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Outer approximation for integer nonlinear programs via decision diagrams

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Abstract

As an alternative to traditional integer programming (IP), decision diagrams (DDs) provide a new solution technology for discrete problems exploiting their combinatorial structure and dynamic programming representation. While the literature mainly focuses on the competitive aspects of DDs as a stand-alone solver, we investigate their complementary role by introducing IP techniques that can be derived from DDs and used in conjunction with IP to enhance the overall performance. This perspective allows for studying problems with more general structure than those typically modeled via recursive formulations. In particular, we develop linear programming and subgradient-type methods to generate valid inequalities for the convex hull of the feasible region described by DDs. For convex IPs, these cutting planes dominate the so-called linearized cuts used in the outer approximation framework. These cutting planes can also be derived for nonconvex IPs, which leads to a generalization of the outer approximation framework. Computational experiments show significant optimality gap improvement for integer nonlinear programs over the traditional cutting plane methods employed in the state-of-the-art solvers.

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Acknowledgements

We thank Nick Sahinidis and Michael Bussieck for providing GAMS license for our experiments. We also thank the anonymous referees and the Associate Editor for their helpful comments that contributed to improving the paper.

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Correspondence to Danial Davarnia.

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Davarnia, D., van Hoeve, W. Outer approximation for integer nonlinear programs via decision diagrams. Math. Program. (2020). https://doi.org/10.1007/s10107-020-01475-4

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Keywords

  • Decision diagrams
  • Cutting planes
  • Outer approximation
  • Dynamic programming
  • Integer nonlinear programs

Mathematics Subject Classification

  • 90-08
  • 90C10
  • 90C26
  • 90C30
  • 90C35
  • 90C57
  • 90C90