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On inexact projected gradient methods for solving variable vector optimization problems

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Abstract

Variable order structures model situations in which the comparison between two points depends on a point-to-cone map. In this paper, inexact projected gradient methods for solving smooth constrained vector optimization problems on variable ordered spaces are presented. It is shown that every accumulation point of the generated sequences satisfies the first-order necessary optimality condition. Moreover, under suitable convexity assumptions for the objective function, it is proved that all accumulation points of any generated sequences are weakly efficient points. The convergence results are also derived in the particular case in which the problem is unconstrained and even if inexact directions are taken as descent directions. Furthermore, we investigate the application of the proposed method to optimization models where the domain of the variable order map coincides with the image of the objective function. In this case, similar concepts and convergence results are presented. Finally, some computational experiments designed to illustrate the behavior of the proposed inexact methods versus the exact ones (in terms of CPU time) are performed.

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Acknowledgements

JYBC was partially supported by the National Science Foundation (NSF) Grant DMS-1816449 and by Research & Artistry grant from NIU. The results of this paper were developed during the stay of the second author at the University of Halle-Wittenberg supported by the Humboldt Foundation.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Northern Illinois University, Watson Hall 366, DeKalb, IL, 60115, USA

    J. Y. Bello-Cruz

  2. Facultad de Matemática y Computación, Universidad de La Habana, 10400, La Habana, Cuba

    G. Bouza Allende

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  1. J. Y. Bello-Cruz
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  2. G. Bouza Allende
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Correspondence to J. Y. Bello-Cruz.

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Bello-Cruz, J.Y., Bouza Allende, G. On inexact projected gradient methods for solving variable vector optimization problems. Optim Eng 23, 201–232 (2022). https://doi.org/10.1007/s11081-020-09579-8

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