Rate of Convergence of the Bundle Method

TL;DR

This paper establishes a convergence rate for the bundle method in nonsmooth optimization, showing it reaches a specified accuracy within a logarithmic and inverse proportional number of iterations for strongly convex functions.

Contribution

It provides the first proven iteration complexity bounds for the bundle method with multiple cuts and cut aggregation in strongly convex nonsmooth optimization.

Findings

Convergence rate of O(ln(1/ε)/ε) for strongly convex functions

Applicable to bundle method variants with multiple cuts and cut aggregation

Improves understanding of efficiency in nonsmooth optimization algorithms

Abstract

We prove that the bundle method for nonsmooth optimization achieves solution accuracy $\varepsilon$ in at most $\mathcal{O}\big(\ln(1/\varepsilon)/\varepsilon\big)$ iterations, if the function is strongly convex. The result is true for the versions of the method with multiple cuts and with cut aggregation.