Penalty Alternating Direction Methods for Mixed-Integer Optimization: A New View on Feasibility Pumps

Feasibility pumps are highly effective primal heuristics for mixedinteger linear and nonlinear optimization. However, despite their success in practice there are only few works considering their theoretical properties. We show that feasibility pumps can be seen as alternating direction methods applied to special reformulations of the original problem, inheriting the convergence theory of these methods. Moreover, we propose a novel penalty framework that encompasses this alternating direction method, which allows us to refrain from random perturbations that are applied in standard versions of feasibility pumps in case of failure. We present a convergence theory for the new penalty based alternating direction method and compare the new variant of the feasibility pump with existing versions in an extensive numerical study for mixed-integer linear and nonlinear problems. Due to their practical relevance, mixed-integer nonlinear problems (MINLPs) form a very important class of optimization problems. One important part of successful algorithms for the solution of such problems is finding feasible solutions quickly. For this, typically heuristics are employed. These can be roughly divided into heuristics that improve known feasible solutions (e.g., local branching [25] or RINS [16]) and heuristics that construct feasible solutions from scratch. This article discusses a heuristic of the latter type: The algorithm of interest in this article is the so-called feasibility pump that has originally been proposed by Fischetti et al. in [24] for MIPs and that has been extended by many other researchers, e.g., in [1–3, 6, 7, 17–19, 26, 34]. In addition, feasibility pumps have also been applied to MINLPs during the last years; see, e.g., [4, 8, 9, 14, 15, 39, 40]. A more detailed review of the literature about feasibility pumps is given in Section 1. For a comprehensive overview over primal heuristics for mixed-integer linear and nonlinear problems in general, we refer the interested reader to Berthold [4, 5] and the references therein. In a nutshell, feasibility pumps work as follows: given an optimal solution of the continuous relaxation of the problem, the methods construct two sequences. The first one contains integer-feasible points, the second one contains points that are feasible w.r.t. the continuous relaxation. Thus, one has found an overall feasible point if these sequences converge to a common point. To escape from situations where the construction of the sequences gets stuck and thus do not converge to a common point, feasibility pumps usually incorporate randomized restarts. The feasibility pumps described in the literature are difficult to analyze theoretically due to the use of random perturbations. These random perturbations are, however, crucial to the practical performance of the methods. The main object of the existing theoretical analysis is the idealized feasibility pump, i.e., the method without random perturbations. This is the method analyzed in the publications [17] Date: March 22, 2017. 2010 Mathematics Subject Classification. 65K05, 90-08, 90C10, 90C11, 90C59.