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A stabilized filter SQP algorithm for nonlinear programming

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Abstract

This paper presents a stabilized filter sequential quadratic programming (SQP) method for the general nonlinear optimization problems. The technique of stabilizing the inner quadratic programmings is an efficient strategy for the degenerate problem and brings the local superlinear convergence, while the integrated filter technique works effectively and guarantees the global convergence. The new algorithm works on both the primal and dual variables and solves the problem within the computational complexity comparable to the classical SQP algorithm. For the convergence, we show that (1) it converges either to a Karush–Kuhn–Tucker point at which the cone-continuity property holds, or to a stationary point in the sense of minimizing the constraint violation, and (2) under some second-order sufficient conditions, it converges locally superlinearly without any constraint qualifications. Our preliminary numerical results on a set of CUTEr test problems as well as on degenerate problems demonstrate the efficiency of the new algorithm.

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Notes

  1. In terms of the SC condition, the SOSC is equivalent to the stronger SOSC [8].

  2. In other words, \((\hat{x},\hat{\mu })\) is said to be g-acceptable to \((x^l,\mu ^l)\) if at least one of (3.8) and (3.9) is satisfied.

  3. Here, \(\tau \) and \(\vartheta \) can be replaced by a single parameter. The reason that they are referred to separately here is for the convenience of the later references (3.15) and (3.18b).

  4. In our implementations, we simply set \(\mu _{\max }=10^6\) and it works well in our numerical tests.

  5. Inner loop G may find a step size \(\alpha ^k\ge \alpha _{\min }^k\) such that \((\hat{x},\hat{\mu })\) is accepted as a next iterate, or return \((\hat{x},\hat{\mu })\) and \(\alpha (< \alpha _{\min }^k)\).

  6. The proof of Lemma 4.5 needs Assumption 4.2\(^*\) explicitly, and all subsequent lemmas or theorems based on Lemma 4.5 need Assumption 4.2\(^*\) implicitly.

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Acknowledgments

We would like to thank the Editor and the anonymous referees for their comments and suggestions that have improved the quality of this paper greatly. We also thank Professor Mikhail Solodov for suggesting the filter-type strategy in globalizing the stabilized SQP to the first author.

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Authors and Affiliations

  1. College of Science, University of Shanghai for Science and Technology, Shanghai, 200093, China

    Chungen Shen

  2. School of Mathematics, Shanghai University of Finance and Economics, Shanghai, 200433, China

    Lei-Hong Zhang

  3. Shanghai Key Laboratory of Financial Information Technology (Shanghai University of Finance and Economics), Shanghai, 200433, China

    Lei-Hong Zhang & Wei Liu

  4. School of Statistics and Mathematics, Shanghai Finance University, Shanghai, 201209, China

    Wei Liu

Authors
  1. Chungen Shen
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  2. Lei-Hong Zhang
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  3. Wei Liu
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Correspondence to Chungen Shen.

Additional information

The work of the first author is supported in part by the National Natural Science Foundation of China (No. 11271259). The work of the second author is supported in part by the National Natural Science Foundation of China (No. 11371102) and by the Basic Academic Discipline Program, the 11th five year plan of the 211 Project for Shanghai University of Finance and Economics. The work of the third author is supported in part by the National Natural Science Foundation of China (No. 71401105), by the National Natural Science Foundation of Shanghai (No. 13ZR1427200), and by the Innovation Program of the Shanghai Municipal Education Commission (No. 13YZ126).

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Shen, C., Zhang, LH. & Liu, W. A stabilized filter SQP algorithm for nonlinear programming. J Glob Optim 65, 677–708 (2016). https://doi.org/10.1007/s10898-015-0400-6

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