Abstract
The Jacobian consistency of smoothing functions plays an important role for achieving the rapid convergence of Newton methods or Newton-like methods with an appropriate parameter control. In this paper, we study the properties, derive the computable formula for the Jacobian matrix and prove the Jacobian consistency of a one-parametric class of smoothing Fischer–Burmeister functions for second-order cone complementarity problems proposed by Tang et al. (Comput Appl Math 33:655–669, 2014). Then we apply its Jacobian consistency to a smoothing Newton method with the appropriate parameter control presented by Chen et al. (Math Comput 67:519–540, 1998), and show the global convergence and local quadratic convergence of the algorithm for solving the SOCCP under rather weak assumptions.
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Acknowledgments
This research is supported by the National Natural Science Foundation of China (Nos. 11401126, 11361018, 71471140, 71461005), Guangxi Natural Science Foundation (No. 2014GXNSFFA118001), the Scientific Research Foundation of the Higher Education Institutions of Guangxi (No. ZD2014050), and Guangxi Key Laboratory of Automatic Detecting Technology and Instruments, China (Nos. YQ15112, YQ16112). The authors are grateful to the editor and the anonymous referees for their valuable comments, which have greatly improved this paper.
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Chi, X., Wang, Y., Zhu, Z. et al. Jacobian consistency of a one-parametric class of smoothing Fischer–Burmeister functions for SOCCP. Comp. Appl. Math. 37, 439–455 (2018). https://doi.org/10.1007/s40314-016-0352-6
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DOI: https://doi.org/10.1007/s40314-016-0352-6
Keywords
- Second-order cone complementarity problem
- Smoothing Fischer–Burmeister function
- Jacobian consistency
- Smoothing Newton method
- Quadratic convergence