Extension of M-convexity and L-convexity to polyhedral convex functions

Kazuo Murota, Akiyoshi Shioura

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33 Citations (Scopus)


The concepts of M-convex and L-convex functions were proposed by Murota in 1996 as two mutually conjugate classes of discrete functions over integer lattice points. M/L-convex functions are deeply connected with the well-solvability in nonlinear combinatorial optimization with integer variables. In this paper, we extend the concept of M-convexity and L-convexity to polyhedral convex functions, aiming at clarifying the well-behaved structure in well-solved nonlinear combinatorial optimization problems in real variables. The extended M/L-convexity often appears in nonlinear combinatorial optimization problems with piecewise-linear convex cost. We investigate the structure of polyhedral M-convex and L-convex functions from the dual viewpoint of analysis and combinatorics and provide some properties and characterizations. It is also shown that polyhedral M/L-convex functions have nice conjugacy relationships.

Original languageEnglish
Pages (from-to)352-427
Number of pages76
JournalAdvances in Applied Mathematics
Issue number4
Publication statusPublished - 2000 Nov
Externally publishedYes


  • Combinatorial optimization; matroid; base polyhedron; convex analysis; polyhedral convex function

ASJC Scopus subject areas

  • Applied Mathematics


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