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A Discrete Convex Min-Max Formula for Box-TDI Polyhedra

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  • András Frank

    (MTA-ELTE Egerváry Research Group, Department of Operations Research, Eötvös University, Budapest H-1117, Hungary)

  • Kazuo Murota

    (School of Business Administration, Tokyo Metropolitan University, Tokyo 192-0397, Japan)

Abstract

A min-max formula is proved for the minimum of an integer-valued separable discrete convex function in which the minimum is taken over the set of integral elements of a box total dual integral polyhedron. One variant of the theorem uses the notion of conjugate function (a fundamental concept in nonlinear optimization), but we also provide another version that avoids conjugates, and its spirit is conceptually closer to the standard form of classic min-max theorems in combinatorial optimization. The presented framework provides a unified background for separable convex minimization over the set of integral elements of the intersection of two integral base-polyhedra, submodular flows, L-convex sets, and polyhedra defined by totally unimodular matrices. As an unexpected application, we show how a wide class of inverse combinatorial optimization problems can be covered by this new framework.

Suggested Citation

  • András Frank & Kazuo Murota, 2022. "A Discrete Convex Min-Max Formula for Box-TDI Polyhedra," Mathematics of Operations Research, INFORMS, vol. 47(2), pages 1026-1047, May.
    • Handle: RePEc:inm:ormoor:v:47:y:2022:i:2:p:1026-1047
    DOI: 10.1287/moor.2021.1160
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    References listed on IDEAS

    as
    1. EDMONDS, Jack & GILES, Rick, 1977. "A min-max relation for submodular functions on graphs," LIDAM Reprints CORE 301, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Satoru Fujishige, 1980. "Lexicographically Optimal Base of a Polymatroid with Respect to a Weight Vector," Mathematics of Operations Research, INFORMS, vol. 5(2), pages 186-196, May.
    3. Clemens Heuberger, 2004. "Inverse Combinatorial Optimization: A Survey on Problems, Methods, and Results," Journal of Combinatorial Optimization, Springer, vol. 8(3), pages 329-361, September.
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