Improved convergence analysis of a smoothing Newton method for the circular cone programming

ABSTRACT

In this paper, we propose a new smoothing Newton method to solve the circular cone programming (denoted by CCP). The proposed method is designed based on a non-monotone derivative-free line search scheme. We show that any accumulation point of the iteration sequence generated by this method is a solution of the CCP. Moreover, we prove that the proposed method is locally quadratically convergent without requiring strict complementarity conditions. Compared with existing smoothing Newton methods for solving the CCP, our method has three new features: (i) the generated iteration sequence is bounded; (ii) the value of the merit function converges to zero; (iii) the whole iteration sequence converges to an accumulation point if this point is isolated. Some numerical results are reported.

Keywords:

• Circular cone programming
• smoothing Newton method
• derivative-free line search
• quadratical convergence

Mathematics Subject Classifications 2000:

• 90C05
• 90C22
• 90C25
• 90C30

Acknowledgments

This paper was partly supported by National Natural Science Foundation of China (11771255), Young Innovation Teams of Shandong Province (2019KJI013), Program of Science and Technology Activities for Overseas Students in Henan Province in 2020 and Nanhu Scholars Program for Young Scholars of Xinyang Normal University. The authors would like to thank two referees for their valuable suggestions that greatly improved the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by National Natural Science Foundation of China [grant number 11771255], [grant number 11801325], Young Innovation Teams of Shandong Province [grant number 2019KJI013]