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A general framework for convexity analysis in deterministic global optimization

Author

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  • Ralph Kearfott
  • Jessie Castille
  • Gaurav Tyagi

Abstract

In previous work, we, and also Epperly and Pistikopoulos, proposed an analysis of general nonlinear programs that identified certain variables as convex, not ever needing subdivision, and non-convex, or possibly needing subdivision in branch and bound algorithms. We proposed a specific algorithm, based on a generated computational graph of the problem, for identifying such variables. In our previous work, we identified only independent variables in the computational graph. Here, we examine alternative sets of non-convex variables consisting not just of independent variables, but of a possibly smaller number of intermediate variables. We do so with examples and theorems. We also apply variants of our proposed analysis to the well-known COCONUT Lib-1 test set. If the number of such non-convex variables is sufficiently small, it may be possible to fully subdivide them before analysis of ranges of objective and constraints, thus dispensing totally with the branch and bound process. Advantages to such a non-adaptive process include higher predictability and easier parallizability. We present an algorithm and exploratory results here, with a more complete empirical study in a subsequent paper. Copyright Springer Science+Business Media, LLC. 2013

Suggested Citation

  • Ralph Kearfott & Jessie Castille & Gaurav Tyagi, 2013. "A general framework for convexity analysis in deterministic global optimization," Journal of Global Optimization, Springer, vol. 56(3), pages 765-785, July.
    • Handle: RePEc:spr:jglopt:v:56:y:2013:i:3:p:765-785
    DOI: 10.1007/s10898-012-9905-4
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    Cited by:

    1. Jared T. Guilbeau & Md. Istiaq Hossain & Sam D. Karhbet & Ralph Baker Kearfott & Temitope S. Sanusi & Lihong Zhao, 2017. "A review of computation of mathematically rigorous bounds on optima of linear programs," Journal of Global Optimization, Springer, vol. 68(3), pages 677-683, July.
    2. Matthew E. Wilhelm & Chenyu Wang & Matthew D. Stuber, 2023. "Convex and concave envelopes of artificial neural network activation functions for deterministic global optimization," Journal of Global Optimization, Springer, vol. 85(3), pages 569-594, March.

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