Mathematical Programming

, Volume 58, Issue 1–3, pp 353–367

# A nonsmooth version of Newton's method

• Liqun Qi
• Jie Sun
Article

## Abstract

Newton's method for solving a nonlinear equation of several variables is extended to a nonsmooth case by using the generalized Jacobian instead of the derivative. This extension includes the B-derivative version of Newton's method as a special case. Convergence theorems are proved under the condition of semismoothness. It is shown that the gradient function of the augmented Lagrangian forC2-nonlinear programming is semismooth. Thus, the extended Newton's method can be used in the augmented Lagrangian method for solving nonlinear programs.

## Key words

Newton's methods generalized Jacobian semismoothness

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© The Mathematical Programming Society, Inc. 1993

## Authors and Affiliations

• Liqun Qi
• 1
• Jie Sun
• 2
1. 1.School of MathematicsThe University of New South WalesKensingtonAustralia
2. 2.Department of Industrial Engineering and Management SciencesNorthwestern UniversityEvanstonUSA