On the Composition of Convex Envelopes for Quadrilinear Terms

Belotti, Pietro; Cafieri, Sonia; Lee, Jon; Liberti, Leo; Miller, Andrew J.

doi:10.1007/978-1-4614-5131-0_1

Cited by 5 publications

(6 citation statements)

References 40 publications

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“…For a quadrilinear function x 1 x 2 x 3 x 4 , the two groupings ((x 1 x 2 )x 3 )x 4 and (x 1 x 2 x 3 )x 4 result in two different convex relaxations. In [3], we establish that the first composition results in a stronger relaxation and provide empirical evidence to that fact as well.…”

Section: Convex Envelopessupporting

confidence: 60%

See 1 more Smart Citation

Final Report---Next-Generation Solvers for Mixed-Integer Nonlinear Programs: Structure, Search, and Implementation

Linderoth¹,

Luedtke²

2013

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The mathematical modeling of systems often requires the use of both nonlinear and discrete components. Problems involving both discrete and nonlinear components are known as mixed-integer nonlinear programs (MINLPs) and are among the most challenging computational optimization problems. This research project added to the understanding of this area by making a number of fundamental advances. First, the work demonstrated many novel, strong, tractable relaxations designed to deal with non-convexities arising in mathematical formulation. Second, the research implemented the ideas in software that is available to the public. Finally, the work demonstrated the importance of these ideas on practical applications and disseminated the work through scholarly journals, survey publications, and conference presentations.

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Section: Convex Envelopessupporting

confidence: 60%

“…The publications [5,3,4,13,19,20,18,14,10,11,21,1,9,2,6,7,12] were all produced with support of from the grant award DE-FG02-08ER25861.…”

Section: Publicationsmentioning

confidence: 99%

Final Report---Next-Generation Solvers for Mixed-Integer Nonlinear Programs: Structure, Search, and Implementation

Linderoth¹,

Luedtke²

2013

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“…The class of functions considered herein can be summarized as G(z) = t∈T c t i∈I t f i (z), where T and I t ⊂ I are index sets and c t are constants. Such functions can be handled by recursive application of McCormick's product rule and these approaches give weaker than possible relaxations, compare for instance [4,5,9,25]. In contrast, Theorem 2 provides the framework to directly handle such terms and provide tighter relaxations.…”

Section: Corollary 5 Let G(z)mentioning

confidence: 99%

Multivariate McCormick relaxations

Tsoukalas

Mitsos

2014

J Glob Optim

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McCormick (Math Prog 10(1): 1976) provides the framework for convex/concave relaxations of factorable functions, via rules for the product of functions and compositions of the form F • f , where F is a univariate function. Herein, the composition theorem is generalized to allow multivariate outer functions F, and theory for the propagation of subgradients is presented. The generalization interprets the McCormick relaxation approach as a decomposition method for the auxiliary variable method. In addition to extending the framework, the new result provides a tool for the proof of relaxations of specific functions. Moreover, a direct consequence is an improved relaxation for the product of two functions, at least as tight as McCormick's result, and often tighter. The result also allows the direct relaxation of multilinear products of functions. Furthermore, the composition result is applied to obtain improved convex underestimators for the minimum/maximum and the division of two functions for which current relaxations are often weak. These cases can be extended to allow composition of a variety of functions for which relaxations have been proposed.

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“…Convex hull Floudas (2003, 2004) Quadrilinear Cafieri et al (2010);Belotti et al (2013a) Multilinear…”

Section: Floudas Andmentioning

confidence: 99%

Global optimization advances in Mixed-Integer Nonlinear Programming, MINLP, and Constrained Derivative-Free Optimization, CDFO

Boukouvala

Misener

Floudas

2016

European Journal of Operational Research

194

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This manuscript reviews recent advances in deterministic global optimization for Mixed-Integer Nonlinear Programming (MINLP), as well as Constrained DerivativeFree Optimization (CDFO). This work provides a comprehensive and detailed literature review in terms of significant theoretical contributions, algorithmic developments, software implementations and applications for both MINLP and CDFO. Both research areas have experienced rapid growth, with a common aim to solve a wide range of real-world problems. We show their individual prerequisites, formulations and applicability, but also point out possible points of interaction in problems which contain hybrid characteristics. Finally, an inclusive and complete test suite is provided for both MINLP and CDFO algorithms, which is useful for future benchmarking.

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On the Composition of Convex Envelopes for Quadrilinear Terms

Cited by 5 publications

References 40 publications

Final Report---Next-Generation Solvers for Mixed-Integer Nonlinear Programs: Structure, Search, and Implementation

Final Report---Next-Generation Solvers for Mixed-Integer Nonlinear Programs: Structure, Search, and Implementation

Multivariate McCormick relaxations

Global optimization advances in Mixed-Integer Nonlinear Programming, MINLP, and Constrained Derivative-Free Optimization, CDFO

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