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{cT x : Ax ≥ b, xj integer ∀j ∈ I}. Trivially, a feasible solution can be defined as a point x∗ ∈ P := {x : Ax ≥ b} that is equal to its rounding x̃, where the rounded point x̃ is defined by x̃j := [xj ] if j ∈ I and x̃j := xj otherwise, and [·] represents scalar rounding to the nearest integer. Replacing “equal” with “as close as possible” relative to a suitable distance function ∆(x∗, x̃), 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 x̃. At each FP iteration we look for a point x∗ ∈ P that is as close as possible to the current x̃ by solving the problem min{∆(x, x̃) : x ∈ P}. Assuming ∆(x, x̃) is chosen appropriately, this is an easily solvable LP problem. If ∆(x∗, x̃) = 0, then x∗ is a feasible MIP solution and we are done. Otherwise, we replace x̃ by the rounding of x∗, and repeat. From a geometric point of view, the FP generates two trajectories of points x∗ and x̃ that satisfy feasibility in a complementary but partial way—one satisfies the linear constraints, the other the integer requirement. The FP can also be viewed as a strategy for making a heuristic sequence of roundings that yields a feasible MIP point. 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.