TY - JOUR
T1 - Extension of M-convexity and L-convexity to polyhedral convex functions
AU - Murota, Kazuo
AU - Shioura, Akiyoshi
N1 - Funding Information:
1This work is supported by Grant-in-Aid of the Ministry of Education, Science, Sports and Culture of Japan.
PY - 2000/11
Y1 - 2000/11
N2 - 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.
AB - 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.
KW - Combinatorial optimization; matroid; base polyhedron; convex analysis; polyhedral convex function
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U2 - 10.1006/aama.2000.0702
DO - 10.1006/aama.2000.0702
M3 - Article
AN - SCOPUS:0013533304
VL - 25
SP - 352
EP - 427
JO - Advances in Applied Mathematics
JF - Advances in Applied Mathematics
SN - 0196-8858
IS - 4
ER -