Convex proximal bundle methods in depth: a unified analysis for inexact oracles (Q484139)

The paper of W. de Oliveira, C. Sagastizabal and C. Lemarechal is a valuable contribution to the nonlinear functional analysis and algorithms for minimizing a convex function in n-dimensional Euclidean space. In fact, this deterministic problem is treated as a stochastic one by that the authors employ an oracle, refining the well-known bundle method, whereby the oracle involves noise (stochasticity). Since the paper also is from numerical mathematics, that step of generalization requires from the authors a careful new work on proving convergence. In fact, recent years have witnessed the rise of a new paradigm of bundle methods which is able to handle inexact oracles, strongly affected by ``noise''. Providing a convergence proof of a bundle method is never easy and the technicalities strongly increase even when additionally coping with inexact oracles. In addition, there are several versions to handle noise, each of them requires an ad hoc proof for showing convergence. The authors present a synthetic convergence theory where they highlight the core arguments and specify which supposition is employed to provide each intermediate result. The setting is comprehensive and in various ways it extends several algorithms which have been proposed in the literature. Based on the ingredients of the authors' synthetic theory, they reflect on various bundle methods that are adapted to oracles for which a high accuracy is possible. But it is preferable yet to not often make exact calculations, as they are too time consuming. This excellent paper is well structured, mathematically deep, demonstrated carefully and well written. The seven sections of this paper are as follows: 1. Introduction and general aim, 2. Oracle examples, 3. Parametric characterization of bundle methods, 4. Main ingredients in the algorithm, 5. On noise management and a concrete instance, 6. Convergence analysis, and 7. Instances. Further analytic and algorithmic generalizations and refinements, strong results and numerical codes may be expected in future, inspired by this research article in our scientific community. This progress could initiate and support important achievements in science, economics, engineering and Operational Research and, eventually, lead to improvements of living conditions of our people.