Abstract
In this paper we consider the NP-hard problem of finding a feasible solution (if any exists) for a generic MIP problem of the form min{c T x:A x≥b,x j integer ∀j ∈ }. Trivially, a feasible solution can be defined as a point x* ∈ P:={x:A x≥b} that is equal to its rounding , where the rounded point is defined by := x* j if j ∈ and := x* j otherwise, and [·] represents scalar rounding to the nearest integer. Replacing “equal” with “as close as possible” relative to a suitable distance function Δ(x*, ), suggests the following Feasibility Pump (FP) heuristic for finding a feasible solution of a given MIP.
We start from any x* ∈ P, and define its rounding . At each FP iteration we look for a point x* ∈ P that is as close as possible to the current by solving the problem min {Δ(x, ): x ∈ P}. Assuming Δ(x, ) is chosen appropriately, this is an easily solvable LP problem. If Δ(x*, )=0, then x* is a feasible MIP solution and we are done. Otherwise, we replace by the rounding of x*, and repeat.
We report computational results on a set of 83 difficult 0-1 MIPs, using the commercial software ILOG-Cplex 8.1 as a benchmark. The outcome is that FP, in spite of its simple foundation, proves competitive with ILOG-Cplex both in terms of speed and quality of the first solution delivered. Interestingly, ILOG-Cplex could not find any feasible solution at the root node for 19 problems in our test-bed, whereas FP was unsuccessful in just 3 cases.
Similar content being viewed by others
References
Achterberg, T., Koch, T., Martin, A.: The mixed integer programming library: MIPLIB 2003. http://miplib.zib.de.
Balas, E., Ceria, S., Dawande, M., Margot, F., Pataki, G.: OCTANE: A New Heuristic For Pure 0-1 Programs. Oper. Res. 49, 207–225 (2001)
Balas, E., Martin, C.H.: Pivot-And-Complement: A Heuristic For 0-1 Programming. Management Science 26, 86–96 (1980)
Balas, E., Schmieta, S., Wallace, C.: Pivot and Shift–A Mixed Integer Programming Heuristic. Discrete Optimization 1, 3–12 (2004)
Bixby, R.E.: Personal communication. 2003
Chinneck, J.W.: The constraint consesus method for finding approximately feasible points in nonlinear programs. Technical Report Carleton University, Ottawa, Ontario, Canada, October 2002
Danna, E., Rothberg, E., Le Paper, C.: Exploring relaxation induced neighborhoods to improve MIP solutions. Mathematical Programming, 102, 71–90 (2005)
Double-Click sas.: Personal communication. 2001
Fischetti, M., Lodi, A.: Local Branching. Mathematical Programming 98, 23–47 (2003)
Glover, F., Laguna, M.: General Purpose Heuristics For Integer Programming: Part I. J. Heuristics 2, 343–358 (1997)
Glover, F., Laguna, M.: General Purpose Heuristics For Integer Programming: Part II. J. Heuristics 3, 161–179 (1997)
Glover, F., Laguna, M.: Tabu Search. Kluwer Academic Publisher, Boston, Dordrecht, London, 1997
Hillier, F.S.: Effcient Heuristic Procedures For Integer Linear Programming With An Interior. Oper. Res. 17, 600–637 (1969)
Ibaraki, T., Ohashi, T., Mine, H.: A Heuristic Algorithm For Mixed-Integer Programming Problems. Mathematical Programming Study 2, 115–136 (1974)
G.W. Klau. Personal communication, 2002.
Løkketangen, A.: Heuristics for 0-1 Mixed-Integer Programming. In: P.M. Pardalos, M.G.C. Resende (eds.), Handbook of Applied Optimization, Oxford University Press, 2002, pp. 474–477
Løkketangen, A., Glover, F.: Solving Zero/One Mixed Integer Programming Problems Using Tabu Search. European J. Oper. Res. 106, 624–658 (1998)
Lübbecke, M.: Personal communication. 2002
Miller, A.J.: Personal communication. 2003
Nediak, M., Eckstein, J.: Pivot, Cut, and Dive: A Heuristic for 0-1 Mixed Integer Programming. Research Report RRR 53-2001, RUTCOR, Rutgers University, October 2001
Patel, J., Chinneck, J.W.: Active-Constraint Variable Ordering for Faster Feasibility of Mixed Integer Linear Programs. Technical Report Carleton University, Ottawa, Ontario, Canada, November 2003
Rothberg, E.: Personal communication. 2002
Rothberg, E.: Personal communication. 2003
Spielberg, K., Guignard, M.: Sequential (Quasi) Hot Start Method for BB (0,1) Mixed Integer Programming. Wharton School Research Report, 2002
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Fischetti, M., Glover, F. & Lodi, A. The feasibility pump. Math. Program. 104, 91–104 (2005). https://doi.org/10.1007/s10107-004-0570-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10107-004-0570-3