The method of codifferential descent for convex and global piecewise affine optimization

TL;DR

This paper analyzes the performance of the codifferential descent method for nonsmooth convex and piecewise affine functions, establishing iteration bounds and proposing a global optimization variant that guarantees finite convergence.

Contribution

It provides the first theoretical iteration complexity bounds for the codifferential descent method on convex functions and introduces a global method for piecewise affine functions with finite convergence.

Findings

MCD has an iteration complexity of O(1/ε) for certain convex functions.

A new global optimality condition for piecewise affine functions is established.

The modified MCD finds a global minimum of nonconvex piecewise affine functions in finite steps.

Abstract

The class of nonsmooth codifferentiable functions was introduced by professor V.F.~Demyanov in the late 1980s. He also proposed a method for minimizing these functions called the method of codifferential descent (MCD). However, until now almost no theoretical results on the performance of this method on particular classes of nonsmooth optimization problems were known. In the first part of the paper, we study the performance of the method of codifferential descent on a class of nonsmooth convex functions satisfying some regularity assumptions, which in the smooth case are reduced to the Lipschitz continuity of the gradient. We prove that in this case the MCD has the iteration complexity bound $\mathcal{O}(1 / \varepsilon)$. In the second part of the paper we obtain new global optimality conditions for piecewise affine functions in terms of codifferentials. With the use of these conditions we propose a modification of the MCD for minimizing piecewise affine functions (called the method of global codifferential descent) that does not use line search, and discards those "pieces" of the objective functions that are no longer useful for the optimization process. Then we prove that the MCD as well as its modification proposed in the article find a point of global minimum of a nonconvex piecewise affine function in a finite number of steps.