The feasibility pump

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{cTx:Ax≥b,xj integer ∀j ∈ **}. Trivially, a feasible solution can be defined as a point x* ∈ P:={x:Ax≥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.