Mathematical programming, ISSN 1436-4646, 2016, Volume 168, Issue 1-2, pp. 279 - 311

We synthesize and unify notions of regularity, both of individual sets and of collections of sets, as they appear in the convergence theory of projection methods for consistent feasibility problems...

Normal cone | Theoretical, Mathematical and Computational Physics | Mathematics | Transversality | Normal qualification condition | Alternating projections | Mathematical Methods in Physics | CHIP | 90C30 | Calculus of Variations and Optimal Control; Optimization | Douglasâ€“Rachford | HĂ¶lder regularity | Weak-sharp minima | 65K10 | 49M05 | Metric regularity | Combinatorics | 65K05 | Clarke regularity | Prox-regularity | Secondary 49K40 | Mathematics of Computing | Numerical Analysis | 49M37 | Primary 49J53 | MATHEMATICS, APPLIED | SUBDIFFERENTIAL CALCULUS | WEAK SHARP MINIMA | CONVEX INEQUALITIES | Douglas-Rachford | COMPUTER SCIENCE, SOFTWARE ENGINEERING | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | GENERALIZED EQUATIONS | NONCONVEX SETS | METRIC SUBREGULARITY | BANACH-SPACES | ERROR-BOUNDS | Holder regularity | LINEAR REGULARITY | Feasibility | Convergence | Mathematics - Optimization and Control

Normal cone | Theoretical, Mathematical and Computational Physics | Mathematics | Transversality | Normal qualification condition | Alternating projections | Mathematical Methods in Physics | CHIP | 90C30 | Calculus of Variations and Optimal Control; Optimization | Douglasâ€“Rachford | HĂ¶lder regularity | Weak-sharp minima | 65K10 | 49M05 | Metric regularity | Combinatorics | 65K05 | Clarke regularity | Prox-regularity | Secondary 49K40 | Mathematics of Computing | Numerical Analysis | 49M37 | Primary 49J53 | MATHEMATICS, APPLIED | SUBDIFFERENTIAL CALCULUS | WEAK SHARP MINIMA | CONVEX INEQUALITIES | Douglas-Rachford | COMPUTER SCIENCE, SOFTWARE ENGINEERING | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | GENERALIZED EQUATIONS | NONCONVEX SETS | METRIC SUBREGULARITY | BANACH-SPACES | ERROR-BOUNDS | Holder regularity | LINEAR REGULARITY | Feasibility | Convergence | Mathematics - Optimization and Control

Journal Article

Mathematical programming, ISSN 1436-4646, 2015, Volume 159, Issue 1-2, pp. 371 - 401

We adapt the Douglasâ€“Rachford (DR) splitting method to solve nonconvex feasibility problems by studying this method for a class of nonconvex optimization problem...

Mathematical Methods in Physics | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | Numerical Analysis | Theoretical, Mathematical and Computational Physics | 90C06 | 90C90 | Mathematics | 90C26 | Combinatorics | COMPUTER SCIENCE, SOFTWARE ENGINEERING | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | APPROXIMATION | CONVEX | ALGORITHMS | ALTERNATING PROJECTIONS | CONVERGENCE RATE | Studies | Mathematical analysis | Optimization | Convergence | Mathematical programming | Splitting | Thresholds | Direct reduction | Clusters | Mathematical models

Mathematical Methods in Physics | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | Numerical Analysis | Theoretical, Mathematical and Computational Physics | 90C06 | 90C90 | Mathematics | 90C26 | Combinatorics | COMPUTER SCIENCE, SOFTWARE ENGINEERING | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | APPROXIMATION | CONVEX | ALGORITHMS | ALTERNATING PROJECTIONS | CONVERGENCE RATE | Studies | Mathematical analysis | Optimization | Convergence | Mathematical programming | Splitting | Thresholds | Direct reduction | Clusters | Mathematical models

Journal Article

Journal of global optimization, ISSN 1573-2916, 2015, Volume 65, Issue 2, pp. 309 - 327

In recent times the Douglasâ€“Rachford algorithm has been observed empirically to solve a variety of nonconvex feasibility problems including those of a combinatorial nature...

Non-convex | Global convergence | 65K05 | Douglasâ€“Rachford algorithm | Mathematics | 90C26 | Operation Research/Decision Theory | Computer Science, general | Feasibility problem | Optimization | Real Functions | Half-space | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Douglas-Rachford algorithm | ALGORITHM | CONVERGENCE | Analysis | Methods | Algorithms | Studies | Feasibility | Combinatorics | Mathematical analysis | Half spaces | Empirical analysis | Combinatorial analysis | Convergence | Mathematics - Optimization and Control

Non-convex | Global convergence | 65K05 | Douglasâ€“Rachford algorithm | Mathematics | 90C26 | Operation Research/Decision Theory | Computer Science, general | Feasibility problem | Optimization | Real Functions | Half-space | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Douglas-Rachford algorithm | ALGORITHM | CONVERGENCE | Analysis | Methods | Algorithms | Studies | Feasibility | Combinatorics | Mathematical analysis | Half spaces | Empirical analysis | Combinatorial analysis | Convergence | Mathematics - Optimization and Control

Journal Article

Journal of global optimization, ISSN 1573-2916, 2018, Volume 72, Issue 3, pp. 443 - 474

... feasibility problems with finitely many closed possibly nonconvex sets under different assumptions...

Quasi coercivity | 65K05 | Mathematics | 90C26 | Optimization | Quasi FejĂ©r monotonicity | Cyclic algorithm | Linear convergence | Linear regularity | Superregularity | Operations Research/Decision Theory | 65K10 | 49M27 | Secondary 41A25 | Computer Science, general | Strong regularity | Primary 47H10 | Real Functions | Affine-hull regularity | Generalized Douglasâ€“Rachford algorithm | MATHEMATICS, APPLIED | FINITE CONVERGENCE | Generalized Douglas-Rachford algorithm | Quasi Fejer monotonicity | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | EUCLIDEAN SPACES | NONCONVEX SETS | CONVEX | REGULARITY | ALTERNATING PROJECTIONS | Algorithms | Magnetic properties | Economic models | Convexity | Feasibility studies | Convergence | Mathematics - Optimization and Control

Quasi coercivity | 65K05 | Mathematics | 90C26 | Optimization | Quasi FejĂ©r monotonicity | Cyclic algorithm | Linear convergence | Linear regularity | Superregularity | Operations Research/Decision Theory | 65K10 | 49M27 | Secondary 41A25 | Computer Science, general | Strong regularity | Primary 47H10 | Real Functions | Affine-hull regularity | Generalized Douglasâ€“Rachford algorithm | MATHEMATICS, APPLIED | FINITE CONVERGENCE | Generalized Douglas-Rachford algorithm | Quasi Fejer monotonicity | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | EUCLIDEAN SPACES | NONCONVEX SETS | CONVEX | REGULARITY | ALTERNATING PROJECTIONS | Algorithms | Magnetic properties | Economic models | Convexity | Feasibility studies | Convergence | Mathematics - Optimization and Control

Journal Article

Applied mathematics & optimization, ISSN 1432-0606, 2017, Volume 78, Issue 3, pp. 613 - 641

...) linear regularity property and the linear convergence property of the projection-based methods for solving the convex feasibility problem...

65J05 | Systems Theory, Control | Theoretical, Mathematical and Computational Physics | Projection algorithm | Mathematics | Mathematical Methods in Physics | Linear regularity | Convex feasibility problem | 41A25 | Calculus of Variations and Optimal Control; Optimization | 90C25 | 47H09 | Numerical and Computational Physics, Simulation | MATHEMATICS, APPLIED | INFINITE SYSTEM | SETS | OPTIMIZATION | ERROR-BOUNDS | ALGORITHMS | HILBERT-SPACE | STRONG CHIP | Medical colleges | Algorithms | Analysis | Methods | Computational geometry | Projection | Feasibility | Hilbert space | Convexity | Regularity | Convergence

65J05 | Systems Theory, Control | Theoretical, Mathematical and Computational Physics | Projection algorithm | Mathematics | Mathematical Methods in Physics | Linear regularity | Convex feasibility problem | 41A25 | Calculus of Variations and Optimal Control; Optimization | 90C25 | 47H09 | Numerical and Computational Physics, Simulation | MATHEMATICS, APPLIED | INFINITE SYSTEM | SETS | OPTIMIZATION | ERROR-BOUNDS | ALGORITHMS | HILBERT-SPACE | STRONG CHIP | Medical colleges | Algorithms | Analysis | Methods | Computational geometry | Projection | Feasibility | Hilbert space | Convexity | Regularity | Convergence

Journal Article

Computers & operations research, ISSN 0305-0548, 03/2020, Volume 115, p. 104724

....â€˘The chemical feasibility problem was solved using non-linear techniques.â€˘The well location problem was solved using genetic algorithms...

Lithium | Feasibility | Mine planning | Optimisation | Non-convex optimisation | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | GLOBAL OPTIMIZATION | ALGORITHM | ENGINEERING, INDUSTRIAL | Organic chemistry | Pumping | Brines | Transportation | Minimum cost | Iterative methods | Wells | Combinatorial analysis | Optimization | Genetic algorithms

Lithium | Feasibility | Mine planning | Optimisation | Non-convex optimisation | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | GLOBAL OPTIMIZATION | ALGORITHM | ENGINEERING, INDUSTRIAL | Organic chemistry | Pumping | Brines | Transportation | Minimum cost | Iterative methods | Wells | Combinatorial analysis | Optimization | Genetic algorithms

Journal Article

Inverse problems, ISSN 1361-6420, 2004, Volume 20, Issue 4, pp. 1261 - 1266

.... The split feasibility problem (SFP) is to find x is an element of C with Ax is an element of Q, if such x exists...

VARIATIONAL-INEQUALITIES | PROJECTION | MATHEMATICS, APPLIED | CONVEX-SETS | PHYSICS, MATHEMATICAL | INEXACT NEWTON METHODS

VARIATIONAL-INEQUALITIES | PROJECTION | MATHEMATICS, APPLIED | CONVEX-SETS | PHYSICS, MATHEMATICAL | INEXACT NEWTON METHODS

Journal Article

Fixed point theory and applications (Hindawi Publishing Corporation), ISSN 1687-1812, 2013, Volume 2013, Issue 1, pp. 201 - 12

...., variational inequality problems. We devote this paper to developing and improving the self-adaptive methods for solving the split feasibility problem...

split feasibility problem | minimum-norm | Mathematical and Computational Biology | Analysis | self-adaptive method | minimization problem | Mathematics, general | Mathematics | Applications of Mathematics | Topology | projection | Differential Geometry | Minimum-norm | Self-adaptive method | Projection | Split feasibility problem | Minimization problem | MATHEMATICS, APPLIED | POLYAK PROJECTION METHOD | VARIATIONAL INEQUALITY PROBLEMS | ITERATIVE ALGORITHMS | MATHEMATICS | SETS | CONVEX MINIMIZATION | OPERATORS | FIXED-POINT PROBLEMS | Fixed point theory | Usage | Convergence (Mathematics) | Adaptive control

split feasibility problem | minimum-norm | Mathematical and Computational Biology | Analysis | self-adaptive method | minimization problem | Mathematics, general | Mathematics | Applications of Mathematics | Topology | projection | Differential Geometry | Minimum-norm | Self-adaptive method | Projection | Split feasibility problem | Minimization problem | MATHEMATICS, APPLIED | POLYAK PROJECTION METHOD | VARIATIONAL INEQUALITY PROBLEMS | ITERATIVE ALGORITHMS | MATHEMATICS | SETS | CONVEX MINIMIZATION | OPERATORS | FIXED-POINT PROBLEMS | Fixed point theory | Usage | Convergence (Mathematics) | Adaptive control

Journal Article

Journal of global optimization, ISSN 1573-2916, 2015, Volume 65, Issue 2, pp. 329 - 349

The Douglasâ€“Rachford algorithm is a classical and very successful method for solving optimization and feasibility problems...

Projector | Convex set | 65K05 | Polyhedral set | Epigraph | Primary 47H09 | Mathematics | Monotone operator | Optimization | Alternating projections | Method of reflectionâ€“projection | Convex feasibility problem | 65F10 | Finite convergence | 90C25 | Slaterâ€™s condition | Douglasâ€“Rachford algorithm | Partial inverse | 65K10 | 49M27 | Operation Research/Decision Theory | Computer Science, general | Secondary 47H05 | Real Functions | MATHEMATICS, APPLIED | Slater's condition | Method of reflection-projection | SUM | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Douglas-Rachford algorithm | MONOTONE | PROJECTION METHOD | Projectors | Analysis | Algorithms | Studies | Feasibility | Euclidean space | Mathematical analysis | Construction | Projection | Mathematical models | Convergence | Optimization and Control

Projector | Convex set | 65K05 | Polyhedral set | Epigraph | Primary 47H09 | Mathematics | Monotone operator | Optimization | Alternating projections | Method of reflectionâ€“projection | Convex feasibility problem | 65F10 | Finite convergence | 90C25 | Slaterâ€™s condition | Douglasâ€“Rachford algorithm | Partial inverse | 65K10 | 49M27 | Operation Research/Decision Theory | Computer Science, general | Secondary 47H05 | Real Functions | MATHEMATICS, APPLIED | Slater's condition | Method of reflection-projection | SUM | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Douglas-Rachford algorithm | MONOTONE | PROJECTION METHOD | Projectors | Analysis | Algorithms | Studies | Feasibility | Euclidean space | Mathematical analysis | Construction | Projection | Mathematical models | Convergence | Optimization and Control

Journal Article

Numerical algorithms, ISSN 1572-9265, 2015, Volume 72, Issue 4, pp. 835 - 864

The purpose of this paper is to study split feasibility problems and fixed point problems concerning left Bregman strongly relatively nonexpansive mappings in p-uniformly convex and uniformly smooth Banach spaces...

Strong convergence | Uniformly smooth | Numeric Computing | Theory of Computation | Left Bregman strongly nonexpansive mappings | Split feasibility problem | Fixed point problem | Algorithms | Algebra | 49J53 | 90C25 | Numerical Analysis | Computer Science | 65K10 | 49M37 | Uniformly convex | PROJECTION | MATHEMATICS, APPLIED | NONEXPANSIVE OPERATORS | THEOREMS | SETS | CQ ALGORITHM

Strong convergence | Uniformly smooth | Numeric Computing | Theory of Computation | Left Bregman strongly nonexpansive mappings | Split feasibility problem | Fixed point problem | Algorithms | Algebra | 49J53 | 90C25 | Numerical Analysis | Computer Science | 65K10 | 49M37 | Uniformly convex | PROJECTION | MATHEMATICS, APPLIED | NONEXPANSIVE OPERATORS | THEOREMS | SETS | CQ ALGORITHM

Journal Article

SIAM review, ISSN 1095-7200, 1996, Volume 38, Issue 3, pp. 367 - 426

... (most notably, computerized tomography), algorithms for solving convex feasibility problems continue to receive great attention...

Linear convergence | Algorithms | Hyperplanes | Hilbert spaces | Censorship | Mathematics | Mathematical inequalities | Mathematical functions | Lent | Perceptron convergence procedure | Convex feasibility problem | Convex set | Averaged mapping | Firmly nonexpansive mapping | FejĂ©r monotone sequence | Angle between two subspaces | Convex function | Convex inequalities | Convex programming | Cimmino's method | Computerized tomography | CONVERGENCE RESULT | image recovery | MATHEMATICS, APPLIED | convex function | nonexpansive mapping | Slater point | subgradient algorithm | Kaczmarz's method | projection algorithm | projection method | averaged mapping | convex inequalities | linear inequalities | convex set | angle between two subspaces | convex feasibility problem | Hilbert space | computerized tomography | Fejer monotone sequence | NONEXPANSIVE-MAPPINGS | successive projections | PRODUCT SPACE | subgradient | IMAGE-RECONSTRUCTION | convex programming | iterative method | linear feasibility problem | ASYMPTOTIC-BEHAVIOR | RADIATION-THERAPY | firmly nonexpansive mapping | SUBGRADIENT PROJECTIONS | subdifferential | orthogonal projection | linear convergence | ITERATIVE METHODS | HILBERT-SPACE | Usage | Convex functions | Analysis | Methods | Feasibility studies

Linear convergence | Algorithms | Hyperplanes | Hilbert spaces | Censorship | Mathematics | Mathematical inequalities | Mathematical functions | Lent | Perceptron convergence procedure | Convex feasibility problem | Convex set | Averaged mapping | Firmly nonexpansive mapping | FejĂ©r monotone sequence | Angle between two subspaces | Convex function | Convex inequalities | Convex programming | Cimmino's method | Computerized tomography | CONVERGENCE RESULT | image recovery | MATHEMATICS, APPLIED | convex function | nonexpansive mapping | Slater point | subgradient algorithm | Kaczmarz's method | projection algorithm | projection method | averaged mapping | convex inequalities | linear inequalities | convex set | angle between two subspaces | convex feasibility problem | Hilbert space | computerized tomography | Fejer monotone sequence | NONEXPANSIVE-MAPPINGS | successive projections | PRODUCT SPACE | subgradient | IMAGE-RECONSTRUCTION | convex programming | iterative method | linear feasibility problem | ASYMPTOTIC-BEHAVIOR | RADIATION-THERAPY | firmly nonexpansive mapping | SUBGRADIENT PROJECTIONS | subdifferential | orthogonal projection | linear convergence | ITERATIVE METHODS | HILBERT-SPACE | Usage | Convex functions | Analysis | Methods | Feasibility studies

Journal Article

Computational Optimization and Applications, ISSN 0926-6003, 5/2017, Volume 67, Issue 1, pp. 175 - 199

In this paper, based on a merit function of the split feasibility problem (SFP), we present a Newton projection method for solving it and analyze the convergence properties of the method...

65K05 | Newton projection method | The split feasibility problem | Mathematics | Statistics, general | Optimization | Projection operator | 90C30 | Global convergence and convergence rate | Convex and Discrete Geometry | 90C90 | Operations Research, Management Science | Operation Research/Decision Theory | Generalized Jacobian | COMPLEMENTARITY-PROBLEMS | MATHEMATICS, APPLIED | VARIATIONAL INEQUALITY PROBLEMS | CONVEX-SETS | CQ ALGORITHM | BOX CONSTRAINTS | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | MINIMIZATION | D-GAP FUNCTION | SEMISMOOTH EQUATIONS | Management science | Algorithms | Analysis | Methods | Computer science | Studies | Regularization methods | Mathematical analysis | Operators | Projection | Feasibility analysis | Mathematical models | Convergence

65K05 | Newton projection method | The split feasibility problem | Mathematics | Statistics, general | Optimization | Projection operator | 90C30 | Global convergence and convergence rate | Convex and Discrete Geometry | 90C90 | Operations Research, Management Science | Operation Research/Decision Theory | Generalized Jacobian | COMPLEMENTARITY-PROBLEMS | MATHEMATICS, APPLIED | VARIATIONAL INEQUALITY PROBLEMS | CONVEX-SETS | CQ ALGORITHM | BOX CONSTRAINTS | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | MINIMIZATION | D-GAP FUNCTION | SEMISMOOTH EQUATIONS | Management science | Algorithms | Analysis | Methods | Computer science | Studies | Regularization methods | Mathematical analysis | Operators | Projection | Feasibility analysis | Mathematical models | Convergence

Journal Article

Inverse problems, ISSN 1361-6420, 2005, Volume 21, Issue 5, pp. 1791 - 1799

.... The split feasibility problem (SFP) is to find x is an element of C with Ax is an element of Q, if such points exist...

VARIATIONAL-INEQUALITIES | PROJECTION | MATHEMATICS, APPLIED | CONVEX-SETS | ALGORITHMS | PHYSICS, MATHEMATICAL

VARIATIONAL-INEQUALITIES | PROJECTION | MATHEMATICS, APPLIED | CONVEX-SETS | ALGORITHMS | PHYSICS, MATHEMATICAL

Journal Article

Computational optimization and applications, ISSN 0926-6003, 04/2012, Volume 51, Issue 3, pp. 1065 - 1088

.... It is shown that they often have a computational advantage over alternatives that have been proposed for solving the same problem and that this makes them successful in many real-world applications...

Linear inequalities | Sparse matrices | Numerical evaluation | Optimization | Projection methods | Convex feasibility problems | MATHEMATICS, APPLIED | SET | BEHAVIOR | IMAGE-RECONSTRUCTION | RESTORATION | ALGORITHMS | RECOVERY | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | FIXED-POINTS | STRONG-CONVERGENCE | SPARSE SYSTEMS | Computer science | Censorship | Analysis | Methods | Studies | Patents | Effectiveness | Computation | Inequalities | Projection | Feasibility | Dealing

Linear inequalities | Sparse matrices | Numerical evaluation | Optimization | Projection methods | Convex feasibility problems | MATHEMATICS, APPLIED | SET | BEHAVIOR | IMAGE-RECONSTRUCTION | RESTORATION | ALGORITHMS | RECOVERY | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | FIXED-POINTS | STRONG-CONVERGENCE | SPARSE SYSTEMS | Computer science | Censorship | Analysis | Methods | Studies | Patents | Effectiveness | Computation | Inequalities | Projection | Feasibility | Dealing

Journal Article

Optimization Letters, ISSN 1862-4472, 10/2014, Volume 8, Issue 7, pp. 2099 - 2110

In this paper our interest is in investigating properties and numerical solutions of Proximal Split feasibility Problems...

Computational Intelligence | 49J53 | 90C25 | Operations Research/Decision Theory | Numerical and Computational Physics | 65K10 | Mathematics | 49M37 | Moreauâ€“Yosida approximate | Optimization | Proximal split feasibility problems | Prox-regularity | Moreau-Yosida approximate | PROJECTION | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | SET | CONVEX | REGULAR FUNCTIONS | Algorithms

Computational Intelligence | 49J53 | 90C25 | Operations Research/Decision Theory | Numerical and Computational Physics | 65K10 | Mathematics | 49M37 | Moreauâ€“Yosida approximate | Optimization | Proximal split feasibility problems | Prox-regularity | Moreau-Yosida approximate | PROJECTION | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | SET | CONVEX | REGULAR FUNCTIONS | Algorithms

Journal Article

Journal of Inequalities and Applications, ISSN 1025-5834, 12/2010, Volume 2010, Issue 1, pp. 1 - 13

Using the idea of Tikhonov's regularization, we present properties of the approximating curve for the split feasibility problem (SFP...

Mathematics, general | Mathematics | Applications of Mathematics | Analysis | MATHEMATICS | MATHEMATICS, APPLIED | CONVEX-SETS | ITERATIVE ALGORITHMS | Studies | Algorithms | Applied mathematics

Mathematics, general | Mathematics | Applications of Mathematics | Analysis | MATHEMATICS | MATHEMATICS, APPLIED | CONVEX-SETS | ITERATIVE ALGORITHMS | Studies | Algorithms | Applied mathematics

Journal Article