# Proximal gradient methods for multiobjective optimization and their applications

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## Abstract

We propose new descent methods for unconstrained multiobjective optimization problems, where each objective function can be written as the sum of a continuously differentiable function and a proper convex but not necessarily differentiable one. The methods extend the well-known proximal gradient algorithms for scalar-valued nonlinear optimization, which are shown to be efficient for particular problems. Here, we consider two types of algorithms: with and without line searches. Under mild assumptions, we prove that each accumulation point of the sequence generated by these algorithms, if exists, is Pareto stationary. Moreover, we present their applications in constrained multiobjective optimization and robust multiobjective optimization, which is a problem that considers uncertainties. In particular, for the robust case, we show that the subproblems of the proximal gradient algorithms can be seen as quadratic programming, second-order cone programming, or semidefinite programming problems. Considering these cases, we also carry out some numerical experiments, showing the validity of the proposed methods.

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## Notes

1. We denote $$A \succeq (\succ ) O$$ when A is positive semidefinite (positive definite). Also, $$A \succeq (\succ ) B$$ if and only if $$A - B \succeq (\succ ) O$$.

2. Here, $$\dim$$ denotes dimension of a space and $$\ker$$ means kernel of a matrix.

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## Acknowledgements

This work was supported by the Kyoto University Foundation, and the Grant-in-Aid for Scientific Research (C) (17K00032) from Japan Society for the Promotion of Science. We are also grateful to the anonymous referees for their useful comments.

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Correspondence to Ellen H. Fukuda.

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Tanabe, H., Fukuda, E.H. & Yamashita, N. Proximal gradient methods for multiobjective optimization and their applications. Comput Optim Appl 72, 339–361 (2019). https://doi.org/10.1007/s10589-018-0043-x