Abstract
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.
Similar content being viewed by others
References
E. Balas and C. Bergthaller, “Benders method revisited”, Management Sciences Report 401, Carnegie-Mellon University (Pittsburgh, PA, 1977).
J.F. Benders, “Partitioning for solving mixed variables programming problems’,Numerische Mathematik 4 (1962), 238–252.
G.B. Dantzig and P. Wolfe, “Decomposition principle for linear programs”,Operations Research 8 (1960) 101–111.
M.L. Fisher, W.D. Northup and J.F. Shapiro, “Using duality to solve discrete optimizations problems: Theory and computational experience”,Mathematical Programming Study 3 (1975) 56–94.
M.L. Fisher, “The Lagrangean relaxation method for solving integer programming problems”,Management Science 27 (1981) 1–18.
A.M. Geoffrion, “Lagrangean relaxation for integer programming”.Mathematical Programming Study 2 (1974) 82–113.
G.W. Graves and T.J. Van Roy, “Decomposition for large scale linear and mixed integer linear programming”, Working Paper, Graduate School of Management. UCLA (Los Angeles, November 1979).
L.S. Lasdon,Optimization theory for large systems (MacMillan, New York, 1970).
D. McDaniel and M. Devine, “A modified Benders' partitioning algorithm for mixed integer programming”,Management Science (1977) 312–379.
D.J. Sweeney and R.A. Murphy, “A method of decomposition for integer programs”,Operations Research 27 (1979) 1128–1141.
T.J. Van Roy, “A cross decomposition algorithm for capacitated facility location”, to appear inOperations Research.
T.J. Van Roy, “Cross decomposition for large-scale mixed integer linear programming with applications to facility location on distribution networks”, doctoral dissertation, Applied Sciences, Katholieke Universiteit Leuven (1980).
L.A. Wolsey, “A resource decomposition algorithm for general mathematical programs”,Mathematical Programming Study 14 (1981) 244–257.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Van Roy, T.J. Cross decomposition for mixed integer programming. Mathematical Programming 25, 46–63 (1983). https://doi.org/10.1007/BF02591718
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02591718