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Conditional gradient method for vector optimization

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Abstract

In this paper, we propose a conditional gradient method for solving constrained vector optimization problems with respect to a partial order induced by a closed, convex and pointed cone with nonempty interior. When the partial order under consideration is the one induced by the non-negative orthant, we regain the method for multiobjective optimization recently proposed by Assunção et al. (Comput Optim Appl 78(3):741–768, 2021). In our method, the construction of the auxiliary subproblem is based on the well-known oriented distance function. Three different types of step size strategies (Armijo, adaptative and nonmonotone) are considered. Without convexity assumption related to the objective function, we obtain the stationarity of accumulation points of the sequences produced by the proposed method equipped with the Armijo or the nonmonotone step size rule. To obtain the convergence result of the method with the adaptative step size strategy, we introduce a useful cone convexity condition which allows us to circumvent the intricate question of the Lipschitz continuity of Jocabian for the objective function. This condition helps us to generalize the classical descent lemma to the vector optimization case. Under convexity assumption for the objective function, it is proved that all accumulation points of any generated sequences obtained by our method are weakly efficient solutions. Numerical experiments illustrating the practical behavior of the methods are presented.

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  1. The performance profiles were in this paper generated using the MATLAB code perfprof.m freely available in the website https://github.com/higham/matlab-guide-3ed/blob/master/perfprof.m.

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Acknowledgements

We thank the anonymous referees for their constructive comments and suggestions to improve the quality and contributions of the paper. Also, we would like to thank Prof. L.P. Tang for the helpful suggestions and discussions during the writing of this paper.

Funding

This research was supported by the Major Program of the National Natural Science Foundation of China (11991020, 11991024), the National Natural Science Foundation of China (11971084, 12001072, 12271067), NSFC-RGC (Hong Kong) Joint Research Program (12261160365), the Team Project of Innovation Leading Talent in Chongqing (CQYC20210309536), the Natural Science Foundation of Chongqing (CSTB2022NSCQ-MSX1318, cstc2019jcyj-zdxmX0016), the China Postdoctoral Science Foundation Project (2019M653332).

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Authors and Affiliations

  1. College of Mathematics, Sichuan University, Sichuan, 610065, China

    Wang Chen

  2. National Center for Applied Mathematics of Chongqing, Chongqing Normal University, Chongqing, 401331, China

    Xinmin Yang

  3. School of Mathematical Sciences, Chongqing Normal University, Chongqing, 401331, China

    Xinmin Yang

  4. College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing, 610101, China

    Yong Zhao

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  1. Wang Chen
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WC conceived of the study and drafted the manuscript. XY participated in its analysis and coordination and helped to draft the manuscript. YZ participated in its analysis and was responsible for checking the content and grammar of the article.

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Correspondence to Xinmin Yang.

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Chen, W., Yang, X. & Zhao, Y. Conditional gradient method for vector optimization. Comput Optim Appl 85, 857–896 (2023). https://doi.org/10.1007/s10589-023-00478-z

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