2025/03/23 by Oikonomidis, Konstantinos, Laude, Emanuel, Patrinos, Panagiotis · 3 citations
#FOS: Mathematics #Optimization and Control (math.OC)
paper · doi:10.48550/arxiv.2503.18098
In this paper we present a unifying framework for continuous optimization methods grounded in the concept of generalized convexity. Utilizing the powerful theory of Φ-convexity, we propose a conceptual algorithm that extends the classical difference-of-convex method, encompassing a broad spectrum of optimization algorithms. Relying exclusively on the tools of generalized convexity we develop a gap function analysis that strictly characterizes the decrease of the function values, leading to simplified and unified convergence results. As an outcome of this analysis, we naturally obtain a generalized PL inequality which ensures q-linear convergence rates of the proposed method, incorporating various well-established conditions from the existing literature. Moreover we propose a Φ-Bregman proximal point interpretation of the scheme that allows us to capture conditions that lead to sublinear rates under convexity.