The Annals of Statistics, ISSN 0090-5364, 10/2007, Volume 35, Issue 5, pp. 2173 - 2192

We study the effective degrees of freedom of the lasso in the framework of Stein's unbiased risk estimation (SURE). We show that the number of nonzero...

Degrees of freedom | Transition points | Estimation theory | Mathematical vectors | Unbiased estimators | Diabetes | Coefficients | Modeling | Parametric models | Estimation methods | SURE | Unbiased estimate | Model selection | LARS algorithm | Lasso | REGRESSION | NONCONCAVE PENALIZED LIKELIHOOD | lasso | SHRINKAGE | degrees of freedom | unbiased estimate | ADAPTIVE MODEL SELECTION | model selection | STATISTICS & PROBABILITY | VARIABLE SELECTION | 62J05 | 90C46 | 62J07

Degrees of freedom | Transition points | Estimation theory | Mathematical vectors | Unbiased estimators | Diabetes | Coefficients | Modeling | Parametric models | Estimation methods | SURE | Unbiased estimate | Model selection | LARS algorithm | Lasso | REGRESSION | NONCONCAVE PENALIZED LIKELIHOOD | lasso | SHRINKAGE | degrees of freedom | unbiased estimate | ADAPTIVE MODEL SELECTION | model selection | STATISTICS & PROBABILITY | VARIABLE SELECTION | 62J05 | 90C46 | 62J07

Journal Article

Electronic Journal of Statistics, ISSN 1935-7524, 2013, Volume 7, Issue 1, pp. 1456 - 1490

The lasso is a popular tool for sparse linear regression, especially for problems in which the number of variables p exceeds the number of observations n . But...

LARS | High-dimensional | Lasso | Uniqueness | SPARSITY | high-dimensional | L MINIMIZATION | RECOVERY | DANTZIG SELECTOR | uniqueness | MODEL SELECTION | STATISTICS & PROBABILITY | VARIABLE SELECTION

LARS | High-dimensional | Lasso | Uniqueness | SPARSITY | high-dimensional | L MINIMIZATION | RECOVERY | DANTZIG SELECTOR | uniqueness | MODEL SELECTION | STATISTICS & PROBABILITY | VARIABLE SELECTION

Journal Article

The Annals of Statistics, ISSN 0090-5364, 4/2012, Volume 40, Issue 2, pp. 1198 - 1232

We derive the degrees of freedom of the lasso fit, placing no assumptions on the predictor matrix X. Like the well-known result of Zou, Hastie and Tibshirani...

Regression coefficients | Degrees of freedom | Gaussian distributions | Linear regression | Inner products | Analytical estimating | Polyhedrons | Mathematical vectors | Matrices | Unbiased estimators | Generalized lasso | High-dimensional | Lasso | PATH | high-dimensional | degrees of freedom | LEAST ANGLE REGRESSION | STATISTICS & PROBABILITY | generalized lasso | VARIABLE SELECTION | 90C46 | 62J07

Regression coefficients | Degrees of freedom | Gaussian distributions | Linear regression | Inner products | Analytical estimating | Polyhedrons | Mathematical vectors | Matrices | Unbiased estimators | Generalized lasso | High-dimensional | Lasso | PATH | high-dimensional | degrees of freedom | LEAST ANGLE REGRESSION | STATISTICS & PROBABILITY | generalized lasso | VARIABLE SELECTION | 90C46 | 62J07

Journal Article

Mathematical Methods of Operations Research, ISSN 1432-2994, 10/2019, Volume 90, Issue 2, pp. 255 - 270

The aim of this work is to locate a semi-obnoxious facility, i.e. to minimize the distances to a given set of customers in order to save transportation costs...

Obnoxious facility location | Global optimization | 90B85 | 90C46 | Calculus of Variations and Optimal Control; Optimization | Operations Research/Decision Theory | Primal and dual algorithms | Mathematics | Dc problems | 90C26 | Business and Management, general | Algorithms

Obnoxious facility location | Global optimization | 90B85 | 90C46 | Calculus of Variations and Optimal Control; Optimization | Operations Research/Decision Theory | Primal and dual algorithms | Mathematics | Dc problems | 90C26 | Business and Management, general | Algorithms

Journal Article

Journal of Optimization Theory and Applications, ISSN 0022-3239, 2019, Volume 181, Issue 3, pp. 787 - 816

Motivated by many applications (for instance, some production models in finance require infinity-dimensional commodity spaces, and the preference is defined in...

Nonconvex vector optimization | Proper efficiency | Quasi-relative interior | Saddle point | 90C31 | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | 90C46 | 49B27 | 90C29 | 49A52 | Economic models | Constraints | Nonlinear programming | Efficiency | Saddle points | Optimization

Nonconvex vector optimization | Proper efficiency | Quasi-relative interior | Saddle point | 90C31 | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | 90C46 | 49B27 | 90C29 | 49A52 | Economic models | Constraints | Nonlinear programming | Efficiency | Saddle points | Optimization

Journal Article

Arabian Journal of Mathematics, ISSN 2193-5343, 9/2013, Volume 2, Issue 3, pp. 321 - 332

In this paper, we introduce some new kinds of generalized convexity, which include (semistrict) G-semipreinvexity and (semistrict) G-semipreincavity. Moreover,...

Mathematics, general | Mathematics | 90C26 | 90C46

Mathematics, general | Mathematics | 90C26 | 90C46

Journal Article

Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, 12/2019, Volume 58, Issue 6, pp. 1 - 28

The optimal weak transport problem has recently been introduced by Gozlan et al. (J Funct Anal 273(11):3327â€“3405, 2017). We provide general existence and...

60G42 | 90C46 | Systems Theory, Control | Calculus of Variations and Optimal Control; Optimization | Analysis | Theoretical, Mathematical and Computational Physics | Mathematics | 58E30 | MATHEMATICS | MATHEMATICS, APPLIED | INEQUALITIES

60G42 | 90C46 | Systems Theory, Control | Calculus of Variations and Optimal Control; Optimization | Analysis | Theoretical, Mathematical and Computational Physics | Mathematics | 58E30 | MATHEMATICS | MATHEMATICS, APPLIED | INEQUALITIES

Journal Article

Journal of Optimization Theory and Applications, ISSN 0022-3239, 9/2019, Volume 182, Issue 3, pp. 984 - 1000

In this paper, we deal with robust optimal solution sets for a class of optimization problems with data uncertainty in both the objective and constraints. We...

Uncertain optimization | Robust optimal solution set | Mathematics | Theory of Computation | Optimization | 90C31 | 90C46 | Calculus of Variations and Optimal Control; Optimization | Operations Research/Decision Theory | Mixed-type duality | Lagrangian-type function | Applications of Mathematics | Engineering, general | 49K35 | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | CONVEX-PROGRAMS | Electrical engineering | Electric properties | Robustness (mathematics)

Uncertain optimization | Robust optimal solution set | Mathematics | Theory of Computation | Optimization | 90C31 | 90C46 | Calculus of Variations and Optimal Control; Optimization | Operations Research/Decision Theory | Mixed-type duality | Lagrangian-type function | Applications of Mathematics | Engineering, general | 49K35 | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | CONVEX-PROGRAMS | Electrical engineering | Electric properties | Robustness (mathematics)

Journal Article

Annals of Operations Research, ISSN 0254-5330, 10/2018, Volume 269, Issue 1, pp. 419 - 438

In this paper, we consider a semi-infinite multiobjective optimization problem with more than two differentiable objective functions and uncertain constraint...

90C29 | Theory of Computation | Semi-infinite programming | Optimality conditions | Duality results | Business and Management | Robust optimization | 90C46 | Operations Research/Decision Theory | Multiobjective optimization | 90C34 | Weakly robust efficient solution | Combinatorics | Mu ltiobjective optimization | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | PROGRAMS | Management science | Research | Mathematical research | Mathematical optimization | Studies | Multiple objective analysis | Theorems | Operations research | Robustness (mathematics) | Nonlinear programming | Optimization | Mathematical programming

90C29 | Theory of Computation | Semi-infinite programming | Optimality conditions | Duality results | Business and Management | Robust optimization | 90C46 | Operations Research/Decision Theory | Multiobjective optimization | 90C34 | Weakly robust efficient solution | Combinatorics | Mu ltiobjective optimization | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | PROGRAMS | Management science | Research | Mathematical research | Mathematical optimization | Studies | Multiple objective analysis | Theorems | Operations research | Robustness (mathematics) | Nonlinear programming | Optimization | Mathematical programming

Journal Article

SIAM Journal on Optimization, ISSN 1052-6234, 2013, Volume 23, Issue 4, pp. 2124 - 2149

The paper addresses the problem of low-rank trace norm minimization. We propose an algorithm that alternates between fixed-rank optimization and rank-one...

Trust region | Trace norm | Predictorcorrector on quotient manifold | Multivariate linear regression | Regularization path | Matrix completion | Riemannian optimization | trust region | predictor-corrector on quotient manifold | MATHEMATICS, APPLIED | regularization path | matrix completion | multivariate linear regression | ALGORITHM | trace norm | Mathematics | Optimization and Control | Machine Learning | Computer Science

Trust region | Trace norm | Predictorcorrector on quotient manifold | Multivariate linear regression | Regularization path | Matrix completion | Riemannian optimization | trust region | predictor-corrector on quotient manifold | MATHEMATICS, APPLIED | regularization path | matrix completion | multivariate linear regression | ALGORITHM | trace norm | Mathematics | Optimization and Control | Machine Learning | Computer Science

Journal Article

SIAM Journal on Optimization, ISSN 1052-6234, 2017, Volume 27, Issue 1, pp. 146 - 172

Conic linear programs, among them semidefinite programs, often behave pathologically: the optimal values of the primal and dual programs may differ, and may...

Conic linear programming | Semidefinite programming | Duality | Closedness of the linear image of a closed convex cone | Pathological semidefinite programs | MATHEMATICS, APPLIED | SET | pathological semidefinite programs | CONES | conic linear programming | FACIAL REDUCTION | duality | CLOSEDNESS | semidefinite programming | closedness of the linear image of a closed convex cone | IMAGE | EXACT DUALITY-THEORY | NICE | OPTIMIZATION | Mathematics - Optimization and Control

Conic linear programming | Semidefinite programming | Duality | Closedness of the linear image of a closed convex cone | Pathological semidefinite programs | MATHEMATICS, APPLIED | SET | pathological semidefinite programs | CONES | conic linear programming | FACIAL REDUCTION | duality | CLOSEDNESS | semidefinite programming | closedness of the linear image of a closed convex cone | IMAGE | EXACT DUALITY-THEORY | NICE | OPTIMIZATION | Mathematics - Optimization and Control

Journal Article

Mathematical Programming, ISSN 0025-5610, 6/2018, Volume 169, Issue 2, pp. 489 - 529

A key step in solving minimax distributionally robust optimization (DRO) problems is to reformulate the inner maximization w.r.t. probability measure as a...

Strong duality | Matrix moment constraints | Cutting plane methods | Theoretical, Mathematical and Computational Physics | Mathematics | 90C15 | Slater type conditions | Random discretization | Mathematical Methods in Physics | 90C46 | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | 90C47 | Numerical Analysis | Lower semicontinuity conditions | Combinatorics | MATHEMATICS, APPLIED | INEQUALITIES | APPROXIMATION | COMPUTER SCIENCE, SOFTWARE ENGINEERING | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | PROGRAMS | CONVERGENCE | Lower bounds | Cutting | Minimax technique | Maximization | Robustness (mathematics) | Discretization | Upper bounds | Nonlinear programming | Optimization | Ambiguity

Strong duality | Matrix moment constraints | Cutting plane methods | Theoretical, Mathematical and Computational Physics | Mathematics | 90C15 | Slater type conditions | Random discretization | Mathematical Methods in Physics | 90C46 | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | 90C47 | Numerical Analysis | Lower semicontinuity conditions | Combinatorics | MATHEMATICS, APPLIED | INEQUALITIES | APPROXIMATION | COMPUTER SCIENCE, SOFTWARE ENGINEERING | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | PROGRAMS | CONVERGENCE | Lower bounds | Cutting | Minimax technique | Maximization | Robustness (mathematics) | Discretization | Upper bounds | Nonlinear programming | Optimization | Ambiguity

Journal Article

Journal of Global Optimization, ISSN 0925-5001, 3/2012, Volume 52, Issue 3, pp. 509 - 535

In the present paper, we are concerned with conditions ensuring the exact penalty for nonconvex programming. Firstly, we consider problems with concave...

Concave programming | DCA | Subdifferential | DC programming | 90C26 | Optimization | Economics / Management Science | 90C20 | Exact penalty | 90C30 | 90C46 | 49J52 | Operations Research/Decision Theory | Reformulation | Computer Science, general | Local and global error bounds | Real Functions | LOWER SEMICONTINUOUS FUNCTIONS | MATHEMATICS, APPLIED | EXACT PENALIZATION | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | SYSTEMS | OPTIMIZATION | EFFICIENT | Studies | Mathematical programming | Computer Science

Concave programming | DCA | Subdifferential | DC programming | 90C26 | Optimization | Economics / Management Science | 90C20 | Exact penalty | 90C30 | 90C46 | 49J52 | Operations Research/Decision Theory | Reformulation | Computer Science, general | Local and global error bounds | Real Functions | LOWER SEMICONTINUOUS FUNCTIONS | MATHEMATICS, APPLIED | EXACT PENALIZATION | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | SYSTEMS | OPTIMIZATION | EFFICIENT | Studies | Mathematical programming | Computer Science

Journal Article

Mathematical Programming, ISSN 0025-5610, 2015, Volume 150, Issue 2, pp. 281 - 319

In this paper, we study the performance of static solutions for two-stage adjustable robust linear optimization problems with uncertain constraint and...

90C46 | 90C47 | 49K35 | ADAPTIVE OPTIMIZATION | COMPUTER SCIENCE, SOFTWARE ENGINEERING | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | COMBINATORIAL OPTIMIZATION | COMPLEXITY | UNCERTAINTY | POWER | STOCHASTIC-PROGRAMMING PROBLEMS | CONVEX-OPTIMIZATION | Management science | Chemical properties | Studies | Mathematical analysis | Optimization | Uncertainty | Approximation | Computation | Perception | Mathematical models | Transformations | Adjustable

90C46 | 90C47 | 49K35 | ADAPTIVE OPTIMIZATION | COMPUTER SCIENCE, SOFTWARE ENGINEERING | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | COMBINATORIAL OPTIMIZATION | COMPLEXITY | UNCERTAINTY | POWER | STOCHASTIC-PROGRAMMING PROBLEMS | CONVEX-OPTIMIZATION | Management science | Chemical properties | Studies | Mathematical analysis | Optimization | Uncertainty | Approximation | Computation | Perception | Mathematical models | Transformations | Adjustable

Journal Article

Journal of Optimization Theory and Applications, ISSN 0022-3239, 6/2018, Volume 177, Issue 3, pp. 743 - 769

The main purpose of this paper is to study the duality and penalty method for a constrained nonconvex vector optimization problem. Following along with the...

Image regularity condition | Mathematics | Theory of Computation | Exact penalization | Image space analysis | Optimization | 90C30 | 90C46 | Calculus of Variations and Optimal Control; Optimization | Operations Research/Decision Theory | Lagrange-type duality | 49N15 | Applications of Mathematics | Engineering, general | Vector optimization | MATHEMATICS, APPLIED | NONLINEAR SEPARATION | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | PENALTY METHODS | THEOREMS | EXTREMUM PROBLEMS | SEPARATION APPROACH | OPTIMALITY CONDITIONS | Lagrange multiplier | Separation | Lagrange multipliers | Saddle points | Images | Nonlinear programming

Image regularity condition | Mathematics | Theory of Computation | Exact penalization | Image space analysis | Optimization | 90C30 | 90C46 | Calculus of Variations and Optimal Control; Optimization | Operations Research/Decision Theory | Lagrange-type duality | 49N15 | Applications of Mathematics | Engineering, general | Vector optimization | MATHEMATICS, APPLIED | NONLINEAR SEPARATION | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | PENALTY METHODS | THEOREMS | EXTREMUM PROBLEMS | SEPARATION APPROACH | OPTIMALITY CONDITIONS | Lagrange multiplier | Separation | Lagrange multipliers | Saddle points | Images | Nonlinear programming

Journal Article

Journal of Optimization Theory and Applications, ISSN 0022-3239, 4/2019, Volume 181, Issue 1, pp. 144 - 162

In this paper, we study some problems with a continuously differentiable and quasiconvex objective function. We prove that exactly one of the following two...

Pseudoconvex function | Characterizations of the solution set | Mathematics | Theory of Computation | 90C26 | Quasiconvex function | Optimization | Quasiconvex program | 90C46 | Calculus of Variations and Optimal Control; Optimization | Operations Research/Decision Theory | KKT Conditions | Applications of Mathematics | Engineering, general | 26B25 | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | OPTIMIZATION | Economic models | Lagrange multipliers | Lagrange multiplier

Pseudoconvex function | Characterizations of the solution set | Mathematics | Theory of Computation | 90C26 | Quasiconvex function | Optimization | Quasiconvex program | 90C46 | Calculus of Variations and Optimal Control; Optimization | Operations Research/Decision Theory | KKT Conditions | Applications of Mathematics | Engineering, general | 26B25 | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | OPTIMIZATION | Economic models | Lagrange multipliers | Lagrange multiplier

Journal Article

Mathematical Programming, ISSN 0025-5610, 5/2019, Volume 175, Issue 1, pp. 503 - 536

We contribute improvements to a Lagrangian dual solution approach applied to large-scale optimization problems whose objective functions are convex,...

Nonlinear block Gaussâ€“Seidel method | Augmented Lagrangian method | Proximal bundle method | Theoretical, Mathematical and Computational Physics | 90-08 | Simplicial decomposition method | 90C06 | Mathematics | 90C15 | 90C26 | Mathematical Methods in Physics | Parallel computing | 90C30 | 90C46 | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | 90C25 | Numerical Analysis | 90C11 | Combinatorics | COMPUTER SCIENCE, SOFTWARE ENGINEERING | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Nonlinear block Gauss-Seidel method | Parallel processing | Lagrange multiplier | Approximation | Optimization

Nonlinear block Gaussâ€“Seidel method | Augmented Lagrangian method | Proximal bundle method | Theoretical, Mathematical and Computational Physics | 90-08 | Simplicial decomposition method | 90C06 | Mathematics | 90C15 | 90C26 | Mathematical Methods in Physics | Parallel computing | 90C30 | 90C46 | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | 90C25 | Numerical Analysis | 90C11 | Combinatorics | COMPUTER SCIENCE, SOFTWARE ENGINEERING | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Nonlinear block Gauss-Seidel method | Parallel processing | Lagrange multiplier | Approximation | Optimization

Journal Article

Journal of Applied Analysis, ISSN 1425-6908, 12/2018, Volume 24, Issue 2, pp. 223 - 231

In this paper, sufficient optimality conditions are established for the multiobjective control problem using efficiency of higher order as a criterion for...

generalized invexity of higher order | optimality conditions | 90C46 | 90C29 | 49N15 | efficiency of higher order | 26B25 | duality | Control problem | Nonlinear programming

generalized invexity of higher order | optimality conditions | 90C46 | 90C29 | 49N15 | efficiency of higher order | 26B25 | duality | Control problem | Nonlinear programming

Journal Article

Mathematical Programming, ISSN 0025-5610, 7/2018, Volume 170, Issue 1, pp. 357 - 386

An important step in a multi-sensor surveillance system is to estimate sensor biases from their noisy asynchronous measurements. This estimation problem is...

Theoretical, Mathematical and Computational Physics | Block coordinate decent algorithm | Mathematics | Tightness of semidefinite relaxation | 90C26 | Mathematical Methods in Physics | 90C30 | Sensor registration problem | 90C46 | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | Numerical Analysis | Nonlinear least squares | 49M37 | Combinatorics | COMPUTER SCIENCE, SOFTWARE ENGINEERING | TARGET TRACKING | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | ERRORS | ALGORITHM | REGISTRATION | Algorithms | Surveillance equipment | Sensors | Mathematical optimization | Measuring instruments | Economic models | Computer simulation | Multisensor fusion | Data integration | Least squares | Estimating techniques | Noise measurement

Theoretical, Mathematical and Computational Physics | Block coordinate decent algorithm | Mathematics | Tightness of semidefinite relaxation | 90C26 | Mathematical Methods in Physics | 90C30 | Sensor registration problem | 90C46 | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | Numerical Analysis | Nonlinear least squares | 49M37 | Combinatorics | COMPUTER SCIENCE, SOFTWARE ENGINEERING | TARGET TRACKING | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | ERRORS | ALGORITHM | REGISTRATION | Algorithms | Surveillance equipment | Sensors | Mathematical optimization | Measuring instruments | Economic models | Computer simulation | Multisensor fusion | Data integration | Least squares | Estimating techniques | Noise measurement

Journal Article