Augmented Lagrangian method for nonlinear circular conic programs: a local convergence analysis

Abstract

In this paper, we analyse a local convergence of augmented Lagrangian method (ALM) for a class of nonlinear circular conic optimization problems. In light of the singular value decomposition, the Debreu theorem and the implicit function theorem, we prove that the sequence generated by ALM converges to a local minimizer in the linear convergence rate under the constraint nondegeneracy condition and the strong second-order sufficient condition, in which the ratio constant is proportional to $1/\tau$, where τ is the associated penalty parameter with a given lower threshold. As a byproduct, we also derive explicit expressions of critical cone and its affine hull for the given nonlinear circular conic program.

Keywords:

• Circular cone
• augmented Lagrangian method
• local convergence

2020 Mathematics Subject Classifications:

• 26A27
• 26B35
• 49J52
• 65K10

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first author's work is supported by National Natural Science Foundation of China (Grant Number: 11601389). The second author's work is supported by Postgraduate Research Innovation Project of Tianjin Normal University (Grant No. 2023KYCX054Y). The third author's work is supported by Natural Science Foundation of Tianjin (Grant Number: 19JCQNJC03700). The fourth author's work is supported by National Science and Technology Council, Taiwan.