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A primal-dual interior-point relaxation method with global and rapidly local convergence for nonlinear programs

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Abstract

Based on solving an equivalent parametric equality constrained mini-max problem of the classic logarithmic-barrier subproblem, we present a novel primal-dual interior-point relaxation method for nonlinear programs with general equality and nonnegative constraints. In each iteration, our method approximately solves the KKT system of a parametric equality constrained mini-max subproblem, which avoids the requirement that any primal or dual iterate is an interior-point. The method has some similarities to the warmstarting interior-point methods in relaxing the interior-point requirement and is easily extended for solving problems with general inequality constraints. In particular, it has the potential to circumvent the jamming difficulty that appears with many interior-point methods for nonlinear programs and improve the ill conditioning of existing primal-dual interior-point methods as the barrier parameter is small. A new smoothing approach is introduced to develop our relaxation method and promote convergence of the method. Under suitable conditions, it is proved that our method can be globally convergent and locally quadratically convergent to the KKT point of the original problem. The preliminary numerical results on a well-posed problem for which many interior-point methods fail to find the minimizer and a set of test problems from the CUTEr collection show that our method is efficient.

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Notes

  1. A little change is that both z and y are divided by \(\rho \) in this paper.

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Acknowledgements

The research is supported by the NSFC grants (nos. 12071108, 11671116, 12021001, 11991021, 11991020, 11971372, and 11701137), National Key R &D Program of China (nos. 2021YFA1000300 and 2021YFA1000301), the Strategic Priority Research Program of Chinese Academy of Sciences (no. XDA27000000), and the Natural Science Foundation of Hebei Province (no. A2021202010).

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Authors and Affiliations

  1. Institute of Mathematics, Hebei University of Technology, Xiping Road No. 5340, Beichen District, 300401, Tianjin, China

    Xin-Wei Liu & Ya-Kui Huang

  2. LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Zhongguancun East Road No. 55, Haidian District, 100190, Beijing, China

    Yu-Hong Dai

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  1. Xin-Wei Liu
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Correspondence to Xin-Wei Liu.

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Liu, XW., Dai, YH. & Huang, YK. A primal-dual interior-point relaxation method with global and rapidly local convergence for nonlinear programs. Math Meth Oper Res 96, 351–382 (2022). https://doi.org/10.1007/s00186-022-00797-7

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  • DOI: https://doi.org/10.1007/s00186-022-00797-7

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