of Algorithms 2 (1981) 328–336.
B. Gavish and H. Pirkul, “Efficient Algorithms for Solving Multiconstraint Zero-One Knapsack Problems to Optimality,”Mathematical Programming 31 (1985) 78–105.
A. Geoffrion, “Lagrangean Relaxation and its Uses in Integer Programming,”Mathematical Programming Study 2 (1974) 82–114.
F. Glover and J. Mulvey, “The Equivalence of the 0–1 Integer Programming Problem to Discrete Generalized and Pure Network Models,” Report HBS 75-46, Harvard University (1975), alsoOperations Research 28 (1980) 829–933.
F. Glover and D. Klingman, “Layering Strategies for Creating Exploitable Structure in Linear and Integer Programs,” Center for Business Decision Analysis Report 119 (1984, revised 1985).
H.J. Greenberg and W.P. Pierskalla, “Surrogate Mathematical Programming,”Operations Research 18 (1970) 924–939.
M. Guignard, “A Lagrangean Dual Ascent Method Based on (Separable) Relaxation Strengthened by Valid Inequalities (Illustrated for the Matching Problem),” Department of Statistics Report #54, University of Pennsylvania (1983).
M. Guignard, “A Lagrangean Dual Ascent Method for Simple Plant Location Problems,” Department of Statistics Report #59, University of Pennsylvania (1984a), to appear inEuropean J. Oper. Res.
M. Guignard, “Lagrangean Decomposition: An Improvement over Lagrangean and Surrogate Duals,” Department of Statistics Report #62, University of Pennsylvania (1984b).
M. Guignard and S. Kim, “A Strong Lagrangean Relaxation for Capacitated Plant Locations Problems,” Department of Statistics Report #56, University of Pennsylvania (1983).
M. Guignard and K. Opaswongkarn, “Lagrangean Ascent for the Capacitated Plant Location Problem,” Department of Statistics Report #57, University of Pennsylvania (1984).
M. Guignard and M. Rosenwein, “An Application of Lagrangean Decomposition to the Resource Constrained Minimum Weighted Arborescence Problem,” AT&T Bell Laboratories Report (1987).
M. Guignard and K. Spielberg “Separable Lagrangean Relaxation for Weakly-Linked Mixed Integer Programming Problems,”Methods of Operations Research 49 (1985) 239–253.
G.Y. Handler and I. Zang, “A Dual Algorithm for the Constrained Shortest Path Problem,”Networks 10 (1980) 293–310.
K.O. Jörnsten, M. Näsberg and P.A. Smeds, “Variable splitting—A New Lagrangean Relaxation Approach to Some Mathematical Programming Models,” Department of Mathematics Report LiTHMAT-R-85-04, Linköping Institute of Technology, Sweden (1985).
M.H. Karwan and R.L. Rardin, “Some Relationships between Lagrangean and Surrogate Duality in Integer Programming,”Mathematical Programming 18 (1979) 320–334.
S. Kim, oral communication (1984).
S. Kim, “Solving Single Side Constraint Problems in Integer Programming: the Inequality Case,” Department of Statistics, Report #60, University of Pennsylvania (1984).
S. Kim, “Primal Interpretation of Lagrangean Decomposition” Department of Statistics Report #72, University of Pennsylvania (1985).
M. Minoux, “Plus courts chemins avec contraintes: algorithmes et applications,”Annales des Télécommunications 30 (1975) 383–394.
M. Minoux, “Lagrangean Decomposition,” oral communication (1983).
C. Ribeiro, “Algorithmes de Recherche de Plus Courts Chemins avec Contraintes: Etude Théorique, Implémentation et Parallélisation,” Doctoral Dissertation, Paris (1983).
C. Ribeiro and M. Minoux, “Solving Hard Constrained Shortest Path Problems,” ORSA-TIMS Meeting, Dallas (1984).
C. Ribeiro and M. Minoux, “Solving Hard Constrained Shortest Path Problems by Lagrangean Relaxation and Branch and Bound Algorithms,”Methods of Operations Research 53 (1986) 303–316.
F. Shepardson and R. Marsten, “A Lagrangean Relaxation Algorithm for the Two-Duty Period Scheduling Problem,”Management Science 26 (1980) 274–281.
R. Wong, “A Dual Ascent Approach for Steiner Tree Problems on a Directed Graph,”Mathematical Programming 28 (1984) 271–287.
Author information
Authors and Affiliations
Additional information
Research supported by the National Science Foundation under grant ECS-8508142.
Rights and permissions
About this article
Cite this article
Guignard, M., Kim, S. Lagrangean decomposition: A model yielding stronger lagrangean bounds. Mathematical Programming 39, 215–228 (1987). https://doi.org/10.1007/BF02592954
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02592954