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

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 { σ k } has an upper bound and the convergence rate is superlinear when { σ k } is increasing to infinity.