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An adaptive, multivariate partitioning algorithm for global optimization of nonconvex programs

  • Harsha NagarajanEmail author
  • Mowen Lu
  • Site Wang
  • Russell Bent
  • Kaarthik Sundar
Article
  • 191 Downloads

Abstract

In this work, we develop an adaptive, multivariate partitioning algorithm for solving nonconvex, Mixed-Integer Nonlinear Programs (MINLPs) with polynomial functions to global optimality. In particular, we present an iterative algorithm that exploits piecewise, convex relaxation approaches via disjunctive formulations to solve MINLPs that is different than conventional spatial branch-and-bound approaches. The algorithm partitions the domains of variables in an adaptive and non-uniform manner at every iteration to focus on productive areas of the search space. Furthermore, domain reduction techniques based on sequential, optimization-based bound-tightening and piecewise relaxation techniques, as a part of a presolve step, are integrated into the main algorithm. Finally, we demonstrate the effectiveness of the algorithm on well-known benchmark problems (including Pooling and Blending instances) from MINLPLib and compare our algorithm with state-of-the-art global optimization solvers. With our novel approach, we solve several large-scale instances, some of which are not solvable by state-of-the-art solvers. We also succeed in reducing the best known optimality gap for a hard, generalized pooling problem instance.

Keywords

Global optimization Adaptive partitioning MILP-based methods Mixed-integer nonlinear programs McCormick Piecewise relaxations Sequential optimization-based bound-tightening 

Notes

Acknowledgements

The work was funded by the Center for Nonlinear Studies (CNLS) at LANL and the LANL’s directed research and development project “POD: A Polyhedral Outer-approximation, Dynamic-discretization optimization solver”. Work was carried out under the auspices of the U.S. DOE under Contract No. DE-AC52-06NA25396.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Center for Nonlinear StudiesLos Alamos National LaboratoryNew MexicoUSA
  2. 2.Department of Industrial EngineeringClemson UniversityClemsonUSA

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