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Optimality Conditions, Qualifications and Approximation Method for a Class of Non-Lipschitz Mathematical Programs with Switching Constraints

Author

Listed:
  • Jinman Lv

    (School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China)

  • Zhenhua Peng

    (Department of Mathematics, School of Sciences, Nanchang University, Nanchang 330031, China)

  • Zhongping Wan

    (School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China)

Abstract

In this paper, we consider a class of mathematical programs with switching constraints (MPSCs) where the objective involves a non-Lipschitz term. Due to the non-Lipschitz continuity of the objective function, the existing constraint qualifications for local Lipschitz MPSCs are invalid to ensure that necessary conditions hold at the local minimizer. Therefore, we propose some MPSC-tailored qualifications which are related to the constraints and the non-Lipschitz term to ensure that local minimizers satisfy the necessary optimality conditions. Moreover, we study the weak, Mordukhovich, Bouligand, strongly (W-, M-, B-, S-) stationay, analyze what qualifications making local minimizers satisfy the (M-, B-, S-) stationay, and discuss the relationship between the given MPSC-tailored qualifications. Finally, an approximation method for solving the non-Lipschitz MPSCs is given, and we show that the accumulation point of the sequence generated by the approximation method satisfies S-stationary under the second-order necessary condition and MPSC Mangasarian-Fromovitz (MF) qualification.

Suggested Citation

  • Jinman Lv & Zhenhua Peng & Zhongping Wan, 2021. "Optimality Conditions, Qualifications and Approximation Method for a Class of Non-Lipschitz Mathematical Programs with Switching Constraints," Mathematics, MDPI, vol. 9(22), pages 1-20, November.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:22:p:2915-:d:680067
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    References listed on IDEAS

    as
    1. Yakui Huang & Hongwei Liu, 2016. "Smoothing projected Barzilai–Borwein method for constrained non-Lipschitz optimization," Computational Optimization and Applications, Springer, vol. 65(3), pages 671-698, December.
    2. Lijuan Wang & Qishu Yan, 2015. "Time Optimal Controls of Semilinear Heat Equation with Switching Control," Journal of Optimization Theory and Applications, Springer, vol. 165(1), pages 263-278, April.
    3. Yan-Chao Liang & Jane J. Ye, 2021. "Optimality Conditions and Exact Penalty for Mathematical Programs with Switching Constraints," Journal of Optimization Theory and Applications, Springer, vol. 190(1), pages 1-31, July.
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