Quasi-difference-convexity: Modernization of Quasi-differentiable Optimization

TL;DR

This paper modernizes the concept of quasi-differentiable functions, renaming them quasi-difference-convex functions, and develops unified algorithms with convergence analysis for optimizing this broad class of nonconvex, nondifferentiable functions.

Contribution

It introduces the quasi-difference-convex (quasi-dc) class, unifies iterative convex programming algorithms for their optimization, and establishes convergence properties and rates.

Findings

Unified treatment of descent algorithms for quasi-dc programs

Proven subsequential and sequential convergence of algorithms

Established convergence rates for optimization methods

Abstract

Quasi-differentiable functions were introduced by Pshenichnyi in a 1969 monograph written in Russian and translated in an English version in 1971. This class of nonsmooth functions was studied extensively in two decades since but has not received much attention in today's wide optimization literature. This regrettable omission is in spite of the fact that many functions in modern day applications of optimization can be shown to be quasi-differentiable. In essence, a quasi-differentiable function is one whose directional derivative at an arbitrary reference vector, as a function of the direction, is the difference of two positively homogenous, convex functions. Thus, to bring quasi-differentiable functions closer to the class of difference-of-convex functions that has received fast growing attention in recent years in connection with many applied subjects, we propose to rename quasi-differentiable functions as quasi-difference-convex (quasi-dc) functions. Besides modernizing and advancing this class of nonconvex and nondifferentiable functions, our research aims to put together a unified treatment of iterative convex-programming based descent algorithms for solving a broad class of composite quasi-dc programs and to establish their subsequential convergence, sequential convergence, and rates of convergence; the latter two topics are in line with the modern focus of such analysis for convex programs and some extensions and are departures from the sole emphasis of subsequential convergence in the traditional studies of quasi-differentiable optimization. Through this research, we have gained significant new insights and understanding, advanced the fundamentals, and broadened the applications of this neglected yet pervasive class of nonconvex and nondifferentiable functions and their optimization.