Work

Characterization of the Convergence Rate of the Augmented Lagrange for the Nonlinear Semidefinite Optimization Problem

Year: 2025

Type: article

Abstract: The convergence rate of the augmented Lagrangian method (ALM) for solving the nonlinear semidefinite optimization problem is studied. Under the Jacobian uniqueness conditions, when a multiplier vector (π,Y) and the penalty parameter σ are chosen such that σ is larger than a threshold σ*>0 and the ratio ∥(π,Y)−(π*,Y*)∥/σ is small enough, it is demonstrated that the convergence rate of the augmented Lagrange method is linear with respect to ∥(π,Y)−(π*,Y*)∥ and the ratio constant is proportional to 1/σ, where (π*,Y*) is the multiplier corresponding to a local minimizer. Furthermore, by analyzing the second-order derivative of the perturbation function of the nonlinear semidefinite optimization problem, we characterize the rate constant of local linear convergence of the sequence of Lagrange multiplier vectors produced by the augmented Lagrange method. This characterization shows that the sequence of Lagrange multiplier vectors has a Q-linear convergence rate when the sequence of penalty parameters {σk} has an upper bound and the convergence rate is superlinear when {σk} is increasing to infinity.

Source: Mathematics

DOI: 10.3390/math13121946

Authors: Yule Zhang, Jia Wu, Jihong Zhang, Haoyang Liu

Institutions: Dalian Maritime University, Dalian University of Technology, Shenyang Ligong University, Dongbei University of Finance and Economics

Language: English

Topic: Advanced Optimization Algorithms Research

Subfield: Numerical Analysis

Field: Mathematics

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $3,137

Funders: National Natural Science Foundation of China, National Key Research and Development Program of China