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Quasi-contingent derivatives and studies of higher-orders in nonsmooth optimization

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Abstract

We consider higher-order conditions and sensitivity analysis for solutions to equilibrium problems. The conditions for solutions are in terms of quasi-contingent derivatives and involve higher-order complementarity slackness for both the objective and the constraints and under Hölder metric subregularity assumptions. For sensitivity analysis, a formula of this type of derivative of the solution map to a parametric equilibrium problem is established in terms of the same types of derivatives of the data of the problem. Here, the concepts of a quasi-contingent derivative and critical directions are new. We consider open-cone solutions and proper solutions. We also study an important and typical special case: weak solutions of a vector minimization problem with mixed constraints. The results are significantly new and improve recent corresponding results in many aspects.

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Acknowledgements

The work was supported by the National Foundation for Science and Technology Developments (NAFOSTED) of Vietnam under Grant 101.01-2021.13. A part of this work was completed during a scientific stay of the second and third authors at the Vietnam Institute for Advanced Study in Mathematics (VIASM). They would like to thank VIASM for its hospitality and support. The authors are warmly grateful to the handling editor and the anonymous referee for their helpful remarks and valuable suggestions.

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, University of Science, Ho Chi Minh City, Vietnam

    Nguyen Xuan Duy Bao

  2. Vietnam National University, Ho Chi Minh City, Vietnam

    Nguyen Xuan Duy Bao

  3. Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Phan Quoc Khanh

  4. Faculty of Mathematical Economics, Banking University of Ho Chi Minh City, Ho Chi Minh City, Vietnam

    Nguyen Minh Tung

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  1. Nguyen Xuan Duy Bao
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  3. Nguyen Minh Tung
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Correspondence to Phan Quoc Khanh.

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Bao, N.X.D., Khanh, P.Q. & Tung, N.M. Quasi-contingent derivatives and studies of higher-orders in nonsmooth optimization. J Glob Optim 84, 205–228 (2022). https://doi.org/10.1007/s10898-022-01129-z

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