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Mathematical programming, ISSN 1436-4646, 07/2013, Volume 146, Issue 1-2, pp. 459 - 494

We introduce a proximal alternating linearized minimization (PALM) algorithm for solving a broad class of nonconvex and nonsmooth minimization problems....

Gauss-Seidel method | Kurdyka–Łojasiewicz property | Theoretical, Mathematical and Computational Physics | Block coordinate descent | Alternating minimization | Mathematics | 90C26 | Nonconvex-nonsmooth minimization | Proximal forward-backward | Mathematical Methods in Physics | 90C30 | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | Numerical Analysis | 65K10 | 49M27 | 49M37 | Combinatorics | 47J25 | Sparse nonnegative matrix factorization | Kurdyka-Łojasiewicz property | Operations Research & Management Science | Physical Sciences | Technology | Computer Science | Computer Science, Software Engineering | Mathematics, Applied | Science & Technology | Analysis | Management science | Algorithms | Studies | Data smoothing | Mathematical programming | Functions (mathematics) | Construction | Mathematical analysis | Palm | Byproducts | Minimization | Optimization | Optimization and Control

Gauss-Seidel method | Kurdyka–Łojasiewicz property | Theoretical, Mathematical and Computational Physics | Block coordinate descent | Alternating minimization | Mathematics | 90C26 | Nonconvex-nonsmooth minimization | Proximal forward-backward | Mathematical Methods in Physics | 90C30 | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | Numerical Analysis | 65K10 | 49M27 | 49M37 | Combinatorics | 47J25 | Sparse nonnegative matrix factorization | Kurdyka-Łojasiewicz property | Operations Research & Management Science | Physical Sciences | Technology | Computer Science | Computer Science, Software Engineering | Mathematics, Applied | Science & Technology | Analysis | Management science | Algorithms | Studies | Data smoothing | Mathematical programming | Functions (mathematics) | Construction | Mathematical analysis | Palm | Byproducts | Minimization | Optimization | Optimization and Control

Journal Article

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Full Text
Linear convergence of the generalized Douglas–Rachford algorithm for feasibility problems

Journal of global optimization, ISSN 1573-2916, 04/2018, Volume 72, Issue 3, pp. 443 - 474

In this paper, we study the generalized Douglas–Rachford algorithm and its cyclic variants which include many projection-type methods such as the classical...

Quasi coercivity | 65K05 | Mathematics | 90C26 | Optimization | Quasi Fejér 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 Douglas–Rachford algorithm | Operations Research & Management Science | Physical Sciences | Mathematics, Applied | Technology | Science & Technology | Algorithms | Magnetic properties | Economic models | Convexity | Feasibility studies | Convergence | Mathematics - Optimization and Control

Quasi coercivity | 65K05 | Mathematics | 90C26 | Optimization | Quasi Fejér 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 Douglas–Rachford algorithm | Operations Research & Management Science | Physical Sciences | Mathematics, Applied | Technology | Science & Technology | Algorithms | Magnetic properties | Economic models | Convexity | Feasibility studies | Convergence | Mathematics - Optimization and Control

Journal Article

Mathematical programming, ISSN 1436-4646, 10/2015, Volume 159, Issue 1-2, pp. 253 - 287

We revisit the proofs of convergence for a first order primal–dual algorithm for convex optimization which we have studied a few years ago. In particular, we...

Theoretical, Mathematical and Computational Physics | Convergence rates | Ergodic convergence | Mathematics | 65Y20 | Mathematical Methods in Physics | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | 90C25 | Numerical Analysis | 49M29 | First order algorithms | 65K10 | Combinatorics | Saddle-point problems | Primal–dual algorithms | Operations Research & Management Science | Physical Sciences | Technology | Computer Science | Computer Science, Software Engineering | Mathematics, Applied | Science & Technology | Graphics software | Algorithms | Studies | Mathematical analysis | Mathematical programming | Computational geometry | Operators | Proving | Norms | Nonlinearity | Optimization | Convergence

Theoretical, Mathematical and Computational Physics | Convergence rates | Ergodic convergence | Mathematics | 65Y20 | Mathematical Methods in Physics | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | 90C25 | Numerical Analysis | 49M29 | First order algorithms | 65K10 | Combinatorics | Saddle-point problems | Primal–dual algorithms | Operations Research & Management Science | Physical Sciences | Technology | Computer Science | Computer Science, Software Engineering | Mathematics, Applied | Science & Technology | Graphics software | Algorithms | Studies | Mathematical analysis | Mathematical programming | Computational geometry | Operators | Proving | Norms | Nonlinearity | Optimization | Convergence

Journal Article

Journal of optimization theory and applications, ISSN 1573-2878, 03/2016, Volume 170, Issue 1, pp. 72 - 84

Recently, the tensor complementarity problem has been investigated in the literature. An important question involving the property of global uniqueness and...

Tensor complementarity problem | Mathematics | Theory of Computation | 15A69 | Optimization | Strong P tensor | 15A18 | P tensor | 65F10 | Calculus of Variations and Optimal Control; Optimization | 90C33 | 65K10 | 65F15 | Applications of Mathematics | Engineering, general | Operation Research/Decision Theory | Nonlinear complementarity problem | Global uniqueness and solvability | Operations Research & Management Science | Physical Sciences | Mathematics, Applied | Technology | Science & Technology | Studies | Mathematical analysis | Nonlinear systems | Construction | Tensors | Uniqueness

Tensor complementarity problem | Mathematics | Theory of Computation | 15A69 | Optimization | Strong P tensor | 15A18 | P tensor | 65F10 | Calculus of Variations and Optimal Control; Optimization | 90C33 | 65K10 | 65F15 | Applications of Mathematics | Engineering, general | Operation Research/Decision Theory | Nonlinear complementarity problem | Global uniqueness and solvability | Operations Research & Management Science | Physical Sciences | Mathematics, Applied | Technology | Science & Technology | Studies | Mathematical analysis | Nonlinear systems | Construction | Tensors | Uniqueness

Journal Article

Communications in mathematical sciences, ISSN 1539-6746, 2010, Volume 8, Issue 1, pp. 93 - 111

We propose and analyze an extremely fast, efficient, and simple method for solving the problem:
min{parallel to u parallel to(1) : Au = f, u is an element of...

Basis pursuit | minimization | Compressed sensing | Sparse denoising | Iterative regularization | Physical Sciences | Mathematics | Mathematics, Applied | Science & Technology | ℓ^1-minimization | sparse denoising | compressed sensing | iterative regularization | 90-08 | 65K10 | basis pursuit | 49M99

Basis pursuit | minimization | Compressed sensing | Sparse denoising | Iterative regularization | Physical Sciences | Mathematics | Mathematics, Applied | Science & Technology | ℓ^1-minimization | sparse denoising | compressed sensing | iterative regularization | 90-08 | 65K10 | basis pursuit | 49M99

Journal Article

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Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity

Mathematical programming, ISSN 1436-4646, 03/2016, Volume 168, Issue 1-2, 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 | Operations Research & Management Science | Physical Sciences | Technology | Computer Science | Computer Science, Software Engineering | Mathematics, Applied | Science & Technology | 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 | Operations Research & Management Science | Physical Sciences | Technology | Computer Science | Computer Science, Software Engineering | Mathematics, Applied | Science & Technology | Analysis | Algorithms | Differential equations | Hilbert space | Trajectories | Nonlinear programming | Convergence | Viscous damping | Optimization and Control

Journal Article

Journal of optimization theory and applications, ISSN 1573-2878, 09/2014, Volume 165, Issue 3, pp. 874 - 900

We study the convergence of general descent methods applied to a lower semi-continuous and nonconvex function, which satisfies the Kurdyka–Łojasiewicz...

Nonconvex and nonsmooth optimization | Convergence rates | Variable metric | Mathematics | Theory of Computation | 90C26 | Newton-like method | Optimization | Descent methods | Gauss–Seidel method | 90C30 | Calculus of Variations and Optimal Control; Optimization | Operations Research/Decision Theory | 65K10 | Applications of Mathematics | Engineering, general | 49M37 | Kurdyka–Łojasiewicz inequality

Nonconvex and nonsmooth optimization | Convergence rates | Variable metric | Mathematics | Theory of Computation | 90C26 | Newton-like method | Optimization | Descent methods | Gauss–Seidel method | 90C30 | Calculus of Variations and Optimal Control; Optimization | Operations Research/Decision Theory | 65K10 | Applications of Mathematics | Engineering, general | 49M37 | Kurdyka–Łojasiewicz inequality

Journal Article

Mathematical programming, ISSN 1436-4646, 02/2013, Volume 144, Issue 1-2, pp. 369 - 412

We propose to solve a general quasi-variational inequality by using its Karush–Kuhn–Tucker conditions. To this end we use a globally convergent algorithm based...

Global convergence | KKT conditions | Theoretical, Mathematical and Computational Physics | Mathematics | Mathematical Methods in Physics | 90C51 | 90C30 | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | Numerical Analysis | 90C33 | Quasi-variational inequality | 65K10 | Combinatorics | Interior-point method | Operations Research & Management Science | Physical Sciences | Technology | Computer Science | Computer Science, Software Engineering | Mathematics, Applied | Science & Technology | Algorithms | Studies | Analysis | Mathematical programming | Reduction | Viability | Inequalities | Convergence

Global convergence | KKT conditions | Theoretical, Mathematical and Computational Physics | Mathematics | Mathematical Methods in Physics | 90C51 | 90C30 | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | Numerical Analysis | 90C33 | Quasi-variational inequality | 65K10 | Combinatorics | Interior-point method | Operations Research & Management Science | Physical Sciences | Technology | Computer Science | Computer Science, Software Engineering | Mathematics, Applied | Science & Technology | Algorithms | Studies | Analysis | Mathematical programming | Reduction | Viability | Inequalities | Convergence

Journal Article

Numerical algorithms, ISSN 1572-9265, 10/2015, 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...

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 | Physical Sciences | Mathematics | Mathematics, Applied | Science & Technology

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 | Physical Sciences | Mathematics | Mathematics, Applied | Science & Technology

Journal Article

Journal of optimization theory and applications, ISSN 1573-2878, 03/2015, 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 | Operations Research & Management Science | Physical Sciences | Mathematics, Applied | Technology | Science & Technology | 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 | Operations Research & Management Science | Physical Sciences | Mathematics, Applied | Technology | Science & Technology | Algorithms | Studies | Regularization methods | Mathematical analysis | Convexity | Regularization | Inequalities | Convergence

Journal Article

Journal of optimization theory and applications, ISSN 1573-2878, 11/2018, Volume 181, Issue 2, pp. 411 - 436

In this paper, we investigate a robust nonsmooth multiobjective optimization problem related to a multiobjective optimization with data uncertainty. We firstly...

Robust optimality conditions | Robust duality | Mathematics | Theory of Computation | Robust nonsmooth multiobjective optimization | Optimization | Calculus of Variations and Optimal Control; Optimization | 90C25 | Operations Research/Decision Theory | Uncertain nonsmooth multiobjective optimization | 65K10 | Applications of Mathematics | Engineering, general | Operations Research & Management Science | Physical Sciences | Mathematics, Applied | Technology | Science & Technology | Multiple objective analysis | Economic models | Robustness | Convexity | Nonlinear programming | Mathematical programming

Robust optimality conditions | Robust duality | Mathematics | Theory of Computation | Robust nonsmooth multiobjective optimization | Optimization | Calculus of Variations and Optimal Control; Optimization | 90C25 | Operations Research/Decision Theory | Uncertain nonsmooth multiobjective optimization | 65K10 | Applications of Mathematics | Engineering, general | Operations Research & Management Science | Physical Sciences | Mathematics, Applied | Technology | Science & Technology | Multiple objective analysis | Economic models | Robustness | Convexity | Nonlinear programming | Mathematical programming

Journal Article

Numerische Mathematik, ISSN 0945-3245, 06/2016, Volume 135, Issue 3, pp. 679 - 709

We present and analyze a mixed finite element numerical scheme for the Cahn–Hilliard–Hele–Shaw equation, a modified Cahn–Hilliard equation coupled with the...

Mathematical Methods in Physics | Numerical Analysis | Theoretical, Mathematical and Computational Physics | Appl.Mathematics/Computational Methods of Engineering | Mathematics, general | 65K10 | Mathematics | 65M60 | Numerical and Computational Physics, Simulation | 35K35 | 65M12 | 35K55 | Physical Sciences | Mathematics, Applied | Science & Technology | Finite element method | Analysis | Methods

Mathematical Methods in Physics | Numerical Analysis | Theoretical, Mathematical and Computational Physics | Appl.Mathematics/Computational Methods of Engineering | Mathematics, general | 65K10 | Mathematics | 65M60 | Numerical and Computational Physics, Simulation | 35K35 | 65M12 | 35K55 | Physical Sciences | Mathematics, Applied | Science & Technology | Finite element method | Analysis | Methods

Journal Article

Journal of global optimization, ISSN 1573-2916, 02/2016, Volume 66, Issue 3, pp. 457 - 485

A number of recent works have emphasized the prominent role played by the Kurdyka-Łojasiewicz inequality for proving the convergence of iterative algorithms...

68U10 | Nonconvex optimization | Phase retrieval | Proximity operator | Block coordinate descent | Alternating minimization | Mathematics | 90C26 | 90C05 | Optimization | Inverse problems | 90C25 | 94A08 | 65K10 | Nonsmooth optimization | 49M27 | Operation Research/Decision Theory | Computer Science, general | 65F08 | Real Functions | Majorize–Minimize algorithm | Operations Research & Management Science | Physical Sciences | Mathematics, Applied | Technology | Science & Technology | Usage | Algorithms | Studies | Efficiency | Mathematical analysis | Inequalities | Minimization | Strategy | Convergence | Engineering Sciences | Computer Science | Signal and Image processing

68U10 | Nonconvex optimization | Phase retrieval | Proximity operator | Block coordinate descent | Alternating minimization | Mathematics | 90C26 | 90C05 | Optimization | Inverse problems | 90C25 | 94A08 | 65K10 | Nonsmooth optimization | 49M27 | Operation Research/Decision Theory | Computer Science, general | 65F08 | Real Functions | Majorize–Minimize algorithm | Operations Research & Management Science | Physical Sciences | Mathematics, Applied | Technology | Science & Technology | Usage | Algorithms | Studies | Efficiency | Mathematical analysis | Inequalities | Minimization | Strategy | Convergence | Engineering Sciences | Computer Science | Signal and Image processing

Journal Article