Abstract
During the last decades, much research has been conducted deriving classes of valid inequalities for single-row mixed integer programming polyhedrons. However, no such class has had as much practical success as the MIR inequality when used in cutting plane algorithms for general mixed integer programming problems. In this work we analyze this empirical observation by developing an algorithm which takes as input a point and a single-row mixed integer polyhedron, and either proves the point is in the convex hull of said polyhedron, or finds a separating hyperplane. The main feature of this algorithm is a specialized subroutine for solving the Mixed Integer Knapsack Problem which exploits cost and lexicographic dominance. Separating over the entire closure of single-row systems allows us to establish natural benchmarks by which to evaluate specific classes of knapsack cuts. Using these benchmarks on Miplib 3.0 instances we analyze the performance of MIR inequalities. Computations are performed in exact arithmetic.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Applegate, D., Bixby, R.E., Chvátal, V., Cook, W.: TSP cuts which do not conform to the template paradigm. In: Jünger, M., Naddef, D. (eds.) Computational Combinatorial Optimization. LNCS, vol. 2241, pp. 261–304. Springer, Heidelberg (2001)
Applegate, D., Cook, W., Dash, S., Espinoza, D.: Exact solutions to linear programming problems. Submitted to Operations Research Letters (2006)
Atamtürk, A.: On the facets of the mixed–integer knapsack polyhedron. Mathematical Programming 98, 145–175 (2003)
Balas, E., Perregaard, M.: A precise correspondence between lift-and-project cuts, simple disjuntive cuts, and mixed integer Gomory cuts for 0-1 programming. Mathematical Programming 94, 221–245 (2003)
Balas, E., Saxena, A.: Optimizing over the split closure. Mathematical Programming, To appear
Bellman, R.E.: Dynamic Programming. Princeton University Press, Princeton (1957)
Bixby, R.E., Ceria, S., McZeal, C.M., Savelsbergh, M.W.P.: An updated mixed integer programming library: MIPLIB 3.0. Optima 58, 12–15 (1998)
Boyd, A.E.: Fenchel cutting planes for integer programs. Operations Research 42, 53–64 (1992)
Caprara, A., Letchford, A.: On the separation of split cuts and related inequalities. Mathematical Programming 94(2-3), 279–294 (2003)
Cook, W., Kannan, R., Schrijver, A.: Chvátal closures for mixed integer programming problems. Mathematical Programming 47, 155–174 (1990)
Cornuéjols, G., Li, Y., Vanderbussche, D.: K-cuts: A variation of gomory mixed integer cuts from the LP tableau. Informs Journal On Computing 15, 385–396 (2003)
Crowder, H., Johnson, E.L., Padberg, M.: Solving large-scale zero-one linear-programming problems. Operations Research 31, 803–834 (1983)
Dantzig, G.B.: Discrete variable extremum problems. Operations Research 5(2), 266–277 (1957)
Dash, S., Goycoolea, M., Günlük, O.: Two-step mir inequalities for mixed-integer programs. Optimization Online (Jul. 2006)
Dash, S., Günlük, O.: On the strength of gomory mixed-integer cuts as group cuts. IBM research report RC23967 (2006)
Dash, S., Günlük, O., Lodi, A.: MIR closures of polyhedral sets. Available online at http://www.optimization-online.org/DB_HTML/2007/03/1604.html
Espinoza, D.G.: On Linear Programming, Integer Programming and Cutting Planes. PhD thesis, School of Industrial and Systems Engineering, Georgia Institute of Technology (March 2006)
Fischetti, M., Lodi, A.: On the knapsack closure of 0-1 integer linear problems. Presentation at 10th International Workshop on Combinatorial Optimization, Aussois (2006), Available at http://www-id.imag.fr/IWCO2006/slides/Fischetti.pdf
Gomory, R.E.: Early integer programming (reprinted). Operations Research 50, 78–81 (2002)
Gomory, R.E., Johnson, E.L.: Some continuous functions related to corner polyhedra I. Mathematical Programming 3, 23–85 (1972)
Goycoolea, M.: Cutting Planes for Large Mixed Integer Programming Models. PhD thesis, Georgia Institute of Technology (2006)
Granlund, T.: The GNU multiple precision arithmetic library. Available on-line at http://www.swox.com/gmp/
Gu, Z., Nemhauser, G.L., Savelsbergh, M.W.P.: Lifted cover inequalities for 0-1 integer programs: Computation. INFORMS Journal on Computing 10, 427–437 (1998)
Horowitz, E., Sahni, S.: Computing partitions with applications to the knapsack problem. Journal of the ACM 21, 277–292 (1974)
Kellerer, H., Pferschy, U., Pisinger, D.: Knapsack Problems. Springer, Berlin (2004)
Marchand, H., Wolsey, L.A.: Aggregation and mixed integer rounding to solve MIPs. Operations Research 49, 363–371 (2001)
Martello, S., Toth, P.: Knapsack Problems: Algorithms and Computer Implementations. J. Wiley, New York (1990)
Nemhauser, G.L., Wolsey, L.A.: A recursive procedure for generating all cuts for 0-1 mixed integer programs. Mathematical Programming 46, 379–390 (1990)
Savelsbergh, M.W.P.: Preprocessing and probing for mixed integer programming problems. ORSA Journal on Computing 6, 445–454 (1994)
Yan, X.Q., Boyd, E.A.: Cutting planes for mixed-integer knapsack polyhedra. Mathematical Programming 81, 257–262 (1998)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer Berlin Heidelberg
About this paper
Cite this paper
Fukasawa, R., Goycoolea, M. (2007). On the Exact Separation of Mixed Integer Knapsack Cuts. In: Fischetti, M., Williamson, D.P. (eds) Integer Programming and Combinatorial Optimization. IPCO 2007. Lecture Notes in Computer Science, vol 4513. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72792-7_18
Download citation
DOI: https://doi.org/10.1007/978-3-540-72792-7_18
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-72791-0
Online ISBN: 978-3-540-72792-7
eBook Packages: Computer ScienceComputer Science (R0)