The necessary and sufficient conditions for representing Lipschitz multivariable function as a difference of two convex functions

TL;DR

This paper establishes the necessary and sufficient conditions for representing Lipschitz multivariable functions as differences of two convex functions, providing an algorithm and geometric interpretation for such decompositions.

Contribution

It introduces a complete characterization and an explicit algorithm for representing Lipschitz functions as differences of convex functions in multiple variables.

Findings

Algorithm converges to the convex function pair under given conditions

Provides geometric interpretation of the convex decomposition

Defines a class of curves for the problem's formulation

Abstract

In the article the necessary and sufficient conditions for a representation of Lipschitz function of more than two variables as a difference of two convex functions are formulated. An algorithm of this representation is given. The outcome of this algorithm is a sequence of pairs of convex functions that converge uniformly to a pair of convex functions if the conditions of the formulated theorems are satisfied. A geometric interpretation is also given. In finally, the author considered a class of curves that have the projections on, so called, coordinate planes that bound convex compact sets on such planes. This class of curves is used for formulation the necessary and sufficient conditions for solving this problem.