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Second-Order Enhanced Optimality Conditions and Constraint Qualifications

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Abstract

In this paper, we study second-order necessary optimality conditions for smooth nonlinear programming problems. Employing the second-order variational analysis and generalized differentiation, under the weak constant rank (WCR) condition, we derive an enhanced version of the classical weak second-order Fritz–John condition which contains some new information on multipliers. Based on this enhanced weak second-order Fritz–John condition, we introduce the weak second-order enhanced Karush–Kuhn–Tucker condition and propose some associated second-order constraint qualifications. Finally, using our new second-order constraint qualifications, we establish new sufficient conditions for the existence of a Hölder error bound condition.

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Acknowledgements

The authors are grateful to the anonymous referees for their helpful suggestions and comments. The first author’s work was partially supported by the NSFC Grant 12201531 and the Hong Kong Research Grants Council PolyU153036/22p. The third author’s work was partially supported by the NSFC Grant 12222106, the Shenzhen Science and Technology Program Grant RCYX20200714114700072, and the Guangdong Basic and Applied Basic Research Foundation Grant 2022B1515020082.

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Authors and Affiliations

  1. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China

    Kuang Bai

  2. Department of Mathematics, Southern University of Science and Technology, Shenzhen, China

    Yixia Song

  3. Department of Mathematics, National Center for Applied Mathematics Shenzhen, Southern University of Science and Technology, Peng Cheng Laboratory, Shenzhen, 518055, China

    ** Zhang

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  1. Kuang Bai
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Correspondence to ** Zhang.

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Bai, K., Song, Y. & Zhang, J. Second-Order Enhanced Optimality Conditions and Constraint Qualifications. J Optim Theory Appl 198, 1264–1284 (2023). https://doi.org/10.1007/s10957-023-02276-3

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