Exponential Stability of Partial Primal-Dual Gradient Dynamics with Nonsmooth Objective Functions

TL;DR

This paper proves the exponential stability of a partial primal-dual gradient dynamic method for convex optimization problems with smooth and nonsmooth components, including affine constraints, and shows how to control convergence rates.

Contribution

It establishes exponential stability of P-PDGD for problems with nonsmooth objectives and provides bounds and regulation methods for convergence rates.

Findings

Proves exponential stability of P-PDGD.

Provides bounds on decay rates.

Shows stepsize can regulate convergence speed.

Abstract

In this paper, we investigate the continuous time partial primal-dual gradient dynamics (P-PDGD) for solving convex optimization problems with the form $ \min\limits_{x\in X,y\in\Omega}\ f({x})+h(y),\ \textit{s.t.}\ A{x}+By=C $, where $ f({x}) $ is strongly convex and smooth, but $ h(y) $ is strongly convex and non-smooth. Affine equality and set constraints are included. We prove the exponential stability of P-PDGD, and bounds on decaying rates are provided. Moreover, it is also shown that the decaying rates can be regulated by setting the stepsize.