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Spectral conjugate gradient methods for vector optimization problems

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Abstract

In this work, we present an extension of the spectral conjugate gradient (SCG) methods for solving unconstrained vector optimization problems, with respect to the partial order induced by a pointed, closed and convex cone with a nonempty interior. We first study the direct extension version of the SCG methods and its global convergence without imposing an explicit restriction on parameters. It shows that the methods may lose their good scalar properties, like yielding descent directions, in the vector setting. By using a truncation technique, we then propose a modified self-adjusting SCG algorithm which is more suitable for various parameters. Global convergence of the new scheme covers the vector extensions of three different spectral parameters and the corresponding Perry, Andrei, and Dai–Kou conjugate parameters (SP, N, and JC schemes, respectively) without regular restarts and any convex assumption. Under inexact line searches, we prove that the sequences generated by the proposed methods find points that satisfy the first-order necessary condition for Pareto-optimality. Finally, numerical experiments illustrating the practical behavior of the methods are presented.

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We do not analyse or generate any datasets, because our work proceeds within a theoretical and mathematical approach.

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Acknowledgements

We would like to thank two anonymous referees and the editor for their valuable comments and suggestions, which improved the quality of this paper. We are also deeply indebted to references [24, 37, 38] for their Fortran codes.

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  1. College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China

    Qing-Rui He, Chun-Rong Chen & Sheng-Jie Li

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This work was supported by the National Key Research and Development Program of China (Grant No. 2022YFB3304500) and the National Natural Science Foundation of China (Grant Nos. 11971078 and 12271072).

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He, QR., Chen, CR. & Li, SJ. Spectral conjugate gradient methods for vector optimization problems. Comput Optim Appl 86, 457–489 (2023). https://doi.org/10.1007/s10589-023-00508-w

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