Global optimization of nonconvex problems with multilinear intermediates

Bao, Xiaowei; Khajavirad, Aida; Sahinidis, Nikolaos V.; Tawarmalani, Mohit

doi:10.1007/s12532-014-0073-z

Cited by 42 publications

(33 citation statements)

References 26 publications

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“…In this type of problem, we can apply (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) independently for each trinomial, with no auxiliary variables at all, choosing the best double-McCormick for each trinomial, whenever the associated volume is close to the volume for P H . We have documented that this can happen quite a lot, and so it is a viable approach.…”

Section: 2mentioning

confidence: 99%

Quantifying Double McCormick

Speakman

Lee²

2017

Mathematics of OR

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When using the standard McCormick inequalities twice to convexify trilinear monomials, as is often the practice in modeling and software, there is a choice of which variables to group first. For the important case in which the domain is a nonnegative box, we calculate the volume of the resulting relaxation, as a function of the bounds defining the box. In this manner, we precisely quantify the strength of the different possible relaxations defined by all three groupings, in addition to the trilinear hull itself. As a by product, we characterize the best double-McCormick relaxation.We wish to emphasize that, in the context of spatial branch-and-bound for factorable formulations, our results do not only apply to variables in the input formulation. Our results apply to monomials that involve auxiliary variables as well. So, our results apply to the product of any three (possibly complicated) expressions in a formulation.Key words : global optimization, mixed-integer nonlinear programming, spatial branch-and-bound, convexification, bilinear, trilinear, McCormick inequalities MSC2000 subject classification : 90C26 1. Introduction. Spatial branch-and-bound (sBB) (see [1,20,25]) for so-called factorable mathematical-optimization formulations (see [15]) is the workhorse general-purpose algorithm in the area of global optimization. It works by using additional variables to reformulate every function of the formulation as a (labeled) directed acyclic graph (DAG). Root nodes can be very complicated functions, and leaves are variables that appear in the input formulation, each labeled with its interval domain. Intermediate nodes are labeled with auxiliary variables together with operators from a small dictionary of basic functions of few (often one or two) variables. Also, we have a method for convexifying the graph of each dictionary function. sBB algorithms work by composing convex relaxations of the dictionary functions, according to the DAG, to get relaxations of the root functions. Bounds on the leaves propagate to other nodes and conversely. Branching (subdividing the domain interval of a variable) creates subproblems, which are treated recursively. Objective bounds for subproblems are appropriately combined to achieve a global-optimization algorithm.Much of the research on sBB has focused on developing tight convexifications for basic functions of few variables (many references can be found in [4]). Other research has focused on how bounds can be efficiently propagated and how branching can be judiciously be carried out (see [3], for example). From the viewpoint of good convexifications, much less attention has been paid to how the DAGs are created, but this can have a strong impact on the quality of the resulting convex relaxation of the input formulation; see [12,13,22,29] for some key papers with other viewpoints concerning constructing DAGs.

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Section: 2mentioning

confidence: 99%

Quantifying Double McCormick

Speakman

Lee²

2017

Mathematics of OR

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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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“…If p(x) is a multilinear polynomial (i.e. α j ∈ {0, 1} for all j) and S is a box, then the envelopes are polyhedral and we know exponential sized extended formulations [Rik97;She97], as well as valid inequalities [CRH17;DPK16] and efficient cutting planes [Bao+15;MSF15] in projected spaces. A second method for obtaining lower bounds on the polynomial optimization problem has been to use the moments approach and Lasserre [Las01] hierarchy of semidefinite relaxations (SDPs) that converges to the global optimum [Las15;Lau09].…”

Section: Department Of Mathematical Sciences Clemson Universitymentioning

confidence: 99%

Error bounds for monomial convexification in polynomial optimization

Adams

Gupte

2018

Math. Program.

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Convex hulls of monomials have been widely studied in the literature, and monomial convexifications are implemented in global optimization software for relaxing polynomials. However, there has been no study of the error in the global optimum from such approaches. We give bounds on the worst-case error for convexifying a monomial over subsets of [0, 1] n . This implies additive error bounds for relaxing a polynomial optimization problem by convexifying each monomial separately. Our main error bounds depend primarily on the degree of the monomial, making them easy to compute. Since monomial convexification studies depend on the bounds on the associated variables, in the second part, we conduct an error analysis for a multilinear monomial over two different types of box constraints. As part of this analysis, we also derive the convex hull of a multilinear monomial over [−1, 1] n .

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Global optimization of nonconvex problems with multilinear intermediates

Cited by 42 publications

References 26 publications

Quantifying Double McCormick

Quantifying Double McCormick

Multivariate McCormick relaxations

Error bounds for monomial convexification in polynomial optimization

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