Monomial-wise optimal separable underestimators for mixed-integer polynomial optimization

We introduce a new method for solving box-constrained mixed-integer polynomial problems to global optimality. The approach, a specialized branch-and-bound algorithm, is based on the computation of lower bounds provided by the minimization of separable underestimators of the polynomial objective func...

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Published in:	Journal of global optimization Vol. 67; no. 4; pp. 759 - 786
Main Authors:	Buchheim, Christoph, D’Ambrosio, Claudia
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.04.2017 Springer Springer Nature B.V
Subjects:	Algorithms Branch & bound algorithms Computation Computer Science Linear programming Mathematical analysis Mathematical models Mathematics Mathematics and Statistics Nonlinear programming Operations Research/Decision Theory Optimization Polynomials Real Functions Solvers Studies Polynomial optimization Mixed-integer nonlinear programming Separable underestimators
ISSN:	0925-5001, 1573-2916
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Abstract	We introduce a new method for solving box-constrained mixed-integer polynomial problems to global optimality. The approach, a specialized branch-and-bound algorithm, is based on the computation of lower bounds provided by the minimization of separable underestimators of the polynomial objective function. The underestimators are the novelty of the approach because the standard approaches in global optimization are based on convex relaxations. Thanks to the fact that only simple bound constraints are present, minimizing the separable underestimator can be done very quickly. The underestimators are computed monomial-wise after the original polynomial has been shifted. We show that such valid underestimators exist and their degree can be bounded when the degree of the polynomial objective function is bounded, too. For the quartic case, all optimal monomial underestimators are determined analytically. We implemented and tested the branch-and-bound algorithm where these underestimators are hardcoded. The comparison against standard global optimization and polynomial optimization solvers clearly shows that our method outperforms the others, the only exception being the binary case where, though, it is still competitive. Moreover, our branch-and-bound approach suffers less in case of dense polynomial objective function, i.e., in case of polynomials having a large number of monomials. This paper is an extended and revised version of the preliminary paper [ 4 ].
AbstractList	We introduce a new method for solving box-constrained mixed-integer polynomial problems to global optimality. The approach, a specialized branch-and-bound algorithm, is based on the computation of lower bounds provided by the minimization of separable underestimators of the polynomial objective function. The underestimators are the novelty of the approach because the standard approaches in global optimization are based on convex relaxations. Thanks to the fact that only simple bound constraints are present, minimizing the separable underestimator can be done very quickly. The underestimators are computed monomial-wise after the original polynomial has been shifted. We show that such valid underestimators exist and their degree can be bounded when the degree of the polynomial objective function is bounded, too. For the quartic case, all optimal monomial underestimators are determined analytically. We implemented and tested the branch-and-bound algorithm where these underestimators are hardcoded. The comparison against standard global optimization and polynomial optimization solvers clearly shows that our method outperforms the others, the only exception being the binary case where, though, it is still competitive. Moreover, our branch-and-bound approach suffers less in case of dense polynomial objective function, i.e., in case of polynomials having a large number of monomials. This paper is an extended and revised version of the preliminary paper [ 4 ]. We introduce a new method for solving box-constrained mixed-integer polynomial problems to global optimality. The approach, a specialized branch-and-bound algorithm, is based on the computation of lower bounds provided by the minimization of separable underestimators of the polynomial objective function. The underestimators are the novelty of the approach because the standard approaches in global optimization are based on convex relaxations. Thanks to the fact that only simple bound constraints are present, minimizing the separable underestimator can be done very quickly. The underestimators are computed monomial-wise after the original polynomial has been shifted. We show that such valid underestimators exist and their degree can be bounded when the degree of the polynomial objective function is bounded, too. For the quartic case, all optimal monomial underestimators are determined analytically. We implemented and tested the branch-and-bound algorithm where these underestimators are hardcoded. The comparison against standard global optimization and polynomial optimization solvers clearly shows that our method outperforms the others, the only exception being the binary case where, though, it is still competitive. Moreover, our branch-and-bound approach suffers less in case of dense polynomial objective function, i.e., in case of polynomials having a large number of monomials. This paper is an extended and revised version of the preliminary paper [4 (See CR4)]. We introduce a new method for solving box-constrained mixed-integer polynomial problems to global optimality. The approach, a specialized branch-and-bound algorithm, is based on the computation of lower bounds provided by the minimization of separable underestimators of the polynomial objective function. The underestimators are the novelty of the approach because the standard approaches in global optimization are based on convex relaxations. Thanks to the fact that only simple bound constraints are present, minimizing the separable underestimator can be done very quickly. The underestimators are computed monomial-wise after the original polynomial has been shifted. We show that such valid underestimators exist and their degree can be bounded when the degree of the polynomial objective function is bounded, too. For the quartic case, all optimal monomial underestimators are determined analytically. We implemented and tested the branch-and-bound algorithm where these underestimators are hardcoded. The comparison against standard global optimization and polynomial optimization solvers clearly shows that our method outperforms the others, the only exception being the binary case where, though, it is still competitive. Moreover, our branch-and-bound approach suffers less in case of dense polynomial objective function, i.e., in case of polynomials having a large number of monomials. This paper is an extended and revised version of the preliminary paper [ 4 ].
Audience	Academic
Author	D’Ambrosio, Claudia Buchheim, Christoph
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Cites_doi	10.1007/s10898-014-0166-2 10.1080/10556780903087124 10.1007/s12532-010-0010-8 10.1080/10556780802699201 10.1007/s10107-005-0581-8 10.1007/s10107-004-0550-7 10.1017/S0962492913000032 10.1137/05064504X 10.1137/120878495 10.1007/s10288-011-0181-9 10.1016/j.compchemeng.2014.07.020 10.1007/BF01070233 10.1007/s10479-012-1272-5 10.1007/s10898-012-9974-4 10.1007/s10107-012-0534-y 10.1007/s10107-010-0381-7 10.1137/050646500 10.1090/dimacs/060/08 10.1007/978-3-319-07557-0_17 10.1007/978-3-642-38527-8_22
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Snippet	We introduce a new method for solving box-constrained mixed-integer polynomial problems to global optimality. The approach, a specialized branch-and-bound...
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SubjectTerms	Algorithms Branch & bound algorithms Computation Computer Science Linear programming Mathematical analysis Mathematical models Mathematics Mathematics and Statistics Nonlinear programming Operations Research/Decision Theory Optimization Polynomials Real Functions Solvers Studies
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Title	Monomial-wise optimal separable underestimators for mixed-integer polynomial optimization
URI	https://link.springer.com/article/10.1007/s10898-016-0443-3 https://www.proquest.com/docview/1876683552 https://www.proquest.com/docview/1893881328
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