Topology and its Applications, ISSN 0166-8641, 2002, Volume 117, Issue 3, pp. 259 - 272

Let X be a compact metric space, and let f :X→X be transitive with X infinite. We show that each asymptotic class (or the stable set W s ( x) for each x∈ X) is...

Li–Yorke's chaos | Proximal and asymptotic relation | Scrambled set | Scattering | Devaney's chaos | Li-Yorke's chaos | MATHEMATICS | MATHEMATICS, APPLIED | scattering | scrambled set | proximal and asymptotic relation

Li–Yorke's chaos | Proximal and asymptotic relation | Scrambled set | Scattering | Devaney's chaos | Li-Yorke's chaos | MATHEMATICS | MATHEMATICS, APPLIED | scattering | scrambled set | proximal and asymptotic relation

Journal Article

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Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity

Mathematical Programming, ISSN 0025-5610, 3/2018, Volume 168, Issue 1, pp. 123 - 175

In a Hilbert space setting $${{\mathcal {H}}}$$ H , we study the fast convergence properties as $$t \rightarrow + \infty $$ t→+∞ of the trajectories of the...

65K05 | Inertial dynamics | Theoretical, Mathematical and Computational Physics | Mathematics | Gradient flows | Dynamical systems | 34D05 | Mathematical Methods in Physics | 90C30 | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | Convex optimization | 90C25 | Numerical Analysis | Fast convergent methods | Vanishing viscosity | 65K10 | Combinatorics | 49M25 | Nesterov method | SYSTEM | MATHEMATICS, APPLIED | MAXIMAL MONOTONE-OPERATORS | PROXIMAL METHOD | BEHAVIOR | EQUATIONS | COMPUTER SCIENCE, SOFTWARE ENGINEERING | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Analysis | Algorithms | Differential equations | Hilbert space | Trajectories | Nonlinear programming | Convergence | Viscous damping | Optimization and Control

65K05 | Inertial dynamics | Theoretical, Mathematical and Computational Physics | Mathematics | Gradient flows | Dynamical systems | 34D05 | Mathematical Methods in Physics | 90C30 | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | Convex optimization | 90C25 | Numerical Analysis | Fast convergent methods | Vanishing viscosity | 65K10 | Combinatorics | 49M25 | Nesterov method | SYSTEM | MATHEMATICS, APPLIED | MAXIMAL MONOTONE-OPERATORS | PROXIMAL METHOD | BEHAVIOR | EQUATIONS | COMPUTER SCIENCE, SOFTWARE ENGINEERING | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Analysis | Algorithms | Differential equations | Hilbert space | Trajectories | Nonlinear programming | Convergence | Viscous damping | Optimization and Control

Journal Article

SIAM Journal on Control and Optimization, ISSN 0363-0129, 2011, Volume 49, Issue 2, pp. 574 - 598

We introduce nonautonomous continuous dynamical systems which are linked to the Newton and Levenberg-Marquardt methods. They aim at solving inclusions governed...

Nonautonomous differential equations | Weak asymptotic convergence | Lyapunov analysis | Maximal monotone operators | Newton-like algorithms | Absolutely continuous trajectories | Levenberg-Marquardt algorithms | Dissipative dynamical systems | Numerical convex optimization | dissipative dynamical systems | weak asymptotic convergence | SYSTEM | MATHEMATICS, APPLIED | absolutely continuous trajectories | APPROXIMATIONS | EQUATIONS | nonautonomous differential equations | PROXIMAL POINT ALGORITHM | maximal monotone operators | VARIATIONAL-INEQUALITIES | CONVERGENCE | HILBERT-SPACE | CONVEX MINIMIZATION | OPERATORS | numerical convex optimization | AUTOMATION & CONTROL SYSTEMS | RIEMANNIAN-MANIFOLDS | Studies | Dynamic programming | Asymptotic methods | Convex analysis | Mathematics | Optimization and Control

Nonautonomous differential equations | Weak asymptotic convergence | Lyapunov analysis | Maximal monotone operators | Newton-like algorithms | Absolutely continuous trajectories | Levenberg-Marquardt algorithms | Dissipative dynamical systems | Numerical convex optimization | dissipative dynamical systems | weak asymptotic convergence | SYSTEM | MATHEMATICS, APPLIED | absolutely continuous trajectories | APPROXIMATIONS | EQUATIONS | nonautonomous differential equations | PROXIMAL POINT ALGORITHM | maximal monotone operators | VARIATIONAL-INEQUALITIES | CONVERGENCE | HILBERT-SPACE | CONVEX MINIMIZATION | OPERATORS | numerical convex optimization | AUTOMATION & CONTROL SYSTEMS | RIEMANNIAN-MANIFOLDS | Studies | Dynamic programming | Asymptotic methods | Convex analysis | Mathematics | Optimization and Control

Journal Article

Transactions of the American Mathematical Society, ISSN 0002-9947, 08/2014, Volume 366, Issue 8, pp. 4299 - 4322

-hyperbolic spaces, Busemann spaces and CAT(0) spaces. Furthermore, we apply methods of proof mining to obtain effective rates of asymptotic regularity for the...

Social classes | Mathematical monotonicity | Normed spaces | Applied mathematics | Diagonal lemma | Geodesy | Hilbert spaces | Convexity | Banach space | Curvature | Firmly nonexpansive mappings | Picard iterates | Proof mining | Geodesic spaces | Asymptotic regularity | Effective bounds | Δ-convergence | Minimization problems | Uniform convexity | proof mining | geodesic spaces | MONOTONE VECTOR-FIELDS | ACCRETIVE-OPERATORS | PROXIMAL POINT ALGORITHM | uniform convexity | ASYMPTOTIC-BEHAVIOR | MATHEMATICS | NONLINEAR OPERATORS | Delta-convergence | ITERATIONS | REGULARITY | THEOREMS | asymptotic regularity | SETS | CONVERGENCE | minimization problems | effective bounds

Social classes | Mathematical monotonicity | Normed spaces | Applied mathematics | Diagonal lemma | Geodesy | Hilbert spaces | Convexity | Banach space | Curvature | Firmly nonexpansive mappings | Picard iterates | Proof mining | Geodesic spaces | Asymptotic regularity | Effective bounds | Δ-convergence | Minimization problems | Uniform convexity | proof mining | geodesic spaces | MONOTONE VECTOR-FIELDS | ACCRETIVE-OPERATORS | PROXIMAL POINT ALGORITHM | uniform convexity | ASYMPTOTIC-BEHAVIOR | MATHEMATICS | NONLINEAR OPERATORS | Delta-convergence | ITERATIONS | REGULARITY | THEOREMS | asymptotic regularity | SETS | CONVERGENCE | minimization problems | effective bounds

Journal Article

SIAM Journal on Control and Optimization, ISSN 0363-0129, 2007, Volume 46, Issue 5, pp. 1683 - 1704

Finite dimensional local convergence results for self-adaptive proximal point methods and nonlinear functions with multiple minimizers are generalized and...

Degenerate optimization | Self-adaptive method | Multiple minima | Proximal point | proximal point | MATHEMATICS, APPLIED | MULTIPLIER METHODS | self-adaptive method | ALGORITHM | MONOTONICITY | OPTIMIZATION | degenerate optimization | multiple minima | OPERATORS | AUTOMATION & CONTROL SYSTEMS

Degenerate optimization | Self-adaptive method | Multiple minima | Proximal point | proximal point | MATHEMATICS, APPLIED | MULTIPLIER METHODS | self-adaptive method | ALGORITHM | MONOTONICITY | OPTIMIZATION | degenerate optimization | multiple minima | OPERATORS | AUTOMATION & CONTROL SYSTEMS

Journal Article

Mathematical Programming, ISSN 0025-5610, 9/2016, Volume 159, Issue 1, pp. 403 - 434

In this paper, we present a convergence rate analysis for the inexact Krasnosel’skiĭ–Mann iteration built from non-expansive operators. The presented results...

Monotone inclusion | Theoretical, Mathematical and Computational Physics | Convergence rates | Non-expansive operator | Asymptotic regularity | Mathematics | Mathematical Methods in Physics | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | Convex optimization | 90C25 | Numerical Analysis | Krasnosel’skiĭ–Mann iteration | 47H09 | Combinatorics | 47H05 | MATHEMATICS, APPLIED | Krasnosel'skii-Mann iteration | PROXIMAL POINT ALGORITHM | SUM | EXTRAGRADIENT | COMPUTER SCIENCE, SOFTWARE ENGINEERING | MONOTONE INCLUSIONS | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | REGULARITY | WEAK-CONVERGENCE | CONVEX MINIMIZATION | Studies | Mathematical analysis | Asymptotic methods | Optimization | Convergence | Mathematical programming | Operators (mathematics) | Operators | Splitting | Approximation | Criteria | Iterative methods

Monotone inclusion | Theoretical, Mathematical and Computational Physics | Convergence rates | Non-expansive operator | Asymptotic regularity | Mathematics | Mathematical Methods in Physics | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | Convex optimization | 90C25 | Numerical Analysis | Krasnosel’skiĭ–Mann iteration | 47H09 | Combinatorics | 47H05 | MATHEMATICS, APPLIED | Krasnosel'skii-Mann iteration | PROXIMAL POINT ALGORITHM | SUM | EXTRAGRADIENT | COMPUTER SCIENCE, SOFTWARE ENGINEERING | MONOTONE INCLUSIONS | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | REGULARITY | WEAK-CONVERGENCE | CONVEX MINIMIZATION | Studies | Mathematical analysis | Asymptotic methods | Optimization | Convergence | Mathematical programming | Operators (mathematics) | Operators | Splitting | Approximation | Criteria | Iterative methods

Journal Article

Mathematical Programming, ISSN 0025-5610, 6/2013, Volume 139, Issue 1, pp. 55 - 70

We study nearly equal and nearly convex sets, ranges of maximally monotone operators, and ranges and fixed points of convex combinations of firmly nonexpansive...

52A20 | Theoretical, Mathematical and Computational Physics | Asymptotic regularity | Mathematics | Monotone operator | Mathematical Methods in Physics | 47H10 | Resolvent | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | Nearly convex set | 90C25 | Numerical Analysis | Firmly nonexpansive mapping | 47H09 | Combinatorics | 47H05 | COMPUTER SCIENCE, SOFTWARE ENGINEERING | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | ALGORITHM | PROXIMAL AVERAGE | DUALITY | Studies | Mapping | Asymptotic methods | Analysis | Mathematical programming | Operators | Convexity | Asymptotic properties | Sums

52A20 | Theoretical, Mathematical and Computational Physics | Asymptotic regularity | Mathematics | Monotone operator | Mathematical Methods in Physics | 47H10 | Resolvent | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | Nearly convex set | 90C25 | Numerical Analysis | Firmly nonexpansive mapping | 47H09 | Combinatorics | 47H05 | COMPUTER SCIENCE, SOFTWARE ENGINEERING | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | ALGORITHM | PROXIMAL AVERAGE | DUALITY | Studies | Mapping | Asymptotic methods | Analysis | Mathematical programming | Operators | Convexity | Asymptotic properties | Sums

Journal Article

SIAM Journal on Optimization, ISSN 1052-6234, 2012, Volume 22, Issue 2, pp. 557 - 580

We propose a unifying framework that combines smoothing approximation with fast first order algorithms for solving nonsmooth convex minimization problems. We...

Convex minimization | Infimal convolution | Smoothing methods | First order proximal gradients | Nonsmooth convex minimization | Rate of convergence | Asymptotic functions | asymptotic functions | MATHEMATICS, APPLIED | first order proximal gradients | convex minimization | CONVEX | nonsmooth convex minimization | smoothing methods | rate of convergence | OPTIMIZATION | infimal convolution

Convex minimization | Infimal convolution | Smoothing methods | First order proximal gradients | Nonsmooth convex minimization | Rate of convergence | Asymptotic functions | asymptotic functions | MATHEMATICS, APPLIED | first order proximal gradients | convex minimization | CONVEX | nonsmooth convex minimization | smoothing methods | rate of convergence | OPTIMIZATION | infimal convolution

Journal Article

Optimization, ISSN 0233-1934, 10/2015, Volume 64, Issue 10, pp. 2223 - 2252

In a Hilbert framework, we introduce continuous and discrete dynamical systems which aim at solving inclusions governed by structured monotone operators ,...

proximal-gradient method | weak asymptotic convergence | forward-backward algorithms | structured monotone inclusions | Lyapunov analysis | Levenberg-Marquardt regularization | multiobjective decision | dissipative dynamics | cocoercive operators | subdifferential operators | forward–backward algorithms | HILBERT-SPACES | MATHEMATICS, APPLIED | 65K15 | INCLUSIONS | EVOLUTION-EQUATIONS | 90C53 | ASYMPTOTIC CONVERGENCE | MINIMIZATION | 47J25 | SCHEMES | BEHAVIOR | 34G25 | VARIATIONAL-INEQUALITIES | 47J30 | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | 90C25 | 49M37 | 49M15 | 47J35 | Operators | Splitting | Algorithms | Dynamics | Images | Dynamical systems | Optimization | Convergence | Mathematics

proximal-gradient method | weak asymptotic convergence | forward-backward algorithms | structured monotone inclusions | Lyapunov analysis | Levenberg-Marquardt regularization | multiobjective decision | dissipative dynamics | cocoercive operators | subdifferential operators | forward–backward algorithms | HILBERT-SPACES | MATHEMATICS, APPLIED | 65K15 | INCLUSIONS | EVOLUTION-EQUATIONS | 90C53 | ASYMPTOTIC CONVERGENCE | MINIMIZATION | 47J25 | SCHEMES | BEHAVIOR | 34G25 | VARIATIONAL-INEQUALITIES | 47J30 | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | 90C25 | 49M37 | 49M15 | 47J35 | Operators | Splitting | Algorithms | Dynamics | Images | Dynamical systems | Optimization | Convergence | Mathematics

Journal Article

Journal of Optimization Theory and Applications, ISSN 0022-3239, 6/2012, Volume 153, Issue 3, pp. 769 - 778

In this paper, we obtain some results on the boundedness and asymptotic behavior of the sequence generated by the proximal point algorithm without summability...

Maximal monotone operators | Asymptotic behavior | Mathematics | Theory of Computation | Optimization | Proximal-point algorithm | Monotone bifunctions | Calculus of Variations and Optimal Control; Optimization | Equilibrium problems | Operations Research/Decision Theory | Engineering, general | Applications of Mathematics | Rate of convergence | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Algorithms | Errors | Asymptotic properties | Convergence

Maximal monotone operators | Asymptotic behavior | Mathematics | Theory of Computation | Optimization | Proximal-point algorithm | Monotone bifunctions | Calculus of Variations and Optimal Control; Optimization | Equilibrium problems | Operations Research/Decision Theory | Engineering, general | Applications of Mathematics | Rate of convergence | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Algorithms | Errors | Asymptotic properties | Convergence

Journal Article

Journal of Global Optimization, ISSN 0925-5001, 3/2013, Volume 55, Issue 3, pp. 507 - 520

In this paper, we present a unified approach for studying convex composite multiobjective optimization problems via asymptotic analysis. We characterize the...

Weak Pareto optimal solution | Convex composite multiobjective optimization | 90C29 | 90C48 | Nonemptiness and compactness | Optimization | Economics / Management Science | 90C25 | Operations Research/Decision Theory | Computer Science, general | Asymptotic analysis | Proximal-type method | Real Functions | MATHEMATICS, APPLIED | ALGORITHMS | WEAKLY EFFICIENT SOLUTIONS | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | VECTOR OPTIMIZATION | NONEMPTINESS | SOLUTION SETS | OPTIMALITY CONDITIONS | COMPACTNESS | Studies | Mathematical models | Mathematics | Convex analysis | Pareto optimality | Asymptotic properties | Convergence

Weak Pareto optimal solution | Convex composite multiobjective optimization | 90C29 | 90C48 | Nonemptiness and compactness | Optimization | Economics / Management Science | 90C25 | Operations Research/Decision Theory | Computer Science, general | Asymptotic analysis | Proximal-type method | Real Functions | MATHEMATICS, APPLIED | ALGORITHMS | WEAKLY EFFICIENT SOLUTIONS | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | VECTOR OPTIMIZATION | NONEMPTINESS | SOLUTION SETS | OPTIMALITY CONDITIONS | COMPACTNESS | Studies | Mathematical models | Mathematics | Convex analysis | Pareto optimality | Asymptotic properties | Convergence

Journal Article

Computational Optimization and Applications, ISSN 0926-6003, 5/2011, Volume 49, Issue 1, pp. 179 - 192

In this paper, we consider an extend-valued nonsmooth multiobjective optimization problem of finding weak Pareto optimal solutions. We propose a class of...

Weak Pareto optimal solution | Asymptotic function | Generalized viscosity method | Convex and Discrete Geometry | Operations Research/Decision Theory | Multiobjective optimization | Asymptotic cone | Mathematics | Statistics, general | Operations Research, Mathematical Programming | Optimization | MATHEMATICS, APPLIED | NONEXPANSIVE-MAPPINGS | VARIATIONAL INEQUALITY PROBLEMS | PROXIMAL METHODS | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | VECTOR OPTIMIZATION | EQUILIBRIUM PROBLEMS | FIXED-POINT PROBLEMS | Universities and colleges | Mathematical optimization | Pareto efficiency | Methods | Viscosity | Studies | Pareto optimum | Asymptotic methods | Approximations

Weak Pareto optimal solution | Asymptotic function | Generalized viscosity method | Convex and Discrete Geometry | Operations Research/Decision Theory | Multiobjective optimization | Asymptotic cone | Mathematics | Statistics, general | Operations Research, Mathematical Programming | Optimization | MATHEMATICS, APPLIED | NONEXPANSIVE-MAPPINGS | VARIATIONAL INEQUALITY PROBLEMS | PROXIMAL METHODS | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | VECTOR OPTIMIZATION | EQUILIBRIUM PROBLEMS | FIXED-POINT PROBLEMS | Universities and colleges | Mathematical optimization | Pareto efficiency | Methods | Viscosity | Studies | Pareto optimum | Asymptotic methods | Approximations

Journal Article

Mathematical Programming, ISSN 0025-5610, 1/2018, Volume 167, Issue 1, pp. 99 - 127

We investigate the modeling and the numerical solution of machine learning problems with prediction functions which are linear combinations of elements of a...

62G08 | 68T05 | Theoretical, Mathematical and Computational Physics | Sparse data representation | Mathematics | Forward–backward splitting | Mathematical Methods in Physics | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | Consistent estimator | Convex optimization | 90C25 | Numerical Analysis | Proximal algorithm | 65K10 | Combinatorics | SPARSITY | MATHEMATICS, APPLIED | SIGNAL RECOVERY | ALGORITHMS | INEXACT | COMPUTER SCIENCE, SOFTWARE ENGINEERING | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | MINIMIZATION | Forward-backward splitting | INVERSE PROBLEMS | CONVERGENCE | REGULARIZATION | POINT | Analysis | Algorithms | Machine learning | Error detection | Mathematical models | Regularization | Asymptotic methods | Estimators | Convergence

62G08 | 68T05 | Theoretical, Mathematical and Computational Physics | Sparse data representation | Mathematics | Forward–backward splitting | Mathematical Methods in Physics | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | Consistent estimator | Convex optimization | 90C25 | Numerical Analysis | Proximal algorithm | 65K10 | Combinatorics | SPARSITY | MATHEMATICS, APPLIED | SIGNAL RECOVERY | ALGORITHMS | INEXACT | COMPUTER SCIENCE, SOFTWARE ENGINEERING | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | MINIMIZATION | Forward-backward splitting | INVERSE PROBLEMS | CONVERGENCE | REGULARIZATION | POINT | Analysis | Algorithms | Machine learning | Error detection | Mathematical models | Regularization | Asymptotic methods | Estimators | Convergence

Journal Article

Fixed Point Theory and Applications, ISSN 1687-1820, 12/2018, Volume 2018, Issue 1, pp. 1 - 28

In this paper, we propose two strongly convergent algorithms which combines diagonal subgradient method, projection method and proximal method to solve split...

projected subgradient-proximal algorithm | Mathematical and Computational Biology | equilibrium problem | split equilibrium problem | Mathematics | diagonal subgradient method | Topology | pseudomonotone bifunction | nonexpansive mappings | Analysis | Mathematics, general | Applications of Mathematics | Differential Geometry | common fixed point problem | monotone bifunction | Algorithms | Iterative algorithms | Hilbert space | Queuing theory | Iterative methods | Equilibrium | Asymptotic methods | Convergence

projected subgradient-proximal algorithm | Mathematical and Computational Biology | equilibrium problem | split equilibrium problem | Mathematics | diagonal subgradient method | Topology | pseudomonotone bifunction | nonexpansive mappings | Analysis | Mathematics, general | Applications of Mathematics | Differential Geometry | common fixed point problem | monotone bifunction | Algorithms | Iterative algorithms | Hilbert space | Queuing theory | Iterative methods | Equilibrium | Asymptotic methods | Convergence

Journal Article

Journal of Optimization Theory and Applications, ISSN 0022-3239, 1/2017, Volume 172, Issue 1, pp. 222 - 235

We consider a proximal point algorithm with errors for a maximal monotone operator in a real Hilbert space, previously studied by Boikanyo and Morosanu, where...

90C29 | Mathematics | Theory of Computation | Maximal monotone operator | Resolvent operator | Proximal point algorithm | Optimization | Metric projection | Calculus of Variations and Optimal Control; Optimization | 47H09 | 90C90 | Hilbert space | Applications of Mathematics | Engineering, general | Operation Research/Decision Theory | 47J25 | 47H05 | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | MAXIMAL MONOTONICITY | HILBERT-SPACE | ASYMPTOTIC-BEHAVIOR | Algorithms | Studies | Errors | Operators | Error analysis | Inequalities | Projection | Convergence

90C29 | Mathematics | Theory of Computation | Maximal monotone operator | Resolvent operator | Proximal point algorithm | Optimization | Metric projection | Calculus of Variations and Optimal Control; Optimization | 47H09 | 90C90 | Hilbert space | Applications of Mathematics | Engineering, general | Operation Research/Decision Theory | 47J25 | 47H05 | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | MAXIMAL MONOTONICITY | HILBERT-SPACE | ASYMPTOTIC-BEHAVIOR | Algorithms | Studies | Errors | Operators | Error analysis | Inequalities | Projection | Convergence

Journal Article

Numerical Functional Analysis and Optimization, ISSN 0163-0563, 01/2016, Volume 37, Issue 1, pp. 80 - 91

Let (A, B) be a nonempty bounded closed convex proximal parallel pair in a nearly uniformly convex Banach space and T: A ∪ B → A ∪ B be a continuous and...

Asymptotically nonexpansive maps | property UC | best proximity points | relatively nonexpansive maps | proximal pairs | Asymptoticallynonexpansive maps | proximalpairs | relativelynonexpansive maps | bestproximitypoints | EXISTENCE | MATHEMATICS, APPLIED | CONVERGENCE | Theorems | Proximity | Rectangles | Asymptotic properties | Mapping | Functional analysis | Banach space | Optimization | Mathematics - Functional Analysis

Asymptotically nonexpansive maps | property UC | best proximity points | relatively nonexpansive maps | proximal pairs | Asymptoticallynonexpansive maps | proximalpairs | relativelynonexpansive maps | bestproximitypoints | EXISTENCE | MATHEMATICS, APPLIED | CONVERGENCE | Theorems | Proximity | Rectangles | Asymptotic properties | Mapping | Functional analysis | Banach space | Optimization | Mathematics - Functional Analysis

Journal Article

Journal of Global Optimization, ISSN 0925-5001, 11/2018, Volume 72, Issue 3, pp. 553 - 577

The proximal point algorithm is a widely used tool for solving a variety of convex optimization problems such as finding zeros of maximally monotone operators,...

Jointly firmly nonexpansive families | Uniformly firmly nonexpansive mappings | Mathematics | Proximal point algorithm | Optimization | CAT spaces | Convex optimization | Rates of convergence | 90C25 | Operations Research/Decision Theory | Proof mining | 47H09 | Computer Science, general | 47J25 | 46N10 | 03F10 | Real Functions | MATHEMATICS, APPLIED | HARMONIC MAPS | METRIC-SPACES | GEODESIC SPACES | ASYMPTOTIC-BEHAVIOR | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | VECTOR OPTIMIZATION | FUNCTIONAL-ANALYSIS | FIRMLY NONEXPANSIVE-MAPPINGS | LOGICAL METATHEOREMS | FIXED-POINTS | MONOTONE-OPERATORS | Computer science | Mineral industry | Algorithms | Numerical analysis | Mining industry | Analysis | Computational geometry | Fixed points (mathematics) | Hilbert space | Convexity | Data mining | Convex analysis | Convergence

Jointly firmly nonexpansive families | Uniformly firmly nonexpansive mappings | Mathematics | Proximal point algorithm | Optimization | CAT spaces | Convex optimization | Rates of convergence | 90C25 | Operations Research/Decision Theory | Proof mining | 47H09 | Computer Science, general | 47J25 | 46N10 | 03F10 | Real Functions | MATHEMATICS, APPLIED | HARMONIC MAPS | METRIC-SPACES | GEODESIC SPACES | ASYMPTOTIC-BEHAVIOR | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | VECTOR OPTIMIZATION | FUNCTIONAL-ANALYSIS | FIRMLY NONEXPANSIVE-MAPPINGS | LOGICAL METATHEOREMS | FIXED-POINTS | MONOTONE-OPERATORS | Computer science | Mineral industry | Algorithms | Numerical analysis | Mining industry | Analysis | Computational geometry | Fixed points (mathematics) | Hilbert space | Convexity | Data mining | Convex analysis | Convergence

Journal Article

ESAIM - Control, Optimisation and Calculus of Variations, ISSN 1292-8119, 04/2018, Volume 24, Issue 2, pp. 463 - 477

We address the minimization of the sum of a proper, convex and lower semicontinuous function with a (possibly nonconvex) smooth function from the perspective...

Kurdyka-? ojasiewicz property | Dynamical systems | Continuous forward-backward method | Nonsmooth optimization, limiting subdifferential | continuous forward-backward method | MATHEMATICS, APPLIED | INEQUALITIES | PROXIMAL ALGORITHM | Kurdyka-Lojasiewicz property | nonsmooth optimization | limiting subdifferential | MONOTONE INCLUSIONS | CONVERGENCE | SYSTEMS | OPTIMIZATION | AUTOMATION & CONTROL SYSTEMS | Economic models | Trajectories | Critical point | Regularization | Asymptotic methods | Optimization | Convergence

Kurdyka-? ojasiewicz property | Dynamical systems | Continuous forward-backward method | Nonsmooth optimization, limiting subdifferential | continuous forward-backward method | MATHEMATICS, APPLIED | INEQUALITIES | PROXIMAL ALGORITHM | Kurdyka-Lojasiewicz property | nonsmooth optimization | limiting subdifferential | MONOTONE INCLUSIONS | CONVERGENCE | SYSTEMS | OPTIMIZATION | AUTOMATION & CONTROL SYSTEMS | Economic models | Trajectories | Critical point | Regularization | Asymptotic methods | Optimization | Convergence

Journal Article

Set-Valued and Variational Analysis, ISSN 1877-0533, 9/2011, Volume 19, Issue 3, pp. 361 - 383

Firmly nonexpansive mappings are introduced in Hadamard manifolds, a particular class of Riemannian manifolds with nonpositive sectional curvature. The...

Geometry | Resolvent | Yosida approximation | Analysis | Hadamard manifold | Firmly nonexpansive mapping | Pseudo-contractive mapping | 49J40 | Mathematics | Maximal monotone vector field | 47H05 | EXISTENCE | MATHEMATICS, APPLIED | NONEXPANSIVE-MAPPINGS | ACCRETIVE-OPERATORS | COACCRETIVE OPERATORS | PROXIMAL POINT ALGORITHM | STRONG-CONVERGENCE THEOREMS

Geometry | Resolvent | Yosida approximation | Analysis | Hadamard manifold | Firmly nonexpansive mapping | Pseudo-contractive mapping | 49J40 | Mathematics | Maximal monotone vector field | 47H05 | EXISTENCE | MATHEMATICS, APPLIED | NONEXPANSIVE-MAPPINGS | ACCRETIVE-OPERATORS | COACCRETIVE OPERATORS | PROXIMAL POINT ALGORITHM | STRONG-CONVERGENCE THEOREMS