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Reverse propagation of McCormick relaxations

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Abstract

Constraint propagation techniques have heavily utilized interval arithmetic while the application of convex and concave relaxations has been mostly restricted to the domain of global optimization. Here, reverse McCormick propagation, a method to construct and improve McCormick relaxations using a directed acyclic graph representation of the constraints, is proposed. In particular, this allows the interpretation of constraints as implicitly defining set-valued mappings between variables, and allows the construction and improvement of relaxations of these mappings. Reverse McCormick propagation yields potentially tighter enclosures of the solutions of constraint satisfaction problems than reverse interval propagation. Ultimately, the relaxations of the objective of a non-convex program can be improved by incorporating information about the constraints.

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Notes

  1. This representation of a factorable function is also used in the reverse mode of automatic differentiation [20].

  2. Hereafter, we will not make this distinction explicitly in expressions. Rather it is always assumed tacitly.

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Authors and Affiliations

  1. Process Systems Engineering Laboratory, Massachusetts Institute of Technology, 77 Massachusetts Ave, Rm 66-363, Cambridge, MA, 02139, USA

    Achim Wechsung, Harry A. J. Watson & Paul I. Barton

  2. Department of Chemical and Biomolecular Engineering, Clemson University, 207B Earle Hall, Clemson, SC, 29634, USA

    Joseph K. Scott

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  1. Achim Wechsung
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Wechsung, A., Scott, J.K., Watson, H.A.J. et al. Reverse propagation of McCormick relaxations. J Glob Optim 63, 1–36 (2015). https://doi.org/10.1007/s10898-015-0303-6

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