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Simplicial decomposition in nonlinear programming algorithms

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Abstract

Simplicial decomposition is a special version of the Dantzig—Wolfe decomposition principle, based on Carathéodory's theorem. The associated class of algorithms has the following features and advantages: The master and the subprogram are constructed without dual variables; the methods remain therefore well-defined for non-concave objective functions, and pseudo-concavity suffices for convergence to global maxima. The subprogram produces affinely independent sets of feasible generator points defining simplices, which the master program keeps minimal by dropping redundant generator points and finding maximizers in the relative interiors of the resulting subsimplices. The use of parallel subspaces allows the direct application of any unrestricted optimization method in the master program; thus the best unconstrained procedure for any type of objective function can be used to find constrained maximizers for it.

The paper presents the theory for this class of algorithms, the APL-code of a “demonstration” method and some computational experience with Colville's test problems.

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Authors and Affiliations

  1. University of Alberta, Edmonton, Alberta, Canada

    Balder Von Hohenbalken

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  1. Balder Von Hohenbalken
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Additional information

I am grateful to Philip Wolfe for encouraging me to write this paper, and I am indebted to him and a referee for helpful comments.

Research was partially supported by a grant of the University of Alberta.

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Von Hohenbalken, B. Simplicial decomposition in nonlinear programming algorithms. Mathematical Programming 13, 49–68 (1977). https://doi.org/10.1007/BF01584323

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  • DOI: https://doi.org/10.1007/BF01584323

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