A Generalized Primal-Dual Correction Method for Saddle-Point Problems with a Nonlinear Coupling Operator

TL;DR

This paper introduces a generalized primal-dual correction method for saddle-point problems with nonlinear coupling, enabling larger step sizes and achieving optimal regularization with proven convergence rates.

Contribution

It proposes a dynamic regularization adjustment technique within primal-dual methods, achieving minimal regularization bounds and improved convergence for nonlinear saddle-point problems.

Findings

Achieves the minimum theoretical lower bound of regularization factors.

Allows larger step sizes while maintaining convergence.

Demonstrates $O(1/t)$ ergodic convergence rate theoretically and numerically.

Abstract

The saddle-point problems (SPPs) with nonlinear coupling operators frequently arise in various control systems, such as dynamic programming optimization, H-infinity control, and Lyapunov stability analysis. However, traditional primal-dual methods are constrained by fixed regularization factors. In this paper, a novel generalized primal-dual correction method (GPD-CM) is proposed to adjust the values of regularization factors dynamically. It turns out that this method can achieve the minimum theoretical lower bound of regularization factors, allowing for larger step sizes under the convergence condition being satisfied. The convergence of the GPD-CM is directly achieved through a unified variational framework. Theoretical analysis shows that the proposed method can achieve an ergodic convergence rate of $O(1/t)$. Numerical results support our theoretical analysis for an SPP with an exponential coupling operator.