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A strategy of global convergence for the affine scaling algorithm for convex semidefinite programming

Mathematical Programming volume 179, pages 1–19 (2020)Cite this article

Abstract

The affine scaling algorithm is one of the earliest interior point methods developed for linear programming. This algorithm is simple and elegant in terms of its geometric interpretation, but it is notoriously difficult to prove its convergence. It often requires additional restrictive conditions such as nondegeneracy, specific initial solutions, and/or small step length to guarantee its global convergence. This situation is made worse when it comes to applying the affine scaling idea to the solution of semidefinite optimization problems or more general convex optimization problems. In (Math Program 83(1–3):393–406, 1998), Muramatsu presented an example of linear semidefinite programming, for which the affine scaling algorithm with either short or long step converges to a non-optimal point. This paper aims at developing a strategy that guarantees the global convergence for the affine scaling algorithm in the context of linearly constrained convex semidefinite optimization in a least restrictive manner. We propose a new rule of step size, which is similar to the Armijo rule, and prove that such an affine scaling algorithm is globally convergent in the sense that each accumulation point of the sequence generated by the algorithm is an optimal solution as long as the optimal solution set is nonempty and bounded. The algorithm is least restrictive in the sense that it allows the problem to be degenerate and it may start from any interior feasible point.

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Acknowledgements

The authors would like to thank the two anonymous referees for their constructive comments and suggestions on the earlier version of this paper. We really appreciate their valuable inputs.

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Authors and Affiliations

  1. Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Kowloon, Hong Kong, People’s Republic of China

    Xun Qian & Li-Zhi Liao

  2. Department of Mathematics and Statistics, Curtin University, Perth, Australia

    Jie Sun

  3. School of Business, National University of Singapore, Singapore, Singapore

    Jie Sun

Authors
  1. Xun Qian
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  2. Li-Zhi Liao
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  3. Jie Sun
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Corresponding author

Correspondence to Li-Zhi Liao.

Additional information

The work of L.-Z. Liao was supported in part by grants from Hong Kong Baptist University (FRG) and General Research Fund (GRF) of Hong Kong. The work of J. Sun was partially supported by Australia Research Council under grant DP160102819.

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Qian, X., Liao, LZ. & Sun, J. A strategy of global convergence for the affine scaling algorithm for convex semidefinite programming. Math. Program. 179, 1–19 (2020). https://doi.org/10.1007/s10107-018-1314-0

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  • DOI: https://doi.org/10.1007/s10107-018-1314-0

Keywords

  • Affine scaling
  • Interior point method
  • Convex semidefinite programming

Mathematics Subject Classification

  • 90C25
  • 90C51
  • 37C75
  • 37N40