Mathematical Programming

, Volume 136, Issue 2, pp 375–402

A storm of feasibility pumps for nonconvex MINLP

  • Claudia D’Ambrosio
  • Antonio Frangioni
  • Leo Liberti
  • Andrea Lodi
Full Length Paper Series B

DOI: 10.1007/s10107-012-0608-x

Cite this article as:
D’Ambrosio, C., Frangioni, A., Liberti, L. et al. Math. Program. (2012) 136: 375. doi:10.1007/s10107-012-0608-x

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.

Keywords

Feasibility pumpMINLPGlobal optimizationNonconvex NLP

Mathematics Subject Classification

90C11 Mixed integer programming90C26 Nonconvex programming, global optimization90C59 Approximation methods and heuristics

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2012

Authors and Affiliations

  • Claudia D’Ambrosio
    • 1
  • Antonio Frangioni
    • 2
  • Leo Liberti
    • 1
  • Andrea Lodi
    • 3
  1. 1.LIXEcole PolytechniqueParisFrance
  2. 2.Dipartimento di InformaticaUniversità di PisaPisaItaly
  3. 3.DEIUniversitá di BolognaBolognaItaly