Tighter McCormick relaxations through subgradient propagation

Abstract

Tight convex and concave relaxations are of high importance in deterministic global optimization. We present a method to tighten relaxations obtained by the McCormick technique. We use the McCormick subgradient propagation (Mitsos et al. in SIAM J Optim 20(2):573–601, 2009) to construct simple affine under- and overestimators of each factor of the original factorable function. Then, we minimize and maximize these affine relaxations in order to obtain possibly improved range bounds for every factor resulting in possibly tighter final McCormick relaxations. We discuss the method and its limitations, in particular the lack of guarantee for improvement. Subsequently, we provide numerical results for benchmark cases found in the MINLPLib2 library and case studies presented in previous works, where the McCormick technique appears to be advantageous, and discuss computational efficiency. We see that the presented algorithm provides a significant improvement in tightness and decrease in computational time, especially in the case studies using the reduced space formulation presented in (Bongartz and Mitsos in J Glob Optim 69:761–796, 2017).

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Acknowledgements

We would like to thank Dr. Benoît Chachuat for providing MC++ v2.0 and Dominik Bongartz and Artur Schweidtmann for providing numerical case studies. We appreciate the thorough review and helpful comments provided by the anonymous reviewers and editors which resulted in an improved manuscript. This project has received funding from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) Improved McCormick Relaxations for the efficient Global Optimization in the Space of Degrees of Freedom MA 1188/34-1.

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A Appendix

A Appendix

See Tables 3, 4, 5 and 6.

Table 3 The problems for the numerical studies for different numbers of iterations within the heuristic
Table 4 The numerical results. We allowed only 1 iteration within the heuristic and used the middle point of each node as its initial point
Table 5 The numerical results. We allowed only 1 iteration within the heuristic and used the incumbent as its initial point
Table 6 The numerical results for different numbers of iterations and points within the heuristic

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Najman, J., Mitsos, A. Tighter McCormick relaxations through subgradient propagation. J Glob Optim 75, 565–593 (2019). https://doi.org/10.1007/s10898-019-00791-0

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Keywords

  • Global optimization
  • McCormick
  • Range reduction
  • MAiNGO

Mathematics Subject Classification

  • 49M20
  • 49M37
  • 65K05
  • 90C26