Regular Article: Extension of M-Convexity and L-Convexity to Polyhedral Convex Functions

Authors:
Kazuo Murota;Akiyoshi Shioura
Affiliations:
Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japanf1murota@kurims.kyoto-u.ac.jpf1;Department of Mechanical Engineering, Sophia University, 7-1 Kioi-cho, Chiyoda-ku, Tokyo, 102-8554, Japanf2shioura@me.sophia.ac.jpf2
Venue:
Advances in Applied Mathematics
Year:
2000

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Abstract

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.