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Smoothing SQP algorithm for semismooth equations with box constraints

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Abstract

In this paper, in order to solve semismooth equations with box constraints, we present a class of smoothing SQP algorithms using the regularized-smooth techniques. The main difference of our algorithm from some related literature is that the correspondent objective function arising from the equation system is not required to be continuously differentiable. Under the appropriate conditions, we prove the global convergence theorem, in other words, any accumulation point of the iteration point sequence generated by the proposed algorithm is a KKT point of the corresponding optimization problem with box constraints. Particularly, if an accumulation point of the iteration sequence is a vertex of box constraints and additionally, its corresponding KKT multipliers satisfy strictly complementary conditions, the gradient projection of the iteration sequence finitely terminates at this vertex. Furthermore, under local error bound conditions which are weaker than BD-regular conditions, we show that the proposed algorithm converges superlinearly. Finally, the promising numerical results demonstrate that the proposed smoothing SQP algorithm is an effective method.

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Acknowledgements

The authors would like to thank referees for their constructive comments and suggestions, which significantly improved the presentation of the paper.

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Authors and Affiliations

  1. Inst. of Operations Research, Qufu Normal University, Qufu, China

    Changyu Wang

  2. Department of Mathematics, Shandong Normal University, Jinan, China

    Qian Liu

  3. Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong

    Cheng Ma

Authors
  1. Changyu Wang
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  2. Qian Liu
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  3. Cheng Ma
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Corresponding author

Correspondence to Cheng Ma.

Additional information

This research was partially supported by the National Natural Science Foundation of China under Grants (10971118, 10901096, 11271226, 11271233, 11101231) and the Scientific Research Fund for the Excellent Middle-Aged and Youth Scientists of Shandong Province under Grant No. BS2012SF027, BS2010SF010, ZR2012AM016.

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Wang, C., Liu, Q. & Ma, C. Smoothing SQP algorithm for semismooth equations with box constraints. Comput Optim Appl 55, 399–425 (2013). https://doi.org/10.1007/s10589-012-9524-5

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  • DOI: https://doi.org/10.1007/s10589-012-9524-5

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