scispace - formally typeset
Search or ask a question
Topic

Monotone polygon

About: Monotone polygon is a research topic. Over the lifetime, 14573 publications have been published within this topic receiving 270286 citations.


Papers
More filters
Journal ArticleDOI
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

Journal ArticleDOI
Yann Brenier1
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

Journal ArticleDOI
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

Book
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


Network Information
Related Topics (5)
Bounded function
77.2K papers, 1.3M citations
94% related
Polynomial
52.6K papers, 853.1K citations
89% related
Rate of convergence
31.2K papers, 795.3K citations
89% related
Partial differential equation
70.8K papers, 1.6M citations
88% related
Differential equation
88K papers, 2M citations
87% related
Performance
Metrics
No. of papers in the topic in previous years
YearPapers
2023687
20221,632
2021833
2020766
2019740
2018602