Mathematical Programming, ISSN 0025-5610, 09/2016, Volume 159, Issue 1-2, pp. 435 - 467

Journal Article

Mathematical Programming, ISSN 0025-5610, 8/2014, Volume 146, Issue 1, 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 | MATHEMATICS, APPLIED | DECOMPOSITION | ALGORITHMS | Kurdyka-Lojasiewicz property | COMPUTER SCIENCE, SOFTWARE ENGINEERING | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | CONVERGENCE | CONSTRAINTS | NONNEGATIVE MATRIX FACTORIZATION | LOJASIEWICZ INEQUALITY | 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 | MATHEMATICS, APPLIED | DECOMPOSITION | ALGORITHMS | Kurdyka-Lojasiewicz property | COMPUTER SCIENCE, SOFTWARE ENGINEERING | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | CONVERGENCE | CONSTRAINTS | NONNEGATIVE MATRIX FACTORIZATION | LOJASIEWICZ INEQUALITY | Analysis | Management science | Algorithms | Studies | Data smoothing | Mathematical programming | Functions (mathematics) | Construction | Mathematical analysis | Palm | Byproducts | Minimization | Optimization | Optimization and Control

Journal Article

Journal of Information and Optimization Sciences, ISSN 0252-2667, 04/2019, Volume 40, Issue 3, pp. 615 - 632

This paper aims to study how much "generalized" invex properties differ from invexity and to establish whether or not the use of more and more parameters and...

90C26 | 90C30 | 90C46 | Generalized convexity | generalized invexity

90C26 | 90C30 | 90C46 | Generalized convexity | generalized invexity

Journal Article

Numerische Mathematik, ISSN 0029-599X, 7/2015, Volume 130, Issue 3, pp. 567 - 577

This note proposes a novel approach to derive a worst-case $$O(1/k)$$ O ( 1 / k ) convergence rate measured by the iteration complexity in a non-ergodic sense...

Mathematical Methods in Physics | 90C30 | 90C25 | Numerical Analysis | Theoretical, Mathematical and Computational Physics | Appl.Mathematics/Computational Methods of Engineering | Numerical and Computational Physics | Mathematics, general | Mathematics | MATHEMATICS, APPLIED | Yuan (China) | Methods

Mathematical Methods in Physics | 90C30 | 90C25 | Numerical Analysis | Theoretical, Mathematical and Computational Physics | Appl.Mathematics/Computational Methods of Engineering | Numerical and Computational Physics | Mathematics, general | Mathematics | MATHEMATICS, APPLIED | Yuan (China) | Methods

Journal Article

Mathematical Programming, ISSN 0025-5610, 1/2016, Volume 155, Issue 1, pp. 57 - 79

The alternating direction method of multipliers (ADMM) is now widely used in many fields, and its convergence was proved when two blocks of variables are...

Alternating direction method of multipliers | Theoretical, Mathematical and Computational Physics | Mathematics | Convex programming | Convergence analysis | Mathematical Methods in Physics | Splitting methods | 90C30 | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | 90C25 | Numerical Analysis | Combinatorics | 65K13 | COMPUTER SCIENCE, SOFTWARE ENGINEERING | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Yuan (China) | Management techniques | Management science | Management | Analysis | Studies | Mathematical analysis | Convex analysis | Convergence | Mathematical programming | Functions (mathematics) | Multipliers | Divergence | Minimization | Optimization

Alternating direction method of multipliers | Theoretical, Mathematical and Computational Physics | Mathematics | Convex programming | Convergence analysis | Mathematical Methods in Physics | Splitting methods | 90C30 | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | 90C25 | Numerical Analysis | Combinatorics | 65K13 | COMPUTER SCIENCE, SOFTWARE ENGINEERING | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Yuan (China) | Management techniques | Management science | Management | Analysis | Studies | Mathematical analysis | Convex analysis | Convergence | Mathematical programming | Functions (mathematics) | Multipliers | Divergence | Minimization | Optimization

Journal Article

Mathematical Programming, ISSN 0025-5610, 3/2017, Volume 162, Issue 1, pp. 83 - 112

We analyze the stochastic average gradient (SAG) method for optimizing the sum of a finite number of smooth convex functions. Like stochastic gradient (SG)...

68Q25 | 65K05 | Theoretical, Mathematical and Computational Physics | 90C06 | Mathematics | Stochastic gradient methods | 90C15 | First-order methods | Mathematical Methods in Physics | 90C30 | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | Convex optimization | 90C25 | Numerical Analysis | Convergence Rates | Combinatorics | 62L20 | COMPUTER SCIENCE, SOFTWARE ENGINEERING | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | APPROXIMATION | ALGORITHMS | Analysis | Algorithms | Studies | Convex analysis | Optimization | Mathematical analysis | Sag | Texts | Strategy | Mathematical models | Stochasticity | Sampling | Convergence

68Q25 | 65K05 | Theoretical, Mathematical and Computational Physics | 90C06 | Mathematics | Stochastic gradient methods | 90C15 | First-order methods | Mathematical Methods in Physics | 90C30 | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | Convex optimization | 90C25 | Numerical Analysis | Convergence Rates | Combinatorics | 62L20 | COMPUTER SCIENCE, SOFTWARE ENGINEERING | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | APPROXIMATION | ALGORITHMS | Analysis | Algorithms | Studies | Convex analysis | Optimization | Mathematical analysis | Sag | Texts | Strategy | Mathematical models | Stochasticity | Sampling | Convergence

Journal Article

Journal of Information and Optimization Sciences, ISSN 0252-2667, 08/2018, Volume 39, Issue 6, pp. 1223 - 1230

Quadratic programming problem with bounded below (bounded above, respectively) quadratic objective function over a feasible region, defined as Minkowski sum of...

90C20 | 90C30 | Quadratic programming | Existence of solution

90C20 | 90C30 | Quadratic programming | Existence of solution

Journal Article

Mathematical Programming, ISSN 0025-5610, 12/2013, Volume 142, Issue 1, pp. 397 - 434

Minimization with orthogonality constraints (e.g., $$X^\top X = I$$ ) and/or spherical constraints (e.g., $$\Vert x\Vert _2 = 1$$ ) has wide applications in...

Nearest correlation matrix | Invariant subspace | 65K05 | Theoretical, Mathematical and Computational Physics | Curvilinear search | Maxcut SDP | Mathematics | 90C26 | Eigenvalue and eigenvector | Quadratic assignment problem | 90C27 | Orthogonality constraint | Mathematical Methods in Physics | Cayley transformation | 90C30 | Polynomial optimization | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | Stiefel manifold | Numerical Analysis | 90C22 | Spherical constraint | 49Q99 | Combinatorics | MATHEMATICS, APPLIED | ALGORITHM | DECOMPOSITION | FORMULATION | COMPUTER SCIENCE, SOFTWARE ENGINEERING | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | MINIMIZATION | BARZILAI | MATRICES | RANK REDUCTION | FLOWS | SEMIDEFINITE PROGRAMS | Methods | Algorithms | Studies | Polynomials | Topological manifolds | Analysis | Optimization | Mathematical programming

Nearest correlation matrix | Invariant subspace | 65K05 | Theoretical, Mathematical and Computational Physics | Curvilinear search | Maxcut SDP | Mathematics | 90C26 | Eigenvalue and eigenvector | Quadratic assignment problem | 90C27 | Orthogonality constraint | Mathematical Methods in Physics | Cayley transformation | 90C30 | Polynomial optimization | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | Stiefel manifold | Numerical Analysis | 90C22 | Spherical constraint | 49Q99 | Combinatorics | MATHEMATICS, APPLIED | ALGORITHM | DECOMPOSITION | FORMULATION | COMPUTER SCIENCE, SOFTWARE ENGINEERING | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | MINIMIZATION | BARZILAI | MATRICES | RANK REDUCTION | FLOWS | SEMIDEFINITE PROGRAMS | Methods | Algorithms | Studies | Polynomials | Topological manifolds | Analysis | Optimization | Mathematical programming

Journal Article

Mathematical Programming, ISSN 0025-5610, 3/2006, Volume 106, Issue 1, pp. 25 - 57

We present a primal-dual interior-point algorithm with a filter line-search method for nonlinear programming. Local and global convergence properties of this...

65K05 | Mathematical and Computational Physics | Mathematics | Optimization | Filter method | Nonconvex constrained optimization | Line search | Mathematical Methods in Physics | 90C51 | Barrier method | 90C30 | Mathematics of Computing | Numerical Analysis | Calculus of Variations and Optimal Control | Nonlinear programming | 49M37 | Combinatorics | Interior-point method | COMPUTER SCIENCE, SOFTWARE ENGINEERING | interior-point method | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | nonlinear programming | nonconvex constrained optimization | CONSTRAINED OPTIMIZATION | filter method | line search | GLOBAL CONVERGENCE | Algorithms | Comparative studies | Convergence | Heuristic

65K05 | Mathematical and Computational Physics | Mathematics | Optimization | Filter method | Nonconvex constrained optimization | Line search | Mathematical Methods in Physics | 90C51 | Barrier method | 90C30 | Mathematics of Computing | Numerical Analysis | Calculus of Variations and Optimal Control | Nonlinear programming | 49M37 | Combinatorics | Interior-point method | COMPUTER SCIENCE, SOFTWARE ENGINEERING | interior-point method | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | nonlinear programming | nonconvex constrained optimization | CONSTRAINED OPTIMIZATION | filter method | line search | GLOBAL CONVERGENCE | Algorithms | Comparative studies | Convergence | Heuristic

Journal Article

Optimization Methods and Software, ISSN 1055-6788, 11/2019, Volume 34, Issue 6, pp. 1322 - 1341

In this work, we present a new feasible direction algorithm for solving smooth nonlinear second-order cone programs. These consist of minimizing a nonlinear...

Second-order cone programming | Feasible direction | Interior-point methods | 90C51 | 90C30 | 49M15 | Support vector machines | Algorithms

Second-order cone programming | Feasible direction | Interior-point methods | 90C51 | 90C30 | 49M15 | Support vector machines | Algorithms

Journal Article

Mathematics of Operations Research, ISSN 0364-765X, 2/2020, Volume 45, Issue 1, pp. 86 - 98

We study the convergence rate of a hierarchy of upper bounds for polynomial optimization problems, proposed by Lasserre, and a related hierarchy by de Klerk,...

Jacobi polynomials | Primary: programming: nonlinear (theory) | 90C30 | 90C22 | extremal roots of orthogonal polynomials | polynomial optimization | Lasserre hierarchy | 90C26 | semidefinite optimization

Jacobi polynomials | Primary: programming: nonlinear (theory) | 90C30 | 90C22 | extremal roots of orthogonal polynomials | polynomial optimization | Lasserre hierarchy | 90C26 | semidefinite optimization

Journal Article

Optimization Methods and Software, ISSN 1055-6788, 2019, pp. 1 - 16

Journal Article

Mathematical Programming, ISSN 0025-5610, 6/2016, Volume 157, Issue 2, pp. 421 - 449

We consider a three-tier architecture for mobile and pervasive computing scenarios, consisting of a local tier of mobile nodes, a middle tier (cloudlets) of...

Theoretical, Mathematical and Computational Physics | Mobile cloud computing | Mathematics | Generalized Nash equilibrium problem | Mathematical Methods in Physics | 90C30 | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | Numerical Analysis | 90C33 | Distributed algorithm | 68M20 | Combinatorics | COMPUTER SCIENCE, SOFTWARE ENGINEERING | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | ALGORITHMS | Computer science | Cloud computing | Algorithms | Game theory | Analysis | Studies | Wireless communications | Mathematical analysis | Ubiquitous computing | Architecture (computers) | Servers | Mobile computing | Constraining | Mathematical programming

Theoretical, Mathematical and Computational Physics | Mobile cloud computing | Mathematics | Generalized Nash equilibrium problem | Mathematical Methods in Physics | 90C30 | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | Numerical Analysis | 90C33 | Distributed algorithm | 68M20 | Combinatorics | COMPUTER SCIENCE, SOFTWARE ENGINEERING | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | ALGORITHMS | Computer science | Cloud computing | Algorithms | Game theory | Analysis | Studies | Wireless communications | Mathematical analysis | Ubiquitous computing | Architecture (computers) | Servers | Mobile computing | Constraining | Mathematical programming

Journal Article

Mathematical Programming, ISSN 0025-5610, 3/2009, Volume 117, Issue 1, pp. 387 - 423

We consider the problem of minimizing the sum of a smooth function and a separable convex function. This problem includes as special cases bound-constrained...

Global convergence | 65K05 | Mathematical and Computational Physics | 90C06 | Error bound | Mathematics | 90C26 | Coordinate descent | Linear convergence rate | Mathematical Methods in Physics | 90C30 | Mathematics of Computing | Calculus of Variations and Optimal Control; Optimization | 90C25 | Numerical Analysis | 90C55 | Nonsmooth optimization | 49M27 | 49M37 | Combinatorics | REGRESSION | MATHEMATICS, APPLIED | error bound | CONTINUOUSLY DIFFERENTIABLE FUNCTION | linear convergence rate | ASCENT METHODS | global convergence | ALGORITHM | coordinate descent | nonsmooth optimization | LINEAR CONVERGENCE | COMPUTER SCIENCE, SOFTWARE ENGINEERING | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | NONDIFFERENTIABLE OPTIMIZATION PROBLEMS | CONSTRAINTS | UNCONSTRAINED OPTIMIZATION | WAVELET SHRINKAGE | CONVEX MINIMIZATION | Studies | Optimization | Mathematical programming

Global convergence | 65K05 | Mathematical and Computational Physics | 90C06 | Error bound | Mathematics | 90C26 | Coordinate descent | Linear convergence rate | Mathematical Methods in Physics | 90C30 | Mathematics of Computing | Calculus of Variations and Optimal Control; Optimization | 90C25 | Numerical Analysis | 90C55 | Nonsmooth optimization | 49M27 | 49M37 | Combinatorics | REGRESSION | MATHEMATICS, APPLIED | error bound | CONTINUOUSLY DIFFERENTIABLE FUNCTION | linear convergence rate | ASCENT METHODS | global convergence | ALGORITHM | coordinate descent | nonsmooth optimization | LINEAR CONVERGENCE | COMPUTER SCIENCE, SOFTWARE ENGINEERING | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | NONDIFFERENTIABLE OPTIMIZATION PROBLEMS | CONSTRAINTS | UNCONSTRAINED OPTIMIZATION | WAVELET SHRINKAGE | CONVEX MINIMIZATION | Studies | Optimization | Mathematical programming

Journal Article

Mathematical Programming, ISSN 0025-5610, 3/2018, Volume 168, Issue 1, pp. 533 - 554

Our aim in the current article is to extend the developments in Kruger et al. (SIAM J Optim 20(6):3280–3296, 2010. doi:10.1137/100782206) and, more precisely,...

Metric subregularity | Theoretical, Mathematical and Computational Physics | Subdifferential | Error bound | Mathematics | Perturbation | Mathematical Methods in Physics | 90C30 | 49J53 | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | 49J52 | Numerical Analysis | Metric regularity | Combinatorics | Feasibility problem | LOWER SEMICONTINUOUS FUNCTIONS | CONVEX FEASIBILITY PROBLEMS | MATHEMATICS, APPLIED | SUBDIFFERENTIAL CALCULUS | SUFFICIENT CONDITIONS | COMPUTER SCIENCE, SOFTWARE ENGINEERING | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | HOLDER METRIC SUBREGULARITY | GENERALIZED EQUATIONS | CONSTRAINT SYSTEMS | LINEAR INEQUALITIES | CONVERGENCE RATE | PROJECTION ALGORITHMS | Analysis | Management science | Banach space | Error analysis | Mathematics - Optimization and Control

Metric subregularity | Theoretical, Mathematical and Computational Physics | Subdifferential | Error bound | Mathematics | Perturbation | Mathematical Methods in Physics | 90C30 | 49J53 | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | 49J52 | Numerical Analysis | Metric regularity | Combinatorics | Feasibility problem | LOWER SEMICONTINUOUS FUNCTIONS | CONVEX FEASIBILITY PROBLEMS | MATHEMATICS, APPLIED | SUBDIFFERENTIAL CALCULUS | SUFFICIENT CONDITIONS | COMPUTER SCIENCE, SOFTWARE ENGINEERING | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | HOLDER METRIC SUBREGULARITY | GENERALIZED EQUATIONS | CONSTRAINT SYSTEMS | LINEAR INEQUALITIES | CONVERGENCE RATE | PROJECTION ALGORITHMS | Analysis | Management science | Banach space | Error analysis | Mathematics - Optimization and Control

Journal Article

Mathematical Programming Computation, ISSN 1867-2949, 12/2010, Volume 2, Issue 3, pp. 203 - 230

We present an alternating direction dual augmented Lagrangian method for solving semidefinite programming (SDP) problems in standard form. At each iteration,...

Alternating direction method | Augmented Lagrangian method | Semidefinite programming | 90C30 | 90C35 | Operations Research/Decision Theory | 90C22 | 90C06 | Mathematics | Optimization

Alternating direction method | Augmented Lagrangian method | Semidefinite programming | 90C30 | 90C35 | Operations Research/Decision Theory | 90C22 | 90C06 | Mathematics | Optimization

Journal Article

Mathematical Programming, ISSN 0025-5610, 10/2013, Volume 141, Issue 1, pp. 135 - 163

We investigate the behavior of quasi-Newton algorithms applied to minimize a nonsmooth function f, not necessarily convex. We introduce an inexact line search...

65K05 | Theoretical, Mathematical and Computational Physics | Partly smooth | Mathematics | BFGS | Line search | Mathematical Methods in Physics | 90C30 | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | Numerical Analysis | Clarke stationary | R-linear convergence | Combinatorics | Nonconvex | VARIABLE-METRIC METHOD | MATHEMATICS, APPLIED | CONVEX-OPTIMIZATION | COMPUTER SCIENCE, SOFTWARE ENGINEERING | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | BFGS METHOD | GLOBAL CONVERGENCE | Management science | Algorithms | Analysis | Methods | Studies | Functions (mathematics) | Sequences | Searching | Mathematical analysis | Norms | Mathematical models | Dynamical systems | Convergence

65K05 | Theoretical, Mathematical and Computational Physics | Partly smooth | Mathematics | BFGS | Line search | Mathematical Methods in Physics | 90C30 | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | Numerical Analysis | Clarke stationary | R-linear convergence | Combinatorics | Nonconvex | VARIABLE-METRIC METHOD | MATHEMATICS, APPLIED | CONVEX-OPTIMIZATION | COMPUTER SCIENCE, SOFTWARE ENGINEERING | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | BFGS METHOD | GLOBAL CONVERGENCE | Management science | Algorithms | Analysis | Methods | Studies | Functions (mathematics) | Sequences | Searching | Mathematical analysis | Norms | Mathematical models | Dynamical systems | Convergence

Journal Article

Experimental Mathematics, ISSN 1058-6458, 01/2015, Volume 24, Issue 1, pp. 98 - 105

Kepler (1619) and Croft (1980) considered the problem of finding the largest homothetic copies of one regular polyhedron contained in another regular...

polytopes | 90C30 | 51M20 | 52C17 | nonlinear optimization | MATHEMATICS

polytopes | 90C30 | 51M20 | 52C17 | nonlinear optimization | MATHEMATICS

Journal Article