Convergence Rates of Subgradient Methods for Quasi-convex Optimization Problems

TL;DR

This paper analyzes the convergence rates of various subgradient methods for quasi-convex optimization, providing unified theoretical results and establishing linear or sublinear convergence under specific conditions.

Contribution

It offers a unified framework for understanding iteration complexity and convergence rates of subgradient methods in quasi-convex optimization, including new results under weak sharp minima assumptions.

Findings

Established convergence rates for subgradient methods under different stepsize rules.

Proved linear or sublinear convergence under weak sharp minima and Hölderian conditions.

Applied results to standard, inexact, and conditional subgradient methods.

Abstract

Quasi-convex optimization acts a pivotal part in many fields including economics and finance; the subgradient method is an effective iterative algorithm for solving large-scale quasi-convex optimization problems. In this paper, we investigate the iteration complexity and convergence rates of various subgradient methods for solving quasi-convex optimization in a unified framework. In particular, we consider a sequence satisfying a general (inexact) basic inequality, and investigate the global convergence theorem and the iteration complexity when using the constant, diminishing or dynamic stepsize rules. More importantly, we establish the linear (or sublinear) convergence rates of the sequence under an additional assumption of weak sharp minima of H\"{o}lderian order and upper bounded noise. These convergence theorems are applied to establish the iteration complexity and convergence rates of several subgradient methods, including the standard/inexact/conditional subgradient methods, for solving quasi-convex optimization problems under the assumptions of the H\"{o}lder condition and/or the weak sharp minima of H\"{o}lderian order.