On solving generalized convex MINLP problems using supporting hyperplane techniques

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...

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Published in:	Journal of global optimization Vol. 71; no. 4; pp. 987 - 1011
Main Authors:	Westerlund, Tapio, Eronen, Ville-Pekka, Mäkelä, Marko M.
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.08.2018 Springer Springer Nature B.V
Subjects:	Algorithms Calculus Computer Science Convergence Convex analysis Differential calculus Mathematics Mathematics and Statistics Methods Mixed integer Nonlinear programming Operations Research/Decision Theory Optimization Properties (attributes) Real Functions Mixed-integer nonlinear programming Cutting planes 90C25 90C11 Nonsmooth optimization Supporting hyperplanes Generalized convexities
ISSN:	0925-5001, 1573-2916
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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 ∘ -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 ∘ -pseudoconvex function, subject to f ∘ -pseudoconvex constraints. With some additional conditions, the proof is also valid for f ∘ -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 ∘ -pseudoconvex objective function.
AbstractList	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 ∘ -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 ∘ -pseudoconvex function, subject to f ∘ -pseudoconvex constraints. With some additional conditions, the proof is also valid for f ∘ -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 ∘ -pseudoconvex objective function. 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 [Formula omitted]-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 [Formula omitted]-pseudoconvex function, subject to [Formula omitted]-pseudoconvex constraints. With some additional conditions, the proof is also valid for [Formula omitted]-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 (See CR12)) to also solve problems with [Formula omitted]-pseudoconvex objective function. 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∘-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∘-pseudoconvex function, subject to f∘-pseudoconvex constraints. With some additional conditions, the proof is also valid for f∘-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∘-pseudoconvex objective function.
Audience	Academic
Author	Westerlund, Tapio Eronen, Ville-Pekka Mäkelä, Marko M.
Author_xml	– sequence: 1 givenname: Tapio orcidid: 0000-0002-8979-9642 surname: Westerlund fullname: Westerlund, Tapio email: twesterlund@abo.fi organization: Faculty of Science and Engineering, Åbo Akademi University, Department of Mathematics and Statistics, University of Turku – sequence: 2 givenname: Ville-Pekka surname: Eronen fullname: Eronen, Ville-Pekka organization: Department of Mathematics and Statistics, University of Turku – sequence: 3 givenname: Marko M. surname: Mäkelä fullname: Mäkelä, Marko M. organization: Department of Mathematics and Statistics, University of Turku
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CitedBy_id	crossref_primary_10_1016_j_compchemeng_2020_106743 crossref_primary_10_1007_s10957_022_02114_y crossref_primary_10_1007_s10898_022_01128_0 crossref_primary_10_1080_02331934_2021_1939337 crossref_primary_10_1007_s10898_020_00906_y crossref_primary_10_1016_j_ejor_2025_07_016 crossref_primary_10_1137_23M1622635 crossref_primary_10_1007_s11081_018_9411_8
Cites_doi	10.1007/s10898-015-0322-3 10.1007/BF01581153 10.1007/BF00138689 10.1007/s10898-004-2704-9 10.1016/0098-1354(95)87027-X 10.1016/S0098-1354(97)00000-8 10.1007/s10898-017-0528-7 10.1016/0098-1354(92)80028-8 10.1023/A:1011241421041 10.1007/BF01099647 10.1016/j.compchemeng.2005.07.012 10.1007/978-1-4614-1927-3 10.1007/BF00934810 10.1023/A:1021091110342 10.1080/02331934.2012.712118 10.1023/A:1021039126272 10.1142/1493 10.1007/s10898-012-9877-4 10.1007/BF02592064 10.1137/S1052623494268455 10.1007/978-3-319-08114-4 10.1007/s11750-016-0413-4 10.1287/opre.15.1.147 10.1002/9780470400531.eorms0527 10.1137/1.9781611970791 10.1016/S1570-7946(00)80002-4
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Copyright	Springer Science+Business Media, LLC, part of Springer Nature 2018 COPYRIGHT 2018 Springer Journal of Global Optimization is a copyright of Springer, (2018). All Rights Reserved.
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SubjectTerms	Algorithms Calculus Computer Science Convergence Convex analysis Differential calculus Mathematics Mathematics and Statistics Methods Mixed integer Nonlinear programming Operations Research/Decision Theory Optimization Properties (attributes) Real Functions
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Title	On solving generalized convex MINLP problems using supporting hyperplane techniques
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