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Multiobjective optimization with least constraint violation: optimality conditions and exact penalization

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Abstract

Although multiobjective optimization problem (MOP) is useful for solving many practical optimization problems, it is possible that the constraints are inconsistent. In this paper, we reformulate MOP with possible inconsistent constraints into MOP with least constraint violation and provide necessary optimality conditions from the perspective of M-stationary point, Fritz-John stationary point and L-stationary point. A power penalty problem is proposed by using infeasibility measure of constraints. The calmness conditions of order \(\ell \) of the MOP with least constraint violation and the local exact penalization of order \(\ell \) of the power penalty problem are respectively introduced, which do not require the feasibility of the original MOP. We obtain the equivalence between the calmness of order \(\ell \) of the MOP with least constraint violation and the local exact penalization of order \(\ell \) of the power penalty problem. Necessary and sufficient conditions for calmness of order \(\ell \) are also established under suitable conditions.

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Acknowledgements

The authors would like to express their sincere thanks to the associated editor and anonymous referees for their valuable comments and helpful suggestions, which help to improve the paper. They are also grateful to Professor Liwei Zhang for his useful comments and suggestions. The research part of the first author was done during his visiting in Academy of Mathematics and Systems Science, Chinese Academy of Sciences. Also, Jiawei Chen is grateful to Prof. Y. H. Dai for providing excellent research facilities and support during his stay at AMSS. The first author was supported by the National Natural Science Foundation of China (No: 12071379, 12126412), the Natural Science Foundation of Chongqing (cstc2021jcyj-msxmX0925) and the Youth Top Talent Program of Chongqing Talents. The second author was supported by the National Natural Science Foundation of China (Nos. 12021001, 11991020, 11991021, 11631013, 11971372) and the Strategic Priority Research Program of Chinese Academy of Sciences (No. XDA27000000).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China

    Jiawei Chen

  2. LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Yu-Hong Dai

  3. School of Mathematics Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

    Yu-Hong Dai

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  1. Jiawei Chen
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  2. Yu-Hong Dai
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Both authors contributed equally to this article.

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Correspondence to Yu-Hong Dai.

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Chen, J., Dai, YH. Multiobjective optimization with least constraint violation: optimality conditions and exact penalization. J Glob Optim 87, 807–830 (2023). https://doi.org/10.1007/s10898-022-01158-8

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