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An accelerated active-set algorithm for a quadratic semidefinite program with general constraints

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Abstract

In this paper, we are concerned with efficient algorithms for solving the least squares semidefinite programming which contains many equalities and inequalities constraints. Our proposed method is built upon its dual formulation and is a type of active-set approach. In particular, by exploiting the nonnegative constraints in the dual form, our method first uses the information from the Barzlai–Borwein step to estimate the active/inactive sets, and within an adaptive framework, it then accelerates the convergence by switching the L-BFGS iteration and the semi-smooth Newton iteration dynamically. We show the global convergence under mild conditions, and furthermore, the local quadratic convergence under the additional nondegeneracy condition. Various types of synthetic as well as real-world examples are tested, and preliminary but promising numerical experiments are reported.

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Notes

  1. The initial inverse Hessian approximation in L-BFGS is generally set to be \(\zeta ^k I\) with \(\zeta ^k=\frac{(s^{k-1})^Tw^{k-1}}{(w^{k-1})^Tw^{k-1}}\).

  2. Codes of the approaches of ISNM and P-BFGS are available online at http://www.math.nus.edu.sg/~matsundf/, and https://ctk.math.ncsu.edu/matlab_darts.html, respectively.

  3. We remark that \(\varepsilon\) is set to be \(10^{-8}\) only in this subsubsection because it is beneficial to observe the quadratic convergence rate from numerical results. In the rest numerical experiments, \(\varepsilon\) is still set to be \(10^{-6}\).

  4. One may observe that the convergence in the final stage in Tables 2 and 3 is not indeed quadratic. This is due to the use of CG for the generalized Newton system where only a properly accurate approximation solution is computed.

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Acknowledgements

The authors would like to thank the Associate Editor and anonymous referees for their helpful suggestions.

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Authors and Affiliations

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

    Chungen Shen

  2. Antai College of Economics and Management, Shanghai Jiao Tong University, Shanghai, 200030, China

    Yunlong Wang

  3. School of Mathematics and Physics, Shanghai University of Electric Power, Shanghai, 200090, China

    Wenjuan Xue

  4. School of Mathematical Sciences, Soochow University, Suzhou, 215006, Jiangsu, China

    Lei-Hong Zhang

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  1. Chungen Shen
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Wenjuan Xue: The work of this author was supported in part by the National Natural Science Foundation of China NSFC-11601318. Lei-Hong Zhang: The work of this author was supported in part by the National Natural Science Foundation of China (NSFC-11671246, NSFC-12071332), the National Key R&D Program of China (No. 2018YFB0204404) and Double Innovation Program of Jiangsu Province, Year 2018.

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Shen, C., Wang, Y., Xue, W. et al. An accelerated active-set algorithm for a quadratic semidefinite program with general constraints. Comput Optim Appl 78, 1–42 (2021). https://doi.org/10.1007/s10589-020-00228-5

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