Golden ratio algorithms for variational inequalities

The paper presents a fully adaptive algorithm for monotone variational inequalities. In each iteration the method uses two previous iterates for an approximation of the local Lipschitz constant without running a linesearch. Thus, every iteration of the method requires only one evaluation of a monoto...

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Published in	Mathematical programming Vol. 184; no. 1-2; pp. 383 - 410
Main Author	Malitsky, Yura
Format	Journal Article
Language	English
Published	Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2020 Springer Nature B.V
Subjects	Adaptive algorithms Algorithms Applied mathematics Calculus of Variations and Optimal Control; Optimization Combinatorics Control theory Full Length Paper Inequalities Mapping Mathematical and Computational Physics Mathematical Methods in Physics Mathematical programming Mathematics Mathematics and Statistics Mathematics of Computing Methods Numerical Analysis Optimization Theoretical Fixed point problem 65K15 47J20 Variational inequality Composite minimization 90C33 65K10 Saddle point problem 65Y20 Linesearch First-order methods
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Abstract	The paper presents a fully adaptive algorithm for monotone variational inequalities. In each iteration the method uses two previous iterates for an approximation of the local Lipschitz constant without running a linesearch. Thus, every iteration of the method requires only one evaluation of a monotone operator F and a proximal mapping g . The operator F need not be Lipschitz continuous, which also makes the algorithm interesting in the area of composite minimization. The method exhibits an ergodic O (1 / k ) convergence rate and R -linear rate under an error bound condition. We discuss possible applications of the method to fixed point problems as well as its different generalizations.
AbstractList	The paper presents a fully adaptive algorithm for monotone variational inequalities. In each iteration the method uses two previous iterates for an approximation of the local Lipschitz constant without running a linesearch. Thus, every iteration of the method requires only one evaluation of a monotone operator F and a proximal mapping g . The operator F need not be Lipschitz continuous, which also makes the algorithm interesting in the area of composite minimization. The method exhibits an ergodic O (1 / k ) convergence rate and R -linear rate under an error bound condition. We discuss possible applications of the method to fixed point problems as well as its different generalizations. The paper presents a fully adaptive algorithm for monotone variational inequalities. In each iteration the method uses two previous iterates for an approximation of the local Lipschitz constant without running a linesearch. Thus, every iteration of the method requires only one evaluation of a monotone operator F and a proximal mapping g. The operator F need not be Lipschitz continuous, which also makes the algorithm interesting in the area of composite minimization. The method exhibits an ergodic O(1 / k) convergence rate and R-linear rate under an error bound condition. We discuss possible applications of the method to fixed point problems as well as its different generalizations.
Author	Malitsky, Yura
Author_xml	– sequence: 1 givenname: Yura orcidid: 0000-0001-7325-5766 surname: Malitsky fullname: Malitsky, Yura email: y.malitsky@gmail.com organization: Institute for Numerical and Applied Mathematics, University of Göttingen
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Copyright	Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2019 Mathematical Programming is a copyright of Springer, (2019). All Rights Reserved. Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2019.
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Snippet	The paper presents a fully adaptive algorithm for monotone variational inequalities. In each iteration the method uses two previous iterates for an...
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SubjectTerms	Adaptive algorithms Algorithms Applied mathematics Calculus of Variations and Optimal Control; Optimization Combinatorics Control theory Full Length Paper Inequalities Mapping Mathematical and Computational Physics Mathematical Methods in Physics Mathematical programming Mathematics Mathematics and Statistics Mathematics of Computing Methods Numerical Analysis Optimization Theoretical
Title	Golden ratio algorithms for variational inequalities
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