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A reduced proximal-point homotopy method for large-scale non-convex BQP

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Abstract

In this paper, a reduced proximal-point homotopy (RPP-Hom) method is presented for large-scale non-convex box constrained quadratic programming (BQP) problems. As the outer iteration, at each step, the reduced proximal-point (RPP) algorithm applies the proximal point algorithm to a reduced BQP problem. The variables of the reduced subproblem include all free variables and variables at bound with respect to which the optimality conditions are violated. The RPP subproblem is solved by, as the inner iteration, an efficient piecewise linear homotopy path following method. A special termination criterion for the RPP algorithm is given and the global convergence as well as the locally linear convergence to a Karush-Kuhn-Tucker point is proved. Furthermore, a random perturbation procedure is given to modify RPP such that it converges to a local minimizer with probability 1. An accelerated version of RPP is also presented. Numerical experiments show that the RPP-Hom method outperforms the state-of-the-art algorithms for most of the benchmark problems, especially for training non-convex support vector machine.

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Data availability

All data included in this study are available upon request by contact with the corresponding author.

Notes

  1. http://www.minlp.com/nlp-and-minlp-test-problems

  2. http://yann.lecun.com/exdb/mnist/

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Acknowledgements

This research was supported by the National Natural Science Foundation of China (11971092, 11571061) and the China Postdoctoral Science Foundation (2019M660444).

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Authors and Affiliations

  1. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, Liaoning, People’s Republic of China

    Xiubo Liang & Bo Yu

  2. Business Research and Development department JD.com, Beijing, China

    Guoqiang Wang

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  1. Xiubo Liang
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  2. Guoqiang Wang
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  3. Bo Yu
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Correspondence to Bo Yu.

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Liang, X., Wang, G. & Yu, B. A reduced proximal-point homotopy method for large-scale non-convex BQP. Comput Optim Appl 81, 539–567 (2022). https://doi.org/10.1007/s10589-021-00330-2

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