An accelerated proximal gradient method for multiobjective optimization

This paper presents an accelerated proximal gradient method for multiobjective optimization, in which each objective function is the sum of a continuously differentiable, convex function and a closed, proper, convex function. Extending first-order methods for multiobjective problems without scalariz...

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Published in:	Computational optimization and applications Vol. 86; no. 2; pp. 421 - 455
Main Authors:	Tanabe, Hiroki, Fukuda, Ellen H., Yamashita, Nobuo
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.11.2023 Springer Nature B.V
Subjects:	Algorithms Convergence Convex and Discrete Geometry Iterative methods Management Science Mathematics Mathematics and Statistics Methods Multiple objective analysis Operations Research Operations Research/Decision Theory Optimization Statistics FISTA Global rate of convergence 90C25 Multiobjective optimization Pareto optimality 90C29 First-order method Proximal gradient method
ISSN:	0926-6003, 1573-2894
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Abstract	This paper presents an accelerated proximal gradient method for multiobjective optimization, in which each objective function is the sum of a continuously differentiable, convex function and a closed, proper, convex function. Extending first-order methods for multiobjective problems without scalarization has been widely studied, but providing accelerated methods with accurate proofs of convergence rates remains an open problem. Our proposed method is a multiobjective generalization of the accelerated proximal gradient method, also known as the Fast Iterative Shrinkage-Thresholding Algorithm, for scalar optimization. The key to this successful extension is solving a subproblem with terms exclusive to the multiobjective case. This approach allows us to demonstrate the global convergence rate of the proposed method ( O ( 1 / k 2 ) ), using a merit function to measure the complexity. Furthermore, we present an efficient way to solve the subproblem via its dual representation, and we confirm the validity of the proposed method through some numerical experiments.
AbstractList	This paper presents an accelerated proximal gradient method for multiobjective optimization, in which each objective function is the sum of a continuously differentiable, convex function and a closed, proper, convex function. Extending first-order methods for multiobjective problems without scalarization has been widely studied, but providing accelerated methods with accurate proofs of convergence rates remains an open problem. Our proposed method is a multiobjective generalization of the accelerated proximal gradient method, also known as the Fast Iterative Shrinkage-Thresholding Algorithm, for scalar optimization. The key to this successful extension is solving a subproblem with terms exclusive to the multiobjective case. This approach allows us to demonstrate the global convergence rate of the proposed method (O(1/k2)), using a merit function to measure the complexity. Furthermore, we present an efficient way to solve the subproblem via its dual representation, and we confirm the validity of the proposed method through some numerical experiments. This paper presents an accelerated proximal gradient method for multiobjective optimization, in which each objective function is the sum of a continuously differentiable, convex function and a closed, proper, convex function. Extending first-order methods for multiobjective problems without scalarization has been widely studied, but providing accelerated methods with accurate proofs of convergence rates remains an open problem. Our proposed method is a multiobjective generalization of the accelerated proximal gradient method, also known as the Fast Iterative Shrinkage-Thresholding Algorithm, for scalar optimization. The key to this successful extension is solving a subproblem with terms exclusive to the multiobjective case. This approach allows us to demonstrate the global convergence rate of the proposed method ( O ( 1 / k 2 ) ), using a merit function to measure the complexity. Furthermore, we present an efficient way to solve the subproblem via its dual representation, and we confirm the validity of the proposed method through some numerical experiments.
Author	Yamashita, Nobuo Tanabe, Hiroki Fukuda, Ellen H.
Author_xml	– sequence: 1 givenname: Hiroki orcidid: 0000-0003-0042-9221 surname: Tanabe fullname: Tanabe, Hiroki email: tanabe.hiroki.45n@kyoto-u.jp organization: Yahoo Japan Corporation – sequence: 2 givenname: Ellen H. surname: Fukuda fullname: Fukuda, Ellen H. organization: Kyoto University – sequence: 3 givenname: Nobuo surname: Yamashita fullname: Yamashita, Nobuo organization: Kyoto University
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Snippet	This paper presents an accelerated proximal gradient method for multiobjective optimization, in which each objective function is the sum of a continuously...
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SubjectTerms	Algorithms Convergence Convex and Discrete Geometry Iterative methods Management Science Mathematics Mathematics and Statistics Methods Multiple objective analysis Operations Research Operations Research/Decision Theory Optimization Statistics
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Title	An accelerated proximal gradient method for multiobjective optimization
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