2018/01/09 by Jérôme Bolte, Shoham Sabach, Bolte, Jérôme +3 · 2 citations
Decision Sciences · Engineering · Mathematics · #49M37 #65K10 #90C30 #Advanced Bandit Algorithms Research #Advanced Optimization Algorithms Research #FOS: Mathematics #Optimization and Control (math.OC) #Sparse and Compressive Sensing Techniques
paper · pdf · doi:10.48550/arxiv.1801.03013
openalex publication_date 2018/01/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce a novel approach addressing global analysis of a difficult class\nof nonconvex-nonsmooth optimization problems within the important framework of\nLagrangian-based methods. This genuine nonlinear class captures many problems\nin modern disparate fields of applications. It features complex geometries,\nqualification conditions, and other regularity properties do not hold\neverywhere. To address these issues we work along several research lines to\ndevelop an original general Lagrangian methodology which can deal, all at once,\nwith the above obstacles. A first innovative feature of our approach is to\nintroduce the concept of Lagrangian sequences for a broad class of algorithms.\nCentral to this methodology is the idea of turning an arbitrary descent method\ninto a multiplier method. Secondly, we provide these methods with a\ntransitional regime allowing us to identify in finitely many steps a zone where\nwe can tune the step-sizes of the algorithm for the final converging regime.\nThen, despite the min-max nature of Lagrangian methods, using an original\nLyapunov method we prove that each bounded sequence generated by the resulting\nmonitoring schemes are globally convergent to a critical point for some\nfundamental Lagrangian-based methods in the broad semialgebraic setting, which\nto the best of our knowledge, are the first of this kind.\n