Convergence analysis of accelerated algorithms via a mixed-order dynamical system for separable nonsmooth convex optimization

TL;DR

This paper analyzes the convergence of accelerated algorithms for separable nonsmooth convex optimization problems using a mixed-order dynamical system, introducing new algorithms with proven convergence and strong solution convergence.

Contribution

It introduces new primal-dual and splitting algorithms derived from a mixed-order dynamical system, with convergence analysis and strong convergence to minimal norm solutions.

Findings

Algorithms have nonergodic convergence properties.

Sequences strongly converge to minimal norm solutions.

Numerical experiments validate practical effectiveness.

Abstract

For a linear equality constrained convex optimization problem involving two objective functions with a ``nonsmooth" + ``nonsmooth" composite structure, we study two algorithms derived from a mixed-order dynamical system which incorporates time scales and a Tikhonov regularization term. We observe that different types of multipliers lead to distinct algorithms. For the implicit multiplier and semi-implicit multiplier, we develop a new primal-dual joint algorithm and a new splitting algorithm, respectively. Our proposed joint algorithm can reduce to an algorithm for solving the corresponding non-separable linearly constrained convex optimization problem. Then, we establish the nonergodic convergence properties of all our proposed algorithms. Moreover, we derive that the sequences generated by these algorithms strongly converge to the minimal norm solution. Finally, numerical experiments are conducted to validate the practical performance of the proposed algorithms.