A storm of feasibility pumps for nonconvex MINLP

Abstract

One of the foremost difficulties in solving Mixed-Integer Nonlinear Programs, either with exact or heuristic methods, is to find a feasible point. We address this issue with a new feasibility pump algorithm tailored for nonconvex Mixed-Integer Nonlinear Programs. Feasibility pumps are algorithms that iterate between solving a continuous relaxation and a mixed-integer relaxation of the original problems. Such approaches currently exist in the literature for Mixed-Integer Linear Programs and convex Mixed-Integer Nonlinear Programs: both cases exhibit the distinctive property that the continuous relaxation can be solved in polynomial time. In nonconvex Mixed-Integer Nonlinear Programming such a property does not hold, and therefore special care has to be exercised in order to allow feasibility pump algorithms to rely only on local optima of the continuous relaxation. Based on a new, high level view of feasibility pump algorithms as a special case of the well-known successive projection method, we show that many possible different variants of the approach can be developed, depending on how several different (orthogonal) implementation choices are taken. A remarkable twist of feasibility pump algorithms is that, unlike most previous successive projection methods from the literature, projection is "naturally" taken in two different norms in the two different subproblems. To cope with this issue while retaining the local convergence properties of standard successive projection methods we propose the introduction of appropriate norm constraints in the subproblems; these actually seem to significantly improve the practical performance of the approach. We present extensive computational results on the MINLPLib, showing the effectiveness and efficiency of our algorithm.

Towards an objective feasibility pump for convex MINLPs

“…This is supported by promising results from applying the OFP on a nonconvex MINLP scheduling problem presented in [20], which merit further investigation into such an extension of the algorithm. Finally, we remark that improved handling of cycling in the OFP algorithm may be obtained by including for instance a tabu-list as suggested in [14]. It may also be possible to derive a direct updating strategy for the weight parameters in the objective function of the OFP as a means of mitigating stalling, in addition to the two objectives of integer feasibility and solution quality.…”

“…The OA-based variant, however, is proven to not cycle if a constraint qualification holds [12]. An FP heuristic for nonconvex MINLPs is presented in [14], which applies a combination of metaheuristic techniques in conjunction with a scheme resembling the OA-based FP for convex MINLPs.…”

Deterministic upper bounds for spatial branch-and-bound methods in global minimization with nonconvex constraints

“…In the presence of integer variables these difficulties may become even more demanding, but large neighborhood search (Berthold et al 2011) or the feasibility pump (D'Ambrosio et al 2012) may be employed as heuristics. Further, the solver Bonmin can be used as a heuristic to find feasible points in nonconvex optimization problems (Bonami et al 2008).…”

Branch and Bound Based Coordinate Search Filter Algorithm for Nonsmooth Nonconvex Mixed-Integer Nonlinear Programming Problems

“…For a review of the available solvers, we refer the reader to [3]. Recently, the nonconvex MINLP research area became more interesting due to its range of applications and the new techniques for globally solving NLP problems [8]. In the last 10-15 years, innovative heuristic type algorithms have also appeared: genetic algorithm [15], ant colony [22], pattern search algorithms [1], multistart Hooke and Jeeves algorithm [12], and differential evolution [19].…”