Topic
Monotone polygon
About: Monotone polygon is a research topic. Over the lifetime, 14573 publications have been published within this topic receiving 270286 citations.
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TL;DR: This work studies two splitting algorithms for (stationary and evolution) problems involving the sum of two monotone operators with real-time requirements.
Abstract: Splitting algorithms for the sum of two monotone operators.We study two splitting algorithms for (stationary and evolution) problems involving the sum of two monotone operators. These algorithms ar...
1,939 citations
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11 Dec 1989
1,810 citations
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TL;DR: In this paper, it was shown that for every vector-valued function u Lp(X, p; Rd) there is a unique polar factorization u = V$.s, where $ is a convex function defined on R and s is a measure-preserving mapping from (x, p) into (Q, I. I), provided that u is nondegenerate.
Abstract: Given a probability space (X, p) and a bounded domain R in Rd equipped with the Lebesgue measure 1 . I (normalized so that 10 I = I ), it is shown (under additional technical assumptions on X and Q) that for every vector-valued function u E Lp(X, p; Rd) there is a unique “polar factorization” u = V$.s, where $ is a convex function defined on R and s is a measure-preserving mapping from (X, p) into (Q, I . I), provided that u is nondegenerate, in the sense that p(u-’(E)) = 0 for each Lebesgue negligible subset E of Rd. Through this result, the concepts of polar factorization of real matrices, Helmholtz decomposition of vector fields, and nondecreasing rearrangements of real-valued functions are unified. The Monge-Amgre equation is involved in the polar factorization and the proof relies on the study of an appropriate “Monge-Kantorovich” problem.
1,780 citations
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TL;DR: In this paper, a modification of Rockafellar's proximal point algorithm is obtained and proved to be always strongly convergent, and the ideas of these algorithms are applied to solve a quadratic minimization problem.
Abstract: Iterative algorithms for nonexpansive mappings and maximal monotone operators are investigated. Strong convergence theorems are proved for nonexpansive mappings, including an improvement of a result of Lions. A modification of Rockafellar’s proximal point algorithm is obtained and proved to be always strongly convergent. The ideas of these algorithms are applied to solve a quadratic minimization problem.
1,560 citations
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21 Feb 1989
TL;DR: Convex functions on real Banach spaces were studied in this paper, where a generalization of monotone operators, usco maps, were used for convex functions.
Abstract: Convex functions on real Banach spaces.- Monotone operators, subdifferentials and Asplund spaces.- Lower semicontinuous convex functions.- Smooth variational principles, Asplund spaces, weak Asplund spaces.- Asplund spaces, the RNP and perturbed optimization.- Gateaux differentiability spaces.- A generalization of monotone operators: Usco maps.
1,286 citations