A family of projective splitting methods for the sum of two maximal monotone operators |
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Authors: | Jonathan Eckstein B F Svaiter |
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Institution: | (1) Business School (Department of Management Science and Information Systems) and RUTCOR, Rutgers University, 640 Bartholomew Road, Busch Campus, Piscataway, NJ 08854, USA;(2) IMPA, Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina, 110, Rio de Janeiro, RJ, CEP 22460-320, Brazil |
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Abstract: | A splitting method for two monotone operators A and B is an algorithm that attempts to converge to a zero of the sum A + B by solving a sequence of subproblems, each of which involves only the operator A, or only the operator B. Prior algorithms of this type can all in essence be categorized into three main classes, the Douglas/Peaceman-Rachford class,
the forward-backward class, and the little-used double-backward class. Through a certain “extended” solution set in a product
space, we construct a fundamentally new class of splitting methods for pairs of general maximal monotone operators in Hilbert
space. Our algorithms are essentially standard projection methods, using splitting decomposition to construct separators.
We prove convergence through Fejér monotonicity techniques, but showing Fejér convergence of a different sequence to a different
set than in earlier splitting methods. Our projective algorithms converge under more general conditions than prior splitting
methods, allowing the proximal parameter to vary from iteration to iteration, and even from operator to operator, while retaining
convergence for essentially arbitrary pairs of operators. The new projective splitting class also contains noteworthy preexisting
methods either as conventional special cases or excluded boundary cases.
Dedicated to Clovis Gonzaga on the occassion of his 60th birthday. |
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Keywords: | 47H05 90C25 49M27 |
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