Journal of Optimization Theory and Applications, ISSN 0022-3239, 3/2000, Volume 104, Issue 3, pp. 517 - 538

.... The algorithm has polynomial complexity and it converges with asymptotic quadratic rate. When implementing the method to recover images, we take advantage of the underlying sparsity of the problem...

variable-metric proximal-point methods | Calculus of Variations and Optimal Control | Mathematics | Theory of Computation | infeasible-interior-point methods | Applications of Mathematics | Engineering, general | Operation Research/Decision Theory | image restoration in the presence of noise | kinky images | Optimization | predictor-corrector methods | Predictor-corrector methods | Image restoration in the presence of noise | Infeasible-interior-point methods | Kinky images | Variable-metric proximal-point methods | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | LINEAR COMPLEMENTARITY-PROBLEMS | CONVERGENCE | ALGORITHMS

variable-metric proximal-point methods | Calculus of Variations and Optimal Control | Mathematics | Theory of Computation | infeasible-interior-point methods | Applications of Mathematics | Engineering, general | Operation Research/Decision Theory | image restoration in the presence of noise | kinky images | Optimization | predictor-corrector methods | Predictor-corrector methods | Image restoration in the presence of noise | Infeasible-interior-point methods | Kinky images | Variable-metric proximal-point methods | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | LINEAR COMPLEMENTARITY-PROBLEMS | CONVERGENCE | ALGORITHMS

Journal Article

Applied mathematics & optimization, ISSN 1432-0606, 2019, Volume 80, Issue 3, pp. 547 - 598

In a Hilbert space $${{\mathcal {H}}}$$ H , we study the convergence properties of a class of relaxed inertial forward–backward algorithms. They aim to solve...

65K05 | Systems Theory, Control | Inertial Krasnoselskii–Mann iteration | Theoretical, Mathematical and Computational Physics | Mathematics | Relaxation | Mathematical Methods in Physics | Inertial forward–backward algorithms | Calculus of Variations and Optimal Control; Optimization | 90C25 | Nash equilibration | Structured monotone inclusions | Convergence rate | 65K10 | Numerical and Computational Physics, Simulation | 49M37 | Cocoercive operators

65K05 | Systems Theory, Control | Inertial Krasnoselskii–Mann iteration | Theoretical, Mathematical and Computational Physics | Mathematics | Relaxation | Mathematical Methods in Physics | Inertial forward–backward algorithms | Calculus of Variations and Optimal Control; Optimization | 90C25 | Nash equilibration | Structured monotone inclusions | Convergence rate | 65K10 | Numerical and Computational Physics, Simulation | 49M37 | Cocoercive operators

Journal Article

NPJ microgravity, ISSN 2373-8065, 2017, Volume 3, Issue 1, pp. 8 - 10

Without effective countermeasures, the musculoskeletal system is altered by the microgravity environment of long-duration spaceflight, resulting in atrophy of...

Journal Article | PROLONGED BED REST | SIMULATED WEIGHTLESSNESS | LONG-DURATION SPACEFLIGHT | RESISTANCE EXERCISE | MULTIDISCIPLINARY SCIENCES | PROXIMAL FEMORAL STRENGTH | IONIZING-RADIATION | RESISTIVE VIBRATION EXERCISE | HEALING RESPONSES | HUMAN SKELETAL-MUSCLE | WHOLE-BODY VIBRATION | Vertebrae | Animal models | Computer simulation | Immobilization | Sex differences | Space flight | Bone healing | Tendons | Atrophy | Cartilage | Sarcopenia | Musculoskeletal system | Osteoporosis | Fractures | Multinational space ventures | Sleep | Microgravity | Age | Immune system | Life Sciences | Human health and pathology | Endocrinology and metabolism

Journal Article | PROLONGED BED REST | SIMULATED WEIGHTLESSNESS | LONG-DURATION SPACEFLIGHT | RESISTANCE EXERCISE | MULTIDISCIPLINARY SCIENCES | PROXIMAL FEMORAL STRENGTH | IONIZING-RADIATION | RESISTIVE VIBRATION EXERCISE | HEALING RESPONSES | HUMAN SKELETAL-MUSCLE | WHOLE-BODY VIBRATION | Vertebrae | Animal models | Computer simulation | Immobilization | Sex differences | Space flight | Bone healing | Tendons | Atrophy | Cartilage | Sarcopenia | Musculoskeletal system | Osteoporosis | Fractures | Multinational space ventures | Sleep | Microgravity | Age | Immune system | Life Sciences | Human health and pathology | Endocrinology and metabolism

Journal Article

64.
Full Text
Affine Nonexpansive Operators, Attouch–Théra Duality and the Douglas–Rachford Algorithm

Set-valued and variational analysis, ISSN 1877-0541, 2017, Volume 25, Issue 3, pp. 481 - 505

.... In this paper, we revisit the original affine setting. We provide a powerful convergence result for finding a zero of the sum of two maximally monotone affine relations...

Strong convergence | Probability Theory and Stochastic Processes | Paramonotone operator | Mathematics | Toeplitz matrix | Nonexpansive mapping | Maximally monotone operator | Linear convergence | Secondary 49M29, 49N15, 90C25 | Analysis | Affine mapping | Douglas–Rachford algorithm | Primary 47H05, 47H09, 49M27 | Tridiagonal matrix | Attouch–Théra duality | MATHEMATICS, APPLIED | MAXIMAL MONOTONE-OPERATORS | Attouch-Thera duality | PROXIMAL POINT ALGORITHM | SUM | PARAMONOTONICITY | Douglas-Rachford algorithm | CONVERGENCE | SUBSPACES | Algorithms | Differential equations

Strong convergence | Probability Theory and Stochastic Processes | Paramonotone operator | Mathematics | Toeplitz matrix | Nonexpansive mapping | Maximally monotone operator | Linear convergence | Secondary 49M29, 49N15, 90C25 | Analysis | Affine mapping | Douglas–Rachford algorithm | Primary 47H05, 47H09, 49M27 | Tridiagonal matrix | Attouch–Théra duality | MATHEMATICS, APPLIED | MAXIMAL MONOTONE-OPERATORS | Attouch-Thera duality | PROXIMAL POINT ALGORITHM | SUM | PARAMONOTONICITY | Douglas-Rachford algorithm | CONVERGENCE | SUBSPACES | Algorithms | Differential equations

Journal Article

Science China Mathematics, ISSN 1674-7283, 12/2019, Volume 62, Issue 12, pp. 2447 - 2462

We show that group actions on many treelike compact spaces are not too complicated dynamically. We first observe that an old argument of Seidler (1990) implies...

fragmentability | median pretree | proximal action | tame dynamical system | secondary 54H15, 22A25 | dendrite | Mathematics | amenable group | Applications of Mathematics | Rosenthal Banach space | Primary 54H20 | dendron | MATHEMATICS, APPLIED | REPRESENTATIONS | MATHEMATICS | DYNAMICAL-SYSTEMS | BANACH-SPACES

fragmentability | median pretree | proximal action | tame dynamical system | secondary 54H15, 22A25 | dendrite | Mathematics | amenable group | Applications of Mathematics | Rosenthal Banach space | Primary 54H20 | dendron | MATHEMATICS, APPLIED | REPRESENTATIONS | MATHEMATICS | DYNAMICAL-SYSTEMS | BANACH-SPACES

Journal Article

Journal of inequalities and applications, ISSN 1029-242X, 2014, Volume 2014, Issue 1, pp. 1 - 9

Given a monotone operator in a Banach space, we show that an iterative sequence, which is implicitly defined by a fixed point theorem for mappings of firmly...

convex minimization | fixed point | Analysis | Mathematics, general | mapping of firmly nonexpansive type | Mathematics | variational inequality | Applications of Mathematics | Banach space | proximal point method | monotone operator | Convex minimization | Mapping of firmly nonexpansive type | Variational inequality | Monotone operator | Proximal point method | Fixed point | MATHEMATICS | MATHEMATICS, APPLIED | THEOREMS | MAPPINGS | CONVERGENCE | Operators | Theorems | Inequalities | Norms | Minimization | Mapping | Optimization

convex minimization | fixed point | Analysis | Mathematics, general | mapping of firmly nonexpansive type | Mathematics | variational inequality | Applications of Mathematics | Banach space | proximal point method | monotone operator | Convex minimization | Mapping of firmly nonexpansive type | Variational inequality | Monotone operator | Proximal point method | Fixed point | MATHEMATICS | MATHEMATICS, APPLIED | THEOREMS | MAPPINGS | CONVERGENCE | Operators | Theorems | Inequalities | Norms | Minimization | Mapping | Optimization

Journal Article

Transactions of the American Mathematical Society, ISSN 0002-9947, 06/2015, Volume 367, Issue 6, pp. 3929 - 3953

sequence of spaces, which solves a problem of Kuwae and Shioya [Trans. Amer. Math. Soc. , no. 1, 2008].]]>

CONVEX FUNCTIONALS | HARMONIC MAPS | SPACES | APPROXIMATIONS | semigroup of nonexpansive maps | MARKOV OPERATORS | PROXIMAL POINT ALGORITHM | Mosco convergence | MATHEMATICS | WEAK-CONVERGENCE | SETS | DIRICHLET FORMS | Convex function | gradient flow | Hadamard space | weak convergence | resolvent

CONVEX FUNCTIONALS | HARMONIC MAPS | SPACES | APPROXIMATIONS | semigroup of nonexpansive maps | MARKOV OPERATORS | PROXIMAL POINT ALGORITHM | Mosco convergence | MATHEMATICS | WEAK-CONVERGENCE | SETS | DIRICHLET FORMS | Convex function | gradient flow | Hadamard space | weak convergence | resolvent

Journal Article

Numerical Algorithms, ISSN 1017-1398, 4/2019, Volume 80, Issue 4, pp. 1155 - 1179

In this paper, by using products of finitely many resolvents of monotone operators, we propose an iterative algorithm for finding a common zero of a finite...

C A T space | 47H10 | Algorithms | Algebra | Numerical Analysis | Computer Science | Numeric Computing | 47H09 | Theory of Computation | Monotone operator | Proximal point algorithm | CAT space | MATHEMATICS, APPLIED | NONEXPANSIVE-MAPPINGS | RESOLVENTS | PRODUCTS | ALGORITHM | CONVERGENCE

C A T space | 47H10 | Algorithms | Algebra | Numerical Analysis | Computer Science | Numeric Computing | 47H09 | Theory of Computation | Monotone operator | Proximal point algorithm | CAT space | MATHEMATICS, APPLIED | NONEXPANSIVE-MAPPINGS | RESOLVENTS | PRODUCTS | ALGORITHM | CONVERGENCE

Journal Article

Machine Learning, ISSN 0885-6125, 4/2017, Volume 106, Issue 4, pp. 595 - 622

Composite penalties have been widely used for inducing structured properties in the empirical risk minimization (ERM) framework in machine learning. Such...

Empirical risk minimization | Control, Robotics, Mechatronics | Proximal average | Composite regularizer | Computer Science | Incremental gradient descent | Artificial Intelligence (incl. Robotics) | Computing Methodologies | Simulation and Modeling | Language Translation and Linguistics | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | Artificial intelligence | Algorithms | Approximation | Mathematical analysis | Data sets | Machine learning | Minimization | Empirical analysis | Optimization | Convergence

Empirical risk minimization | Control, Robotics, Mechatronics | Proximal average | Composite regularizer | Computer Science | Incremental gradient descent | Artificial Intelligence (incl. Robotics) | Computing Methodologies | Simulation and Modeling | Language Translation and Linguistics | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | Artificial intelligence | Algorithms | Approximation | Mathematical analysis | Data sets | Machine learning | Minimization | Empirical analysis | Optimization | Convergence

Journal Article

Journal of scientific computing, ISSN 1573-7691, 2015, Volume 66, Issue 3, pp. 889 - 916

The formulation $$\begin{aligned} \min _{x,y} ~f(x)+g(y),\quad \text{ subject } \text{ to } Ax+By=b, \end{aligned}$$ min x , y f ( x ) + g ( y ) , subject to A...

Alternating direction method of multipliers | Computational Mathematics and Numerical Analysis | Global convergence | Linear convergence | Algorithms | Strong convexity | Theoretical, Mathematical and Computational Physics | Appl.Mathematics/Computational Methods of Engineering | Mathematics | Distributed computing | MATHEMATICS, APPLIED | MINIMIZATION | Distributed computingb | PROXIMAL POINT ALGORITHM | OPTIMIZATION | SPLITTING ALGORITHMS | FLOW ALGORITHMS | MONOTONE-OPERATORS | Image processing | Methods | Machine learning | Splitting | Multipliers | Mathematical analysis | Mathematical models | Convexity | Statistics | Optimization | Convergence

Alternating direction method of multipliers | Computational Mathematics and Numerical Analysis | Global convergence | Linear convergence | Algorithms | Strong convexity | Theoretical, Mathematical and Computational Physics | Appl.Mathematics/Computational Methods of Engineering | Mathematics | Distributed computing | MATHEMATICS, APPLIED | MINIMIZATION | Distributed computingb | PROXIMAL POINT ALGORITHM | OPTIMIZATION | SPLITTING ALGORITHMS | FLOW ALGORITHMS | MONOTONE-OPERATORS | Image processing | Methods | Machine learning | Splitting | Multipliers | Mathematical analysis | Mathematical models | Convexity | Statistics | Optimization | Convergence

Journal Article

Mathematical programming, ISSN 0025-5610, 2011, Volume 129, Issue 2, pp. 163 - 195

We consider the minimization of a sum $${\sum_{i=1}^mf_i(x)}$$ consisting of a large number of convex component functions f i . For this problem, incremental...

Convex | Theoretical, Mathematical and Computational Physics | Mathematics | Mathematical Methods in Physics | Mathematics of Computing | Calculus of Variations and Optimal Control; Optimization | Numerical Analysis | Proximal algorithm | 90C33 | 90C90 | Combinatorics | Incremental method | Gradient method | MATHEMATICS, APPLIED | GRADIENT METHODS | THRESHOLDING ALGORITHM | NOISE | SUM | LEAST-SQUARES | COMPUTER SCIENCE, SOFTWARE ENGINEERING | PROJECTION | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | SUBGRADIENT METHODS | CONVERGENCE | MONOTONE-OPERATORS | Computer science | Electrical engineering | Algorithms | Analysis | Methods | Studies | Signal processing | Artificial intelligence | Optimization | Mathematical programming | Randomization | Mathematical analysis | Machine learning | Minimization | Iterative methods | Convergence

Convex | Theoretical, Mathematical and Computational Physics | Mathematics | Mathematical Methods in Physics | Mathematics of Computing | Calculus of Variations and Optimal Control; Optimization | Numerical Analysis | Proximal algorithm | 90C33 | 90C90 | Combinatorics | Incremental method | Gradient method | MATHEMATICS, APPLIED | GRADIENT METHODS | THRESHOLDING ALGORITHM | NOISE | SUM | LEAST-SQUARES | COMPUTER SCIENCE, SOFTWARE ENGINEERING | PROJECTION | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | SUBGRADIENT METHODS | CONVERGENCE | MONOTONE-OPERATORS | Computer science | Electrical engineering | Algorithms | Analysis | Methods | Studies | Signal processing | Artificial intelligence | Optimization | Mathematical programming | Randomization | Mathematical analysis | Machine learning | Minimization | Iterative methods | Convergence

Journal Article

Computational optimization and applications, ISSN 1573-2894, 2019, Volume 72, Issue 3, pp. 641 - 674

Successive quadratic approximations, or second-order proximal methods, are useful for minimizing functions that are a sum of a smooth part and a convex,...

Nonconvex optimization | Second-order approximation | Variable metric | Mathematics | Statistics, general | Optimization | Proximal method | Regularized optimization | Inexact method | Convex optimization | Operations Research/Decision Theory | Convex and Discrete Geometry | Operations Research, Management Science | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Analysis | Algorithms | Approximation | Mathematical analysis | Iterative algorithms | Convexity | Nonlinear programming | Regularization | Optimality criteria | Convergence | Complexity

Nonconvex optimization | Second-order approximation | Variable metric | Mathematics | Statistics, general | Optimization | Proximal method | Regularized optimization | Inexact method | Convex optimization | Operations Research/Decision Theory | Convex and Discrete Geometry | Operations Research, Management Science | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Analysis | Algorithms | Approximation | Mathematical analysis | Iterative algorithms | Convexity | Nonlinear programming | Regularization | Optimality criteria | Convergence | Complexity

Journal Article

Journal of global optimization, ISSN 1573-2916, 2017, Volume 70, Issue 3, pp. 517 - 549

We propose an inexact proximal bundle method for constrained nonsmooth nonconvex optimization problems whose objective and constraint functions are known...

Nonconvex optimization | Proximal bundle method | Inexact oracle | Mathematics | 90C26 | Optimization | Constrained optimization | 49J52 | Operations Research/Decision Theory | Nonsmooth optimization | Computer Science, general | Real Functions | 93B40 | MATHEMATICS, APPLIED | PENALTY-FUNCTION | FILTER | ALGORITHM | CONVEX-OPTIMIZATION | APPROXIMATE SUBGRADIENT LINEARIZATIONS | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | MINIMIZATION | CONVERGENCE | Economic models | Nonlinear programming

Nonconvex optimization | Proximal bundle method | Inexact oracle | Mathematics | 90C26 | Optimization | Constrained optimization | 49J52 | Operations Research/Decision Theory | Nonsmooth optimization | Computer Science, general | Real Functions | 93B40 | MATHEMATICS, APPLIED | PENALTY-FUNCTION | FILTER | ALGORITHM | CONVEX-OPTIMIZATION | APPROXIMATE SUBGRADIENT LINEARIZATIONS | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | MINIMIZATION | CONVERGENCE | Economic models | Nonlinear programming

Journal Article

Journal of optimization theory and applications, ISSN 1573-2878, 2015, Volume 171, Issue 2, pp. 600 - 616

We investigate the convergence of a forward–backward–forward proximal-type algorithm with inertial and memory effects when minimizing the sum of a nonsmooth...

Inertial proximal algorithm | Limiting subdifferential | Mathematics | Theory of Computation | 90C26 | Bregman distance | Tseng’s type proximal algorithm | Optimization | 90C30 | Calculus of Variations and Optimal Control; Optimization | 65K10 | Nonsmooth optimization | Applications of Mathematics | Engineering, general | Operation Research/Decision Theory | Kurdyka–Łojasiewicz inequality | MATHEMATICS, APPLIED | Kurdyka-Lojasiewicz inequality | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | INEQUALITIES | MAXIMAL MONOTONE-OPERATORS | MINIMIZATION | Tseng's type proximal algorithm | CONVERGENCE | Algorithms | Studies | Regularization methods | Mathematical analysis | Convexity | Regularization | Inequalities | Convergence

Inertial proximal algorithm | Limiting subdifferential | Mathematics | Theory of Computation | 90C26 | Bregman distance | Tseng’s type proximal algorithm | Optimization | 90C30 | Calculus of Variations and Optimal Control; Optimization | 65K10 | Nonsmooth optimization | Applications of Mathematics | Engineering, general | Operation Research/Decision Theory | Kurdyka–Łojasiewicz inequality | MATHEMATICS, APPLIED | Kurdyka-Lojasiewicz inequality | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | INEQUALITIES | MAXIMAL MONOTONE-OPERATORS | MINIMIZATION | Tseng's type proximal algorithm | CONVERGENCE | Algorithms | Studies | Regularization methods | Mathematical analysis | Convexity | Regularization | Inequalities | Convergence

Journal Article

Journal of inequalities and applications, ISSN 1029-242X, 2018, Volume 2018, Issue 1, pp. 1 - 14

The purpose of this article is to propose a modified viscosity implicit-type proximal point algorithm for approximating a common solution of a monotone...

65K05 | Monotone mapping | Mathematics | Proximal point algorithm | Viscosity implicit approximation methods | 47J20 | Analysis | 47H09 | CAT space | Mathematics, general | 47J05 | Applications of Mathematics | 47J25 | 47H05 | Hadamard space | MATHEMATICS, APPLIED | CAT METRIC-SPACES | ACCRETIVE-OPERATORS | MATHEMATICS | MINIMIZATION | BANACH-SPACES | CURVATURE | CONVERGENCE | FIXED-POINTS | Viscosity | Fixed points (mathematics) | Algorithms | Research

65K05 | Monotone mapping | Mathematics | Proximal point algorithm | Viscosity implicit approximation methods | 47J20 | Analysis | 47H09 | CAT space | Mathematics, general | 47J05 | Applications of Mathematics | 47J25 | 47H05 | Hadamard space | MATHEMATICS, APPLIED | CAT METRIC-SPACES | ACCRETIVE-OPERATORS | MATHEMATICS | MINIMIZATION | BANACH-SPACES | CURVATURE | CONVERGENCE | FIXED-POINTS | Viscosity | Fixed points (mathematics) | Algorithms | Research

Journal Article

Vietnam Journal of Mathematics, ISSN 2305-221X, 3/2018, Volume 46, Issue 1, pp. 53 - 71

We propose a proximal-gradient algorithm with penalization terms and inertial and memory effects for minimizing the sum of a proper, convex, and lower...

65K05 | Penalization | 90C25 | Mathematics, general | Mathematics | Fenchel conjugate | 47H05 | Proximal-gradient algorithm | Inertial algorithm

65K05 | Penalization | 90C25 | Mathematics, general | Mathematics | Fenchel conjugate | 47H05 | Proximal-gradient algorithm | Inertial algorithm

Journal Article