The Necessary and Sufficient Conditions for Representing Lipschitz Bivariate Functions as a Difference of Two Convex Functions(corrected)

TL;DR

This paper establishes the exact conditions under which Lipschitz bivariate functions can be expressed as a difference of two convex functions, providing an algorithm and geometric interpretation for such representations.

Contribution

It formulates necessary and sufficient conditions for representing Lipschitz functions as a difference of convex functions and introduces an algorithm with convergence guarantees.

Findings

Conditions for representation are both necessary and sufficient.

An algorithm for constructing the convex functions is provided.

The algorithm converges uniformly under the specified conditions.

Abstract

In the article the necessary and sufficient conditions for a representation of Lipschitz function of 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.