Quadratic M-convex and L-convex functions

The concepts of L-convexity and M-convexity are introduced by Murota (1996) for functions defined over the integer lattice, and recently extended to polyhedral convex functions by Murota-Shioura (2000). L-convex and M-convex functions are deeply connected with well-solvability in combinatorial optimization problems with convex objective functions. In this paper, we consider these concepts for quadratic functions and the structure of the coefficient matrices of such quadratic functions. It is shown that quadratic L-convex and M-convex functions can be characterized by nice combinatorial properties of their coefficient matrices. The conjugacy relationship between quadratic L-convex and M-convex functions is also discussed.