# On solving generalized convex MINLP problems using supporting hyperplane techniques

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## Abstract

Solution methods for convex mixed integer nonlinear programming (MINLP) problems have, usually, proven convergence properties if the functions involved are differentiable and convex. For other classes of convex MINLP problems fewer results have been given. Classical differential calculus can, though, be generalized to more general classes of functions than differentiable, via subdifferentials and subgradients. In addition, more general than convex functions can be included in a convex problem if the functions involved are defined from convex level sets, instead of being defined as convex functions only. The notion generalized convex, used in the heading of this paper, refers to such additional properties. The generalization for the differentiability is made by using subgradients of Clarke’s subdifferential. Thus, all the functions in the problem are assumed to be locally Lipschitz continuous. The generalization of the functions is done by considering quasiconvex functions. Thus, instead of differentiable convex functions, nondifferentiable $$f^{\circ }$$-quasiconvex functions can be included in the actual problem formulation and a supporting hyperplane approach is given for the solution of the considered MINLP problem. Convergence to a global minimum is proved for the algorithm, when minimizing an $$f^{\circ }$$-pseudoconvex function, subject to $$f^{\circ }$$-pseudoconvex constraints. With some additional conditions, the proof is also valid for $$f^{\circ }$$-quasiconvex functions, which sums up the properties of the method, treated in the paper. The main contribution in this paper is the generalization of the Extended Supporting Hyperplane method in Eronen et al. (J Glob Optim 69(2):443–459, 2017) to also solve problems with $$f^{\circ }$$-pseudoconvex objective function.

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## References

1. Androulakis, I., Maranas, C., Floudas, C.A.: $$\alpha$$BB: A global optimization method for general constrained nonconvex problems. J. Glob. Optim. 7, 337–363 (1995)

2. Bagirov, A., Mäkelä, M.M., Karmitsa, N.: Introduction to Nonsmooth Optimization: Theory Practice and Software. Springer International Publishing, Cham, Heidelberg (2014)

3. Bonami, P., Kilinc, M., Linderoth, J.: Algorithms and software for convex mixed-integer nonlinear programs. In: Lee, J., Leyffer, S. (eds.) Mixed Integer Programming, The IMA Volumes in Mathematics and Its Applications, pp. 1–39. Springer, New York (2012)

4. Bussieck, M.R., Vigerske, S.: MINLP solver software. In: Wiley Encyclopedia of Operations Research and Management Science. Wiley (2011). https://doi.org/10.1002/9780470400531.eorms0527

5. Cambini, A., Martein, L.: Generalized convexity and optimization—theory and applications. In: Lecture Notes in Economics and Mathematical Systems. Springer, Berlin (2009)

6. Castillo, I., Westerlund, J., Emet, S., Westerlund, T.: Optimization of block layout design problems with unequal areas: a comparison of MILP and MINLP optimization methods. Comput. Chem. Eng. 30, 54–69 (2005)

7. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

8. de Oliveira, W.: Regularized optimization methods for convex MINLP problems. TOP 24, 665–692 (2016)

9. Duran, M.A., Grossmann, I.E.: An outer-approximation algorithm for a class of mixed-integer nonlinear programs. Math. Program. 36, 307–339 (1986)

10. Eronen, V.-P., Mäkelä, M.M., Westerlund, T.: On the generalization of ECP and OA methods to nonsmooth MINLP problems. Optimization 63(7), 1057–1073 (2014)

11. Eronen, V.-P., Mäkelä, M.M., Westerlund, T.: Extended cutting plane method for a class of nonsmooth nonconvex MINLP problems. Optimization 64(3), 641–661 (2015)

12. Eronen, V.-P., Kronqvist, J., Westerlund, T., Mäkelä, M.M., Karmitsa, N.: Method for solving generalized convex nonsmooth mixed-integer nonlinear programming problems. J. Glob. Optim. 69(2), 443–459 (2017)

13. Fletcher, R., Leyffer, S.: Solving mixed integer nonlinear programs by outer approximation. Math. Program. 66, 327–349 (1994)

14. Fletcher, R., Leyffer, S.: Numerical experience with lower bounds for MIQP branch-and-bound. SIAM J. Optim. 8, 604–616 (1998)

15. Geoffrion, A.M.: Generalized benders decomposition. J. Optim. Theory Appl. 10, 237–260 (1973)

16. Grossmann, I.E.: Review of nonlinear mixed-integer and disjunctive programming techniques. Optim. Eng. 3, 227–252 (2002)

17. Jain, V., Grossmann, I.: Cyclic scheduling of continuous parallel-process units with decaying performance. AIChE J. 44, 1623–1636 (1999)

18. Kelley, J.E.: The cutting plane method for solving convex programs. J. SIAM 8, 703–712 (1960)

19. Kronqvist, J., Lundell, A., Westerlund, T.: The extended supporting hyperplane algorithm for convex mixed-integer nonlinear programming. J. Glob. Optim. 64, 249–272 (2016)

20. Lee, J., Leyffer, S.: Mixed Integer Nonlinear Programming. Springer, New York (2012)

21. Leyffer, S.: Integrating SQP and branch-and-bound for mixed integer nonlinear programming. Comput. Optim. Appl. 18, 295–309 (2001)

22. Lundell, A., Skjäl, A., Westerlund, T.: A reformulation framework for global optimization. J. Glob. Optim. 57, 115–141 (2013)

23. Meyer, C.A., Floudas, C.A.: Convex underestimation of twice continuously differential functions by piecewise quadratic perturbations: spline $$\alpha$$BB underestimators. J. Glob. Optim. 32, 221–258 (2005)

24. Mäkelä, M.M., Neittaanmäki, P.: Nonsmooth Optimization: Analysis and Algorithms with Applications to Optimal Control. World Scientific Publishing Co., Singapore (1992)

25. Nestorov, Y., Nemirowskii, A.: Interior-point polynomial algorithms in convex programming. In: SIAM Studies in Applied Mathematics, vol. 13. Philadelphia (1994)

26. Pörn, R.: Mixed-Integer Non-Linear Programming: Convexification Techniques and Algorithm Development. Ph.D. Thesis, Åbo Akademi University (2000)

27. Quesada, I., Grossmann, I.E.: An LP/NLP based branch-and-bound algorithm for convex MINLP optimization problems. Comput. Chem. Eng. 16, 937–947 (1999)

28. Roberts, A.W., Varberg, D.E.: Convex Functions. Academic Press, New York, London (1973)

29. Rockafellar, R.T.: Convex Analysis. Princeton Landmarks in Mathematics and Physics. Princeton University Press, Princeton (1997)

30. Ryoo, H.S., Sahinidis, N.V.: A branch-and-reduce approach to global optimization. J. Glob. Optim. 8, 107–138 (1996)

31. Veinott Jr., A.F.: The supporting hyperplane method for unimodal programming. Oper. Res. 15(1), 147–152 (1967)

32. Westerlund, T., Skrifvars, H., Harjunkoski, I., Pörn, R.: An extended cutting plane method for solving a class of non-convex MINLP problems. Comput. Chem. Eng. 22, 357–365 (1998)

33. Westerlund, T., Pörn, R.: Solving pseudo-convex mixed integer optimization problems by cutting plane techniques. Optim. Eng. 3, 253–280 (2002)

34. Westerlund, T., Pettersson, F.: An extended cutting plane method for solving convex MINLP problems. Comput. Chem. Eng. 19, 131–136 (1995)

35. Westerlund, T.: User’s guide for GAECP, version 5.537. An Interactive Solver for Generalized Convex MINLP-Problems Using Cutting Plane and Supporting Hyperplane Techniques. Åbo Akademi University. www.abo.fi/~twesterl/GAECPDocumentation.pdf (2017)

## Acknowledgements

This research was supported by the Grant No. 294002 of the Academy of Finland. The authors also acknowledge GAMS Development Corporation for providing us license to use different GAMS solvers.

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Correspondence to Tapio Westerlund.

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Westerlund, T., Eronen, VP. & Mäkelä, M.M. On solving generalized convex MINLP problems using supporting hyperplane techniques. J Glob Optim 71, 987–1011 (2018). https://doi.org/10.1007/s10898-018-0644-z

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• DOI: https://doi.org/10.1007/s10898-018-0644-z