The feasibility pump is a recent, highly successful heuristic for general mixed integer linear programming problems. We show that the feasibility pump heuristic can be interpreted as a discrete version of the proximal point algorithm. In doing so, we extend and generalize some of the fundamental results in this area to provide new supporting theory. We show that feasibility pump algorithms implicitly minimize a weighted combination of the objective and a term which penalizes lack of integrality. This function has many local minima, some of which correspond to feasible integral solutions; the feasibility pump's use of random restarts can be viewed as seeking to escape these local minima when they are not feasible integral solutions. This interpretation suggests alternative ways of incorporating restarts, one of which is the application of cutting planes. Numerical experiments with cutting planes show encouraging results on standard test libraries. Introduction. In spite of the continuous improvement of both commercialand open source solvers, numerous mixed integer programming (MIP) problems of practical importance remain intractable. In practice, where rigorous algorithms fail, heuristics often succeed to provide feasible solutions of good quality. Apart from their self-evident value, good feasible solutions are also useful in speeding up the search of branch and cut algorithms. General purpose heuristics include [2,3,4,13,14,15,16,19,20,21,22,23]. We direct the reader to [6] for a recent survey.A heuristic that has attracted significant interest in recent years is the feasibility pump (FP) [17]. The FP starts from the linear program (LP) optimum and computes two trajectories of points, one integer and the other LP feasible, by iteratively applying rounding and projection operations. Every LP feasible point is rounded to an integer point and every integer point is projected back onto the LP feasible region using the l 1 norm. As both the rounding and the projection are minimal distance operations (with respect to the l 1 norm), the distance between pairs of points on the two trajectories decreases monotonically until the method cycles, unable to further decrease the distance. If the distance is reduced to zero before cycling, then a feasible point has been obtained. Otherwise a random move is used to restart the method from a new point. In the original FP [17], the only bias of the method toward points of good objective value is the starting point, so the random restarts threaten to escape the good quality area. Thus, the feasibility pump uses a sophisticated restart scheme, attempting to displace the current point in an economic way by "minor perturbations," and performs a "major restart" only in the presence of further failures.

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We present a relaxation-based dynamic programming algorithm for solving resource-constrained shortest-path problems (RCSPPs) in the context of column generation for the dial-a-flight problem. The resulting network formulation and pricing problem require solving RCSPPs on extremely large time-expanded networks having a huge number of local resource constraints, i.e., constraints that apply to small subnetworks. The relaxation-based dynamic programming algorithm alternates between a forward and a backward search. Each search employs bounds derived in the previous search to prune the search space. Between consecutive searches, the relaxation is tightened using a set of critical resources and a set of critical arcs over which these resources are consumed. As a result, a relatively small state space is maintained, and many paths can be pruned while guaranteeing that an optimal path is ultimately found.