Alternated and multi-step inertial approximation methods for solving convex bilevel optimization problems

In this paper, we propose three kinds of inertial approximation methods based on the proximal gradient algorithm to accelerate the convergence of the algorithm for solving convex bilevel optimization problems. Under reasonable parameters, we prove that our algorithms converge strongly to some soluti...

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Bibliographic Details
Published in:Optimization Vol. 72; no. 10; pp. 2517 - 2545
Main Authors: Duan, Peichao, Zhang, Yiqun
Format: Journal Article
Language:English
Published: Philadelphia Taylor & Francis 03.10.2023
Taylor & Francis LLC
Subjects:
Algorithms
alternated inertia
alternated multi-step inertia
Approximation
Convergence
Convex bilevel optimization
multi-step inertia
Optimization
Parameters
proximal operator
ISSN:0233-1934, 1029-4945
Online Access:Get full text
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Summary:In this paper, we propose three kinds of inertial approximation methods based on the proximal gradient algorithm to accelerate the convergence of the algorithm for solving convex bilevel optimization problems. Under reasonable parameters, we prove that our algorithms converge strongly to some solution of the problem, which is the unique solution of a variational inequality problem. Firstly, two alternated inertial methods are presented. Secondly, a multi-step inertial method is proposed to accelerate the convergence of the algorithm. Thirdly, an alternated multi-step inertial method is further introduced. In addition, we also consider that the inertial parameters can be chosen as positive or negative parameters, and we get better results. Our numerical results illustrate the performances of our algorithms and present some comparisons with related algorithms. Our results improve and extend the corresponding results reported by some authors recently.
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ISSN:0233-1934
1029-4945
DOI:10.1080/02331934.2022.2069022