A sequential quadratic penalty method for nonlinear semidefinite programming

Abstract

In this article, a nonlinear semidefinite program is reformulated into a mathematical program with a matrix equality constraint and a sequential quadratic penalty method is proposed to solve the latter problem. We discuss the differentiability and convexity of the penalty function. Necessary and sufficient conditions for the convergence of optimal values of penalty problems to that of the original semidefinite program are obtained. The convergence of optimal solutions of penalty problems to that of the original semidefinite program is also investigated. We show that any limit point of a sequence of stationary points of penalty problems satisfies the KKT optimality condition of the semidefinite program. Smoothed penalty problems that have the same order of smothness as the original semidefinite program are adopted. Corresponding results such as the convexity of the smoothed penalty function, the convergence of optimal values, optimal solutions and the stationary points of the smoothed penalty problems are obtained.

Keywords:

• Semidefinite program
• Penalty method
• Smoothing method
• Optimality condition
• Convergence
• Mathematics Subject Classifications 2000: 90C22
• 49M37

Acknowledgments

This work is supported by the Postdoctoral Fellowship of Hong Kong Polytechnic University. The authors are grateful to a referee for pointing out Ref. [25] and providing constructive remarks on Lemmas 2.1–2.3, which have helped us improve the presentation of this article.

Notes

*Present Address: Department of Applied Mathematics, Hong Kong Polytechnic University, Kowloon, Hong Kong.

Additional information

Notes on contributors

X.X. Huang*

*Present Address: Department of Applied Mathematics, Hong Kong Polytechnic University, Kowloon, Hong Kong.