Convergence analysis of the augmented Lagrangian method for ℓp-norm cone optimization problems with p≥2

• Published in: Numerical algorithms Vol. 99; no. 3; pp. 1237 - 1267
• Main Authors: Liu, Benqi, Gong, Kai, Zhang, Liwei
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.07.2025; Springer Nature B.V
• Subjects: Algebra; Algorithms; Computer Science; Convergence; Lagrange multiplier; Lagrangian function; Numeric Computing; Numerical Analysis; Optimization; Partial differential equations; Sparsity; Theory of Computation; Jacobian uniqueness conditions; Convergence rate analysis; The augmented Lagrangian method; norm cone optimization problems
• ISSN: 1017-1398; 1572-9265
• DOI: 10.1007/s11075-024-01912-x

This paper focuses on the convergence analysis of the augmented Lagrangian method (ALM) for ℓ p -norm cone optimization problems. We investigate some properties of the augmented Lagrangian function and ℓ p -norm cone. Moreover, under the Jacobian uniqueness conditions, we prove that the local convergence rate of ALM for solving ℓ p -norm cone optimization problems with p ≥ 2 is proportional to 1 / r , where the penalty parameter r is not less than a threshold r ^ . In numerical simulations, we successfully validate the effectiveness and convergence properties of ALM.