A Modified Nonsmooth Levenberg–Marquardt Algorithm for the General Mixed Complementarity Problem

As is well known, the mixed complementarity problem is equivalent to a nonsmooth equation by using a median function. By investigating the generalized Jacobi of a composite vector-valued maximum function, a nonsmooth Levenberg–Marquardt algorithm is proposed in this paper. In the present algorithm, we adopt a new LM parameter form and discuss the local convergence rate under the local error bound condition, which is weaker than nonsingularity. Finally, the numerical experiments and the application for the real-time pricing in smart grid illustrate the effectiveness of the algorithm.

Five axioms for location functions on median graphs

In previous work, two axiomatic characterizations were given for the median function on median graphs: one involving the three simple and natural axioms anonymity, betweenness and consistency; the other involving faithfulness, consistency and [Formula: see text]-Condorcet. To date, the independence of these axioms has not been a serious point of study. The aim of this paper is to provide the missing answers. The independent subsets of these five axioms are determined precisely and examples provided in each case on arbitrary median graphs. There are three cases that stand out. Here nontrivial examples and proofs are needed to give a full answer. Extensive use of the structure of median graphs is used throughout.

The median function on Boolean lattices

A median of a sequence π = ( x1, x2, …, xk) of elements of a finite metric space (X, d) is an element x ∈ X for which [Formula: see text] is a minimum. The function Median with domain the set of all finite sequences on X and defined by Med (π) = {x : x is a median of π} is called the median function on X. In this paper, the median function on finite Boolean lattices is axiomatically characterized via location functions.

THE MEDIAN FUNCTION ON TREES

A median of a sequence π = (x1, x2, …, xk) of elements of a finite metric space (X, d) is an element x for which [Formula: see text] is minimum. The function with domain the set of all finite sequences on X and defined by Med(π) = {x | x is a median of π} is called the Median function on X. In this note, an axiomatic characterization of the median function on finite trees is given.

AXIOMATIC CHARACTERIZATION OF THE ANTIMEDIAN FUNCTION ON PATHS AND HYPERCUBES

An antimedian of a profile π = (x1, x2, …, xk) of vertices of a graph G is a vertex maximizing the sum of the distances to the elements of the profile. The antimedian function is defined on the set of all profiles on G and has as output the set of antimedians of a profile. It is a typical location function for finding a location for an obnoxious facility. The 'converse' of the antimedian function is the median function, where the distance sum is minimized. The median function is well studied. For instance it has been characterized axiomatically by three simple axioms on median graphs. The median function behaves nicely on many classes of graphs. In contrast the antimedian function does not have a nice behavior on most classes. So a nice axiomatic characterization may not be expected. In this paper such a characterization is obtained for two classes of graphs on which the antimedian is well behaved: paths and hypercubes.