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

Sai Wang, Yi Gong

TL;DR

The paper addresses saddle-point problems with nonlinear coupling operators by formulating a generalized primal-dual correction method (GPD-CM) that dynamically adjusts regularization factors to permit larger step sizes while ensuring convergence. It establishes a unified variational framework and proves an ergodic convergence rate of $O(1/t)$. The method combines a prediction step with a corrective update, enforcing positive definiteness of key matrices and achieving a minimal lower bound for the regularization product when $\alpha=\tfrac{1}{2}$. Numerical experiments on an SPP with exponential coupling demonstrate faster convergence than traditional Arrow-Hurwicz and PDHGM methods and highlight the practical benefits of adaptive regularization in nonlinear coupling contexts.

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.

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

TL;DR

The paper addresses saddle-point problems with nonlinear coupling operators by formulating a generalized primal-dual correction method (GPD-CM) that dynamically adjusts regularization factors to permit larger step sizes while ensuring convergence. It establishes a unified variational framework and proves an ergodic convergence rate of . The method combines a prediction step with a corrective update, enforcing positive definiteness of key matrices and achieving a minimal lower bound for the regularization product when . Numerical experiments on an SPP with exponential coupling demonstrate faster convergence than traditional Arrow-Hurwicz and PDHGM methods and highlight the practical benefits of adaptive regularization in nonlinear coupling contexts.

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 . Numerical results support our theoretical analysis for an SPP with an exponential coupling operator.
Paper Structure (7 sections, 7 theorems, 40 equations, 3 figures, 1 algorithm)

This paper contains 7 sections, 7 theorems, 40 equations, 3 figures, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. Preliminaries
  3. Generalized Primal-Dual Correction Method
  4. Arrow-Hurwicz Method
  5. Generalized Primal-Dual Correction Method
  6. Numerical Experiments
  7. Conclusion

Key Result

Lemma 1

Let $h_{1}(\mathbf{x})$ and $h_{2}(\mathbf{x})$ be convex, and $h_{2}(\mathbf{x})$ be continuously differentiable. If the solution set of the minimization problem $\min\{h_{1}(\mathbf{x})+h_{2}(\mathbf{x})\vert \mathbf{x}\in \mathcal{X}\}$ is nonempty, for a closed convex set $\mathcal{X}\subseteq \ For $\mathbf{x}^{*}\in \mathcal{X}$, we can get the variational inequality

Figures (3)

  • Figure 1: Saddle-point in a saddle-point problem.
  • Figure 2: Different sequence values: Number of iterations versus values of $\alpha$.
  • Figure 3: Comparsion of methods: Sequence value versus the number of iterations.

Theorems & Definitions (8)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Theorem 1
  • Remark 1
  • Lemma 4
  • Lemma 5
  • Theorem 2