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Equivalence of convex minimization problems over base polytopes

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Abstract

This paper considers convex optimization problems over base polytopes of polymatroids. We show that the decomposition algorithm for the separable convex function minimization problems helps us derive simple sufficient conditions for the rationality of optimal solutions, and leads us to some useful properties, including the equivalence of the lexicographically optimal base problem, introduced by Fujishige, and the submodular utility allocation market problem, introduced by Jain and Vazirani. In addition, we deal with a class of non-separable convex objective functions. Moreover, we describe an algorithm for the lexicographically optimal base problem, which is a variant of the Fujishige–Wolfe algorithm.

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Authors and Affiliations

  1. Collaborative Research Center for Innovative Mathematical Modelling, Institute of Industrial Science, University of Tokyo, Tokyo, 153-8505, Japan

    Kiyohito Nagano

  2. Institute of Industrial Science, University of Tokyo, Tokyo, 153-8505, Japan

    Kazuyuki Aihara

Authors
  1. Kiyohito Nagano
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  2. Kazuyuki Aihara
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Correspondence to Kiyohito Nagano.

Additional information

This research is supported by Aihara Project, the FIRST program from JSPS, and KAKENHI (22700007). A part of this work has appeared in Proceedings of the 12th IPCO Conference, Lecture Notes in Computer Science, LNCS 4513, Springer (2007), pp. 252–266.

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Nagano, K., Aihara, K. Equivalence of convex minimization problems over base polytopes. Japan J. Indust. Appl. Math. 29, 519–534 (2012). https://doi.org/10.1007/s13160-012-0083-z

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  • DOI: https://doi.org/10.1007/s13160-012-0083-z

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