Communications in Nonlinear Science and Numerical Simulation, ISSN 1007-5704, 2012, Volume 17, Issue 4, pp. 1844 - 1851

► Generalized resolvent operator technique with ( , , )-maximal monotonicity is used. ► A class of general nonlinear operator inclusion couples are considered...

General nonlinear operator inclusion couple | ( A, η, m)-resolvent operators and relaxed cocoercive type operators | Variational graphical convergence | New perturbed iterative algorithm framework with errors | Generalized resolvent operator technique | (A,η,m)-resolvent operators and relaxed cocoercive type operators | SYSTEM | MONOTONE OPERATOR | A | MATHEMATICS, APPLIED | PHYSICS, FLUIDS & PLASMAS | VALUED MAPPINGS | ITERATIVE ALGORITHM | PHYSICS, MATHEMATICAL | H-ACCRETIVE OPERATORS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | H(.,.)-ACCRETIVE OPERATOR | BANACH-SPACES | (A,eta,m)-resolvent operators and relaxed cocoercive type operators | SOLVING VARIATIONAL INCLUSIONS | Artificial intelligence | Analysis | Algorithms | Operators | Couples | Error analysis | Approximation | Nonlinearity | Inclusions | Convergence

General nonlinear operator inclusion couple | ( A, η, m)-resolvent operators and relaxed cocoercive type operators | Variational graphical convergence | New perturbed iterative algorithm framework with errors | Generalized resolvent operator technique | (A,η,m)-resolvent operators and relaxed cocoercive type operators | SYSTEM | MONOTONE OPERATOR | A | MATHEMATICS, APPLIED | PHYSICS, FLUIDS & PLASMAS | VALUED MAPPINGS | ITERATIVE ALGORITHM | PHYSICS, MATHEMATICAL | H-ACCRETIVE OPERATORS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | H(.,.)-ACCRETIVE OPERATOR | BANACH-SPACES | (A,eta,m)-resolvent operators and relaxed cocoercive type operators | SOLVING VARIATIONAL INCLUSIONS | Artificial intelligence | Analysis | Algorithms | Operators | Couples | Error analysis | Approximation | Nonlinearity | Inclusions | Convergence

Journal Article

Computers and Mathematics with Applications, ISSN 0898-1221, 2009, Volume 57, Issue 4, pp. 624 - 632

In this paper, we introduce and study a new class of nonlinear random -resolvent operator equations with random relaxed cocoercive operators in Hilbert spaces....

Existence and convergence | Non-monotone random set-valued operator | Nonlinear random [formula omitted]-resolvent operator equation with random relaxed cocoercive operator | New generalized random iterative algorithm | Random resolvent operator technique | Nonlinear random (A, η)-resolvent operator equation with random relaxed cocoercive operator | MATHEMATICS, APPLIED | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | RESOLVENT EQUATIONS | RANDOM FUZZY MAPPINGS | BANACH-SPACES | Nonlinear random (A, eta)-resolvent operator equation with random relaxed cocoercive operator | VALUED VARIATIONAL INCLUSIONS | Operators | Theorems | Approximation | Mathematical analysis | Nonlinearity | Hilbert space | Mathematical models | Convergence

Existence and convergence | Non-monotone random set-valued operator | Nonlinear random [formula omitted]-resolvent operator equation with random relaxed cocoercive operator | New generalized random iterative algorithm | Random resolvent operator technique | Nonlinear random (A, η)-resolvent operator equation with random relaxed cocoercive operator | MATHEMATICS, APPLIED | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | RESOLVENT EQUATIONS | RANDOM FUZZY MAPPINGS | BANACH-SPACES | Nonlinear random (A, eta)-resolvent operator equation with random relaxed cocoercive operator | VALUED VARIATIONAL INCLUSIONS | Operators | Theorems | Approximation | Mathematical analysis | Nonlinearity | Hilbert space | Mathematical models | Convergence

Journal Article

Nonlinear Analysis, ISSN 0362-546X, 2011, Volume 74, Issue 2, pp. 386 - 395

By using Lim’s inequalities, Nadler’s results, the new parametric resolvent operator technique associated with -maximal monotone operators, in this paper, the...

Existence and continuity theorem | Sensitive analysis | Parametric resolvent operator technique | Relaxed cocoercive operator | Generalized nonlinear parametric [formula omitted]-maximal monotone operator inclusion system | Generalized nonlinear parametric (A,η,m)-maximal monotone operator inclusion system | MATHEMATICS | QUASI-VARIATIONAL INCLUSIONS | MATHEMATICS, APPLIED | MAPPINGS | Generalized nonlinear parametric (A, eta, m)-maximal monotone operator inclusion system | OPTIMIZATION | Operators | Sensitivity analysis | Existence theorems | Inequalities | Nonlinearity | Hilbert space | Inclusions

Existence and continuity theorem | Sensitive analysis | Parametric resolvent operator technique | Relaxed cocoercive operator | Generalized nonlinear parametric [formula omitted]-maximal monotone operator inclusion system | Generalized nonlinear parametric (A,η,m)-maximal monotone operator inclusion system | MATHEMATICS | QUASI-VARIATIONAL INCLUSIONS | MATHEMATICS, APPLIED | MAPPINGS | Generalized nonlinear parametric (A, eta, m)-maximal monotone operator inclusion system | OPTIMIZATION | Operators | Sensitivity analysis | Existence theorems | Inequalities | Nonlinearity | Hilbert space | Inclusions

Journal Article

International Journal of Computer Vision, ISSN 0920-5691, 5/2011, Volume 92, Issue 3, pp. 265 - 280

We examine the underlying structure of popular algorithms for variational methods used in image processing. We focus here on operator splittings and Bregman...

Augmented Lagrangian method | Pattern Recognition | Douglas-Rachford splitting | Computer Science | Computer Imaging, Vision, Pattern Recognition and Graphics | Image Processing and Computer Vision | Artificial Intelligence (incl. Robotics) | Forward-backward splitting | Alternating split Bregman algorithm | Bregman methods | Image denoising | Douglas-rachford splitting | Alternating split bregman algorithm | Augmented lagrangian method | APPROXIMATION | RECONSTRUCTION | SIGNAL RECOVERY | RESTORATION | ITERATIVE ALGORITHMS | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | CONVERGENCE | WAVELET SHRINKAGE | TOTAL VARIATION MINIMIZATION | MONOTONE-OPERATORS | SCHEMES | Usage | Image processing equipment industry | Equipment and supplies | Algorithms | Image processing | Methods | Frames | Operators | Splitting | Images | Mathematical models | Shrinkage

Augmented Lagrangian method | Pattern Recognition | Douglas-Rachford splitting | Computer Science | Computer Imaging, Vision, Pattern Recognition and Graphics | Image Processing and Computer Vision | Artificial Intelligence (incl. Robotics) | Forward-backward splitting | Alternating split Bregman algorithm | Bregman methods | Image denoising | Douglas-rachford splitting | Alternating split bregman algorithm | Augmented lagrangian method | APPROXIMATION | RECONSTRUCTION | SIGNAL RECOVERY | RESTORATION | ITERATIVE ALGORITHMS | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | CONVERGENCE | WAVELET SHRINKAGE | TOTAL VARIATION MINIMIZATION | MONOTONE-OPERATORS | SCHEMES | Usage | Image processing equipment industry | Equipment and supplies | Algorithms | Image processing | Methods | Frames | Operators | Splitting | Images | Mathematical models | Shrinkage

Journal Article

Mathematische Zeitschrift, ISSN 0025-5874, 6/2018, Volume 289, Issue 1, pp. 445 - 454

The aim of this paper is to find some sufficient conditions for positivity of block matrices of positive operators. It is shown that for positive operators...

15A45 | Mathematics, general | Operator monotone function | Mathematics | 47A64 | Positive block matrix | 47A63 | Operator mean | MATHEMATICS | INEQUALITIES | MATRICES

15A45 | Mathematics, general | Operator monotone function | Mathematics | 47A64 | Positive block matrix | 47A63 | Operator mean | MATHEMATICS | INEQUALITIES | MATRICES

Journal Article

Linear Algebra and Its Applications, ISSN 0024-3795, 11/2018, Volume 556, pp. 220 - 237

In this paper, we extend some significant Ky Fan type inequalities in a large setting to operators on Hilbert spaces and derive their equality conditions....

Operator monotone function | Ky Fan type inequalities | Integral representation | Operator mean | MATHEMATICS | MATHEMATICS, APPLIED

Operator monotone function | Ky Fan type inequalities | Integral representation | Operator mean | MATHEMATICS | MATHEMATICS, APPLIED

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 2003, Volume 145, Issue 2, pp. 795 - 803

In this paper, we introduce a new class of monotone operators - H-monotone operators. The resolvent operator associated with an H-monotone operator is defined...

Iterative algorithm | Resolvent operator technique | Variational inclusion | H-Monotone operator | H -Monotone operator | variational inclusion | MATHEMATICS, APPLIED | GENERAL-CLASS | resolvent operator technique | NONCOMPACT VALUED MAPPINGS | PROXIMAL POINT ALGORITHMS | MANN | PERTURBED ITERATIVE ALGORITHMS | H-monotone operator | iterative algorithm

Iterative algorithm | Resolvent operator technique | Variational inclusion | H-Monotone operator | H -Monotone operator | variational inclusion | MATHEMATICS, APPLIED | GENERAL-CLASS | resolvent operator technique | NONCOMPACT VALUED MAPPINGS | PROXIMAL POINT ALGORITHMS | MANN | PERTURBED ITERATIVE ALGORITHMS | H-monotone operator | iterative algorithm

Journal Article

MATHEMATICAL INEQUALITIES & APPLICATIONS, ISSN 1331-4343, 04/2019, Volume 22, Issue 2, pp. 565 - 575

For lambda is an element of (0, 1), let psi be a non-constant, non-negative, continuous function on (0, infinity) and let Gamma(lambda) (psi) be the set of all...

MATHEMATICS | Operator means | operator monotone functions | operator convex functions

MATHEMATICS | Operator means | operator monotone functions | operator convex functions

Journal Article

Mathematical Programming, ISSN 0025-5610, 11/2015, Volume 153, Issue 2, pp. 715 - 722

This note provides a simple proof of a worst-case convergence rate measured by the iteration complexity for the Douglas–Rachford operator splitting method for...

Mathematical Methods in Physics | 65N12 | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | 90C25 | Numerical Analysis | Theoretical, Mathematical and Computational Physics | Convergence rate | 65K10 | Mathematics | Combinatorics | Douglas–Rachford operator splitting method | COMPUTER SCIENCE, SOFTWARE ENGINEERING | PROJECTION | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Douglas-Rachford operator splitting method | MONOTONE VARIATIONAL-INEQUALITIES | PROXIMAL POINT ALGORITHM | Yuan (China) | Methods | Studies | Mathematical programming

Mathematical Methods in Physics | 65N12 | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | 90C25 | Numerical Analysis | Theoretical, Mathematical and Computational Physics | Convergence rate | 65K10 | Mathematics | Combinatorics | Douglas–Rachford operator splitting method | COMPUTER SCIENCE, SOFTWARE ENGINEERING | PROJECTION | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Douglas-Rachford operator splitting method | MONOTONE VARIATIONAL-INEQUALITIES | PROXIMAL POINT ALGORITHM | Yuan (China) | Methods | Studies | Mathematical programming

Journal Article

1973, Lecture notes in mathematics, ISBN 9780387064840, Volume 346, 287

Book

Linear Algebra and Its Applications, ISSN 0024-3795, 09/2018, Volume 553, pp. 238 - 251

This paper concerns three classes of real-valued functions on intervals, operator monotone functions, operator convex functions, and strongly operator convex...

Operator convex functions | Strongly operator convex functions | Operator monotone functions | Completely monotone functions | Loewner theorem | Pick functions | MATHEMATICS | MATHEMATICS, APPLIED | ALGEBRAS | MAJORIZATION

Operator convex functions | Strongly operator convex functions | Operator monotone functions | Completely monotone functions | Loewner theorem | Pick functions | MATHEMATICS | MATHEMATICS, APPLIED | ALGEBRAS | MAJORIZATION

Journal Article

Integral Equations and Operator Theory, ISSN 0378-620X, 9/2017, Volume 89, Issue 1, pp. 1 - 42

We develop a general theory of order isomorphisms of operator intervals. In this way we unify and extend several known results, among others the famous...

Operator interval | Self-adjoint operator | Analysis | Effect algebra | Operator monotone function | Mathematics | Order isomorphism | Primary 47B49 | MATHEMATICS | ELEMENTARY PROOF | AUTOMORPHISMS | Algebra

Operator interval | Self-adjoint operator | Analysis | Effect algebra | Operator monotone function | Mathematics | Order isomorphism | Primary 47B49 | MATHEMATICS | ELEMENTARY PROOF | AUTOMORPHISMS | Algebra

Journal Article

1989, 2nd Edition., Lecture notes in mathematics, ISBN 3540507353, Volume 1364., vii, 114

Monotone operators | Differentiable functions | Convex functions | Functions of complex variables | Functions of a Complex Variable | Systems Theory, Control | Calculus of Variations and Optimal Control; Optimization | Analysis | Global analysis (Mathematics) | Systems theory | Mathematical optimization

Book

14.
Full Text
A new system of variational inclusions with ( H, η)-monotone operators in hilbert spaces

Computers and Mathematics with Applications, ISSN 0898-1221, 2005, Volume 49, Issue 2, pp. 365 - 374

In this paper, we introduce and study a new system of variational inclusions involving ( )-monotone operators in Hilbert space. Using the resolvent operator...

Iterative algorithm | System of variational inclusion | Resolvent operator technique | ( H, η)-monotone operator | (H, η)-monotone operator | (H,eta)-monotone operator | MATHEMATICS, APPLIED | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | INEQUALITIES | resolvent operator technique | MANN | system of variational inclusion | PERTURBED ITERATIVE ALGORITHMS | MAPPINGS | iterative algorithm | PROJECTION METHODS | Operators | Algorithms | Approximation | Uniqueness | Hilbert space | Mathematical models | Inclusions | Convergence

Iterative algorithm | System of variational inclusion | Resolvent operator technique | ( H, η)-monotone operator | (H, η)-monotone operator | (H,eta)-monotone operator | MATHEMATICS, APPLIED | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | INEQUALITIES | resolvent operator technique | MANN | system of variational inclusion | PERTURBED ITERATIVE ALGORITHMS | MAPPINGS | iterative algorithm | PROJECTION METHODS | Operators | Algorithms | Approximation | Uniqueness | Hilbert space | Mathematical models | Inclusions | Convergence

Journal Article

Mathematical Programming, ISSN 0025-5610, 7/2018, Volume 170, Issue 1, pp. 177 - 206

Several aspects of the interplay between monotone operator theory and convex optimization are presented. The crucial role played by monotone operators in the...

65K05 | Self-dual class | Theoretical, Mathematical and Computational Physics | Subdifferential | Proximity operator | Mathematics | Monotone operator | Mathematical Methods in Physics | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | 90C25 | Numerical Analysis | Proximal algorithm | Operator splitting | Proximity-preserving transformation | 49M27 | Combinatorics | Firmly nonexpansive operator | 47H25 | MATHEMATICS, APPLIED | SIGNAL RECOVERY | THRESHOLDING ALGORITHM | DECOMPOSITION | PROXIMAL POINT ALGORITHM | LEAST-SQUARES SOLUTIONS | COMPUTER SCIENCE, SOFTWARE ENGINEERING | NONLINEAR OPERATORS | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | ANALYSE FONCTIONNELLE | CONVERGENCE | PARTIAL INVERSES | SPLITTING METHOD | Analysis | Algorithms | Computational geometry | Operators | Proximity | Convexity | Convex analysis | Optimization

65K05 | Self-dual class | Theoretical, Mathematical and Computational Physics | Subdifferential | Proximity operator | Mathematics | Monotone operator | Mathematical Methods in Physics | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | 90C25 | Numerical Analysis | Proximal algorithm | Operator splitting | Proximity-preserving transformation | 49M27 | Combinatorics | Firmly nonexpansive operator | 47H25 | MATHEMATICS, APPLIED | SIGNAL RECOVERY | THRESHOLDING ALGORITHM | DECOMPOSITION | PROXIMAL POINT ALGORITHM | LEAST-SQUARES SOLUTIONS | COMPUTER SCIENCE, SOFTWARE ENGINEERING | NONLINEAR OPERATORS | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | ANALYSE FONCTIONNELLE | CONVERGENCE | PARTIAL INVERSES | SPLITTING METHOD | Analysis | Algorithms | Computational geometry | Operators | Proximity | Convexity | Convex analysis | Optimization

Journal Article

Linear Algebra and Its Applications, ISSN 0024-3795, 12/2015, Volume 486, pp. 389 - 408

We investigate the operator monotonicity of the following functions: where and with ( ). This property for these functions has been considered by V.E.S. Szabó .

Operator monotone functions | Petz–Hasegawa's functions | Pick functions | Petz-Hasegawa's functions | MATHEMATICS | MATHEMATICS, APPLIED | MATRICES | METRICS

Operator monotone functions | Petz–Hasegawa's functions | Pick functions | Petz-Hasegawa's functions | MATHEMATICS | MATHEMATICS, APPLIED | MATRICES | METRICS

Journal Article

Journal of Optimization Theory and Applications, ISSN 0022-3239, 10/2019, Volume 183, Issue 1, pp. 179 - 198

The Douglas–Rachford method is a popular splitting technique for finding a zero of the sum of two subdifferential operators of proper, closed, and convex...

Lipschitz continuous mapping | Secondary 49M29 | Mathematics | Theory of Computation | Strongly monotone operator | Optimization | Strongly convex function | Skew-symmetric operator | Linear convergence | Primary 47H05 | Calculus of Variations and Optimal Control; Optimization | 90C25 | Operations Research/Decision Theory | Douglas–Rachford algorithm | 47H09 | 49M27 | 49N15 | Applications of Mathematics | Engineering, general | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Douglas-Rachford algorithm | INCLUSIONS | ALGORITHMS | Electrical engineering | Algorithms | Operators (mathematics) | Splitting | Convergence

Lipschitz continuous mapping | Secondary 49M29 | Mathematics | Theory of Computation | Strongly monotone operator | Optimization | Strongly convex function | Skew-symmetric operator | Linear convergence | Primary 47H05 | Calculus of Variations and Optimal Control; Optimization | 90C25 | Operations Research/Decision Theory | Douglas–Rachford algorithm | 47H09 | 49M27 | 49N15 | Applications of Mathematics | Engineering, general | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Douglas-Rachford algorithm | INCLUSIONS | ALGORITHMS | Electrical engineering | Algorithms | Operators (mathematics) | Splitting | Convergence

Journal Article

Linear Algebra and Its Applications, ISSN 0024-3795, 11/2017, Volume 532, pp. 127 - 139

Let be positive matrices in and be a continuous real-valued function on . In addition, consider Φ as a positive linear functional on and define as a variables...

Operator convex functions | Operator monotone functions | BMV conjecture | Laplace transform | MATHEMATICS | MATHEMATICS, APPLIED

Operator convex functions | Operator monotone functions | BMV conjecture | Laplace transform | MATHEMATICS | MATHEMATICS, APPLIED

Journal Article

Linear and Multilinear Algebra, ISSN 0308-1087, 12/2016, Volume 64, Issue 12, pp. 2463 - 2473

In this paper, we obtain a new class of functions, which is developed via the Hermite-Hadamard inequality for convex functions. The well-known one-one...

46L07 | invariance of mean | 47A12 | 47A64 | positive operator | 47A63 | operator monotone function | operator mean | MATHEMATICS | INEQUALITIES | Operators (mathematics) | Operators | Algebra | Monotone functions | Inequalities | Images | Joints

46L07 | invariance of mean | 47A12 | 47A64 | positive operator | 47A63 | operator monotone function | operator mean | MATHEMATICS | INEQUALITIES | Operators (mathematics) | Operators | Algebra | Monotone functions | Inequalities | Images | Joints

Journal Article

Journal of the London Mathematical Society, ISSN 0024-6107, 8/2002, Volume 66, Issue 1, pp. 240 - 256

Iterative algorithms for nonexpansive mappings and maximal monotone operators are investigated. Strong convergence theorems are proved for nonexpansive...

MATHEMATICS | NONEXPANSIVE-MAPPINGS | CONVERGENCE THEOREMS | APPROXIMATION | BANACH-SPACES | PROXIMAL POINT ALGORITHM | MONOTONE-OPERATORS | FIXED-POINTS

MATHEMATICS | NONEXPANSIVE-MAPPINGS | CONVERGENCE THEOREMS | APPROXIMATION | BANACH-SPACES | PROXIMAL POINT ALGORITHM | MONOTONE-OPERATORS | FIXED-POINTS

Journal Article