An Accelerated Inexact Proximal Point Algorithm for Convex Minimization

The proximal point algorithm is classical and popular in the community of optimization. In practice, inexact proximal point algorithms which solve the involved proximal subproblems approximately subject to certain inexact criteria are truly implementable. In this paper, we first propose an inexact p...

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Published in:	Journal of optimization theory and applications Vol. 154; no. 2; pp. 536 - 548
Main Authors:	He, Bingsheng, Yuan, Xiaoming
Format:	Journal Article
Language:	English
Published:	Boston Springer US 01.08.2012 Springer Nature B.V
Subjects:	Acceleration Accuracy Algorithms Applications of Mathematics Calculus of Variations and Optimal Control; Optimization Engineering Mathematics Mathematics and Statistics Operations Research/Decision Theory Optimization Studies Theory of Computation Convex minimization Inexact Proximal point algorithm Acceleration
ISSN:	0022-3239, 1573-2878
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Abstract	The proximal point algorithm is classical and popular in the community of optimization. In practice, inexact proximal point algorithms which solve the involved proximal subproblems approximately subject to certain inexact criteria are truly implementable. In this paper, we first propose an inexact proximal point algorithm with a new inexact criterion for solving convex minimization, and show its O (1/ k ) iteration-complexity. Then we show that this inexact proximal point algorithm is eligible for being accelerated by some influential acceleration schemes proposed by Nesterov. Accordingly, an accelerated inexact proximal point algorithm with an iteration-complexity of O (1/ k 2 ) is proposed.
AbstractList	The proximal point algorithm is classical and popular in the community of optimization. In practice, inexact proximal point algorithms which solve the involved proximal subproblems approximately subject to certain inexact criteria are truly implementable. In this paper, we first propose an inexact proximal point algorithm with a new inexact criterion for solving convex minimization, and show its O(1/k) iteration-complexity. Then we show that this inexact proximal point algorithm is eligible for being accelerated by some influential acceleration schemes proposed by Nesterov. Accordingly, an accelerated inexact proximal point algorithm with an iteration-complexity of O(1/k super(2)) is proposed. The proximal point algorithm is classical and popular in the community of optimization. In practice, inexact proximal point algorithms which solve the involved proximal subproblems approximately subject to certain inexact criteria are truly implementable. In this paper, we first propose an inexact proximal point algorithm with a new inexact criterion for solving convex minimization, and show its O(1/k) iteration-complexity. Then we show that this inexact proximal point algorithm is eligible for being accelerated by some influential acceleration schemes proposed by Nesterov. Accordingly, an accelerated inexact proximal point algorithm with an iteration-complexity of O(1/k ^sup 2^) is proposed.[PUBLICATION ABSTRACT] The proximal point algorithm is classical and popular in the community of optimization. In practice, inexact proximal point algorithms which solve the involved proximal subproblems approximately subject to certain inexact criteria are truly implementable. In this paper, we first propose an inexact proximal point algorithm with a new inexact criterion for solving convex minimization, and show its O (1/ k ) iteration-complexity. Then we show that this inexact proximal point algorithm is eligible for being accelerated by some influential acceleration schemes proposed by Nesterov. Accordingly, an accelerated inexact proximal point algorithm with an iteration-complexity of O (1/ k 2 ) is proposed.
Author	Yuan, Xiaoming He, Bingsheng
Author_xml	– sequence: 1 givenname: Bingsheng surname: He fullname: He, Bingsheng organization: Department of Mathematics and National Key Laboratory for Novel Software Technology, Nanjing University – sequence: 2 givenname: Xiaoming surname: Yuan fullname: Yuan, Xiaoming email: xmyuan@hkbu.edu.hk organization: Department of Mathematics, Hong Kong Baptist University
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Cites_doi	10.1137/0314056 10.1137/S0363012992235547 10.1137/080716542 10.1137/0329022 10.1137/0802032 10.24033/bsmf.1625
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SubjectTerms	Acceleration Accuracy Algorithms Applications of Mathematics Calculus of Variations and Optimal Control; Optimization Engineering Mathematics Mathematics and Statistics Operations Research/Decision Theory Optimization Studies Theory of Computation
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Title	An Accelerated Inexact Proximal Point Algorithm for Convex Minimization
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