Convergence rates analysis of Interior Bregman Gradient Method for Vector Optimization Problems

TL;DR

This paper extends the analysis of Bregman gradient methods to vector optimization problems, establishing convergence rates under various convexity conditions and introducing a linear convergence result for problems satisfying a vector Bregman-PL inequality.

Contribution

It generalizes the convergence analysis of Bregman gradient methods from scalar to vector optimization problems, including new linear convergence results.

Findings

Global convergence rate of O(1/k) for convex VOPs

Linear convergence rate for relative strongly convex VOPs

Linear convergence under vector Bregman-PL inequality

Abstract

In recent years, by using Bregman distance, the Lipschitz gradient continuity and strong convexity were lifted and replaced by relative smoothness and relative strong convexity. Under the mild assumptions, it was proved that gradient methods with Bregman regularity converge linearly for single-objective optimization problems (SOPs). In this paper, we extend the relative smoothness and relative strong convexity to vector-valued functions and analyze the convergence of an interior Bregman gradient method for vector optimization problems (VOPs). Specifically, the global convergence rates are $\mathcal{O}(\frac{1}{k})$ and $\mathcal{O}(r^{k})(0<r<1)$ for convex and relative strongly convex VOPs, respectively. Moreover, the proposed method converges linearly for VOPs that satisfy a vector Bregman-PL inequality.