Please use this identifier to cite or link to this item: https://hdl.handle.net/10419/287440 
Year of Publication: 
2021
Citation: 
[Journal:] Journal of Global Optimization [ISSN:] 1573-2916 [Volume:] 80 [Issue:] 4 [Publisher:] Springer US [Place:] New York, NY [Year:] 2021 [Pages:] 731-756
Publisher: 
Springer US, New York, NY
Abstract: 
The computation of lower bounds via the solution of convex lower bounding problems depicts current state-of-the-art in deterministic global optimization. Typically, the nonlinear convex relaxations are further underestimated through linearizations of the convex underestimators at one or several points resulting in a lower bounding linear optimization problem. The selection of linearization points substantially affects the tightness of the lower bounding linear problem. Established methods for the computation of such linearization points, e.g., the sandwich algorithm, are already available for the auxiliary variable method used in state-of-the-art deterministic global optimization solvers. In contrast, no such methods have been proposed for the (multivariate) McCormick relaxations. The difficulty of determining a good set of linearization points for the McCormick technique lies in the fact that no auxiliary variables are introduced and thus, the linearization points have to be determined in the space of original optimization variables. We propose algorithms for the computation of linearization points for convex relaxations constructed via the (multivariate) McCormick theorems. We discuss alternative approaches based on an adaptation of Kelley's algorithm; computation of all vertices of an n-simplex; a combination of the two; and random selection. All algorithms provide substantial speed ups when compared to the single point strategy used in our previous works. Moreover, we provide first results on the hybridization of the auxiliary variable method with the McCormick technique benefiting from the presented linearization strategies resulting in additional computational advantages.
Subjects: 
Global optimization
McCormick
Linearization
MAiNGO
JEL: 
M20
M37
K05
C26
Persistent Identifier of the first edition: 
Creative Commons License: 
cc-by Logo
Document Type: 
Article
Document Version: 
Published Version

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