Lagrange optimality system for a class of nonsmooth convex optimization

Abstract

In this paper, we revisit the augmented Lagrangian method for a class of nonsmooth convex optimization. We present the Lagrange optimality system of the augmented Lagrangian associated with the problems, and establish its connections with the standard optimality condition and the saddle point condition of the augmented Lagrangian, which provides a powerful tool for developing numerical algorithms: we derive a Lagrange–Newton algorithm for the nonsmooth convex optimization, and establish the nonsingularity of the Newton system and the local convergence of the algorithm.

Keywords:

• nonsmooth convex optimization
• augmented Lagrangian method
• Lagrange optimality system
• Newton method

AMS Subject Classifications:

• 90C25
• 46N10
• 49M15

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research was supported by the Aihara Innovative Mathematical Modelling Project, the Japan Society for the Promotion of Science (JSPS) through the ‘Funding Program for World-Leading Innovative R&D on Science and Technology (FIRST Program)’, initiated by the Council for Science and Technology Policy (CSTP).