Penalty Alternating Direction Methods for Mixed-Integer Optimization: A New View on Feasibility Pumps
SIAM Journal on Optimization
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
... hod, 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. Date: April 7, 2016. 2010 Mathematics Subject Classification. 65K05, 90-08, 90C10, 90C11, 90C59. 1 2 B. GEI LER, A. MORSI, L. SCHEWE, M. SCHMIDT and . To be more specific, De Santis et al. show in  that idealized feasibility pumps are a special case of the Frank-Wolfe algorithm applied to a suitable chosen concave and nonsmooth merit function and Boland et al. showed in  that idealized feasibility pumps can be seen as discrete versions of the proximal point algorithm. Our approach is similar to these publications. We show that the idealized variant can be seen as a so-called alternating direction method (ADM) applied to a special reformulation of the mixed-integer problem at hand. To this end, we extend the known theory on feasibility pumps by applying the convergence theory of general ADMs. We then go one step further: The necessity to use random perturbations comes from the need to escape from undesired points. We replace these random perturbations of the original feasibility pump by a penalty framework. This allows us to view feasibility pumps as penalty based alternating direction methods-a new class of optimization methods for which we also present convergence theory. In summary, we are able to give a convergence theory for a class of feasibility pumps that incorporates deterministic restart rules. Another advantage is that our method can be presented in a quite generic way that comprises both the case of mixed-integer linear and nonlinear problems. We further give extensive computational results to show that our replacement of the random restarts does not lead to a degradation in performance. Our method compares favorably with published variants of feasibility pumps for MIPs and MINLPs. The paper is organized as follows: In Section 1 we review the main ingredients of feasibility pumps and give a more detailed literature survey. Afterward, we discuss general ADMs in Section 2 and show that idealized feasibility pumps can be seen as ADMs applied to certain equivalent reformulations of the original problem. In Section 3, we then present a penalty ADM, prove convergence results, and show how this new method can be used to obtain a novel feasibility pump algorithm that replaces random restarts with penalty parameter updates. In Section 4 we discuss important implementation issues and Section 5 finally presents an extensive computational study both for MIPs and MINLPs.