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Using regularization and second order information in outer approximation for convex MINLP

Abstract

In this paper, we present two new methods for solving convex mixed-integer nonlinear programming problems based on the outer approximation method. The first method is inspired by the level method and uses a regularization technique to reduce the step size when choosing new integer combinations. The second method combines ideas from both the level method and the sequential quadratic programming technique and uses a second order approximation of the Lagrangean when choosing the new integer combinations. The main idea behind the methods is to choose the integer combination more carefully at each iteration, in order to obtain the optimal solution in fewer iterations compared to the original outer approximation method. We prove rigorously that both methods will find and verify the optimal solution in a finite number of iterations. Furthermore, we present a numerical comparison of the methods based on 109 test problems to illustrate their advantages.

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Notes

  1. 1.

    http://www.gamsworld.org/minlp/minlplib2/html/index.html.

  2. 2.

    http://www.gamsworld.org/minlp/minlplib2/html/cvxnonsep_nsig40.html.

  3. 3.

    http://www.gamsworld.org/minlp/minlplib2/html/ibs2.html.

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Acknowledgements

Jan Kronqvist is grateful for the grants given by Walter Ahlström foundation, Svenska tekniska vetenskapsakademien i Finland, Tekniikan edistämissäätiö, TFIF and Waldemar von Frenckells stiftelse, which made the research visit at Carnegie Mellon University possible. David E. Bernal and Ignacio E. Grossmann would like to thank the Center Advanced Process Decision Making (CAPD) for its financial support. The authors would like to acknowledge the Dagstuhl Seminar 18081 on Designing and Implementing Algorithms for Mixed-Integer Nonlinear Optimization, where an early version of the results shown in this manuscript were presented and discussed.

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Kronqvist, J., Bernal, D.E. & Grossmann, I.E. Using regularization and second order information in outer approximation for convex MINLP. Math. Program. 180, 285–310 (2020). https://doi.org/10.1007/s10107-018-1356-3

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Mathematics Subject Classification

  • 65K05
  • 90C11
  • 90C30
  • 90C55