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Exponential Stability of Primal-Dual Gradient Dynamics with Non-Strong Convexity

Xin Chen, Na Li

TL;DR

This work analyzes continuous-time primal-dual gradient dynamics for convex problems with equality constraints, where $f$ is strongly convex and smooth while $g$ is convex and smooth. By constructing two quadratic Lyapunov functions and applying Schur-complement arguments, the authors prove global exponential stability when $g$ is quadratic or when a ge-psd regularity condition holds for $g$ and $B$, with $A$ of full row rank. They also establish local exponential convergence under a Hessian-positive condition at the optimum and provide numerical experiments to validate the theory. The results extend exponential stability guarantees to non-strongly convex objectives in the primal variables, offering practical insights for stable PDGD-based algorithms in resource allocation, networks, and related convex optimization settings.

Abstract

This paper studies the exponential stability of primal-dual gradient dynamics (PDGD) for solving convex optimization problems where constraints are in the form of Ax+By= d and the objective is min f(x)+g(y) with strongly convex smooth f but only convex smooth g. We show that when g is a quadratic function or when g and matrix B together satisfy an inequality condition, the PDGD can achieve global exponential stability given that matrix A is of full row rank. These results indicate that the PDGD is locally exponentially stable with respect to any convex smooth g under a regularity condition. To prove the exponential stability, two quadratic Lyapunov functions are designed. Lastly, numerical experiments further complement the theoretical analysis.

Exponential Stability of Primal-Dual Gradient Dynamics with Non-Strong Convexity

TL;DR

This work analyzes continuous-time primal-dual gradient dynamics for convex problems with equality constraints, where is strongly convex and smooth while is convex and smooth. By constructing two quadratic Lyapunov functions and applying Schur-complement arguments, the authors prove global exponential stability when is quadratic or when a ge-psd regularity condition holds for and , with of full row rank. They also establish local exponential convergence under a Hessian-positive condition at the optimum and provide numerical experiments to validate the theory. The results extend exponential stability guarantees to non-strongly convex objectives in the primal variables, offering practical insights for stable PDGD-based algorithms in resource allocation, networks, and related convex optimization settings.

Abstract

This paper studies the exponential stability of primal-dual gradient dynamics (PDGD) for solving convex optimization problems where constraints are in the form of Ax+By= d and the objective is min f(x)+g(y) with strongly convex smooth f but only convex smooth g. We show that when g is a quadratic function or when g and matrix B together satisfy an inequality condition, the PDGD can achieve global exponential stability given that matrix A is of full row rank. These results indicate that the PDGD is locally exponentially stable with respect to any convex smooth g under a regularity condition. To prove the exponential stability, two quadratic Lyapunov functions are designed. Lastly, numerical experiments further complement the theoretical analysis.

Paper Structure

This paper contains 23 sections, 14 theorems, 72 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem Statement and Preliminaries
  3. Problem Statement
  4. Necessary Conditions
  5. Main Results
  6. Quadratic Case of $g(y)$
  7. General Case of $g(y)$
  8. Local Exponential Convergence
  9. Exponential Stability Analysis
  10. Proof of Theorem \ref{['QUA']}
  11. Proof of Theorem \ref{['thm-psd']}
  12. Numerical Examples
  13. Quadratic Case of $g(y)$
  14. General Case of $g(y)$
  15. Conclusion
  16. ...and 8 more sections

Key Result

Proposition 1

Under assumption fa and finite, any equilibrium point $(\bm{x}^*,\bm{y}^*,\bm{\lambda}^*)$ of the primal-dual gradient dynamics (pdg) is an optimal solution of problem (pro).

Figures (2)

  • Figure 1: Convergence results of PDGD with different time constants when $g(\bm{y})$ is a quadratic function.
  • Figure 2: Convergence results of PDGD when $g(\bm{y}) = \sum_{i}^m y_i^4$.

Theorems & Definitions (14)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Theorem 1
  • Proposition 4
  • Theorem 2
  • Corollary 1
  • Theorem 3
  • Lemma 1
  • Lemma 2
  • ...and 4 more