The feasibility pump.

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. [ABSTRACT FROM AUTHOR]