Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 01/2001, Volume 253, Issue 1, pp. 290 - 296

A convex Fréchet differentiable function is minimized subject to a certain hyperplane at a point if the function is minimized in all directions which are defined by a finite set of vectors...

convex function | quadratic programming | Fréchet differentiable | Fré | chet differentiable | Convex function; Fréchet differentiable; quadratic programming | MATHEMATICS | MATHEMATICS, APPLIED | Frechet differentiable

convex function | quadratic programming | Fréchet differentiable | Fré | chet differentiable | Convex function; Fréchet differentiable; quadratic programming | MATHEMATICS | MATHEMATICS, APPLIED | Frechet differentiable

Journal Article

Optimization Methods and Software, ISSN 1055-6788, 07/2014, Volume 29, Issue 4, pp. 720 - 750

...) problem using a regularized gap function approach. For a special class of QVIs, this gap function is continuously differentiable everywhere, in general, however, it has nondifferentiability points...

convex inequalities | Hadamard directional differentiability | Gâteaux differentiability | Fréchet differentiability | finite-dimensional quasi-variational inequalities | generalized Nash equilibrium problem | generalized moving set | regularized gap function | MATHEMATICS, APPLIED | MULTIPLIERS | Gateaux differentiability | COMPUTER SCIENCE, SOFTWARE ENGINEERING | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | IMPULSE CONTROL | OPTIMIZATION | Frechet differentiability | PROGRAMMING ALGORITHM | Mathematical analysis | Inequalities | Software | Mathematical models | Continuity | Smoothness | Optimization | Computer programs

convex inequalities | Hadamard directional differentiability | Gâteaux differentiability | Fréchet differentiability | finite-dimensional quasi-variational inequalities | generalized Nash equilibrium problem | generalized moving set | regularized gap function | MATHEMATICS, APPLIED | MULTIPLIERS | Gateaux differentiability | COMPUTER SCIENCE, SOFTWARE ENGINEERING | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | IMPULSE CONTROL | OPTIMIZATION | Frechet differentiability | PROGRAMMING ALGORITHM | Mathematical analysis | Inequalities | Software | Mathematical models | Continuity | Smoothness | Optimization | Computer programs

Journal Article

Journal of Global Optimization, ISSN 0925-5001, 1/2010, Volume 46, Issue 1, pp. 31 - 47

... Fréchet differentiable functions and locally Lipschitz ones, for the second-order case Fréchet and twice directionally differentiable functions.

90C26 | Optimization | Generalized convexity | Economics / Management Science | Quasiconvex programming | Nonsmooth analysis | 49J52 | Operations Research/Decision Theory | Nonsmooth optimization | 26B25 | Computer Science, general | KT pseudoconvex problems | Real Functions | FJ pseudoconvex problems | CRITERIA | MATHEMATICS, APPLIED | 2ND-ORDER | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | DUALITY | QUASI-CONVEX | SUFFICIENT OPTIMALITY CONDITIONS | Mathematical optimization | Analysis | Studies | Nonlinear programming

90C26 | Optimization | Generalized convexity | Economics / Management Science | Quasiconvex programming | Nonsmooth analysis | 49J52 | Operations Research/Decision Theory | Nonsmooth optimization | 26B25 | Computer Science, general | KT pseudoconvex problems | Real Functions | FJ pseudoconvex problems | CRITERIA | MATHEMATICS, APPLIED | 2ND-ORDER | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | DUALITY | QUASI-CONVEX | SUFFICIENT OPTIMALITY CONDITIONS | Mathematical optimization | Analysis | Studies | Nonlinear programming

Journal Article

Mathematical Programming, ISSN 0025-5610, 2/2015, Volume 149, Issue 1, pp. 195 - 207

.... The distance among mathematical programming problems is defined as the Lipschitz constant of the difference of the corresponding Kojima functions...

90C31 | Mathematical Methods in Physics | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | 90C25 | Numerical Analysis | Theoretical, Mathematical and Computational Physics | Condition number | Condition number theorem | Mathematics | Combinatorics | Convex programming | COMPUTER SCIENCE, SOFTWARE ENGINEERING | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | QUADRATIC OPTIMIZATION | STABILITY | Studies | Convex analysis | Number theory | Mathematical programming | Functions (mathematics) | Theorems | Perturbation methods | Mathematical analysis | Uniqueness | Inequalities | Constants | Optimization

90C31 | Mathematical Methods in Physics | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | 90C25 | Numerical Analysis | Theoretical, Mathematical and Computational Physics | Condition number | Condition number theorem | Mathematics | Combinatorics | Convex programming | COMPUTER SCIENCE, SOFTWARE ENGINEERING | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | QUADRATIC OPTIMIZATION | STABILITY | Studies | Convex analysis | Number theory | Mathematical programming | Functions (mathematics) | Theorems | Perturbation methods | Mathematical analysis | Uniqueness | Inequalities | Constants | Optimization

Journal Article

Computational Optimization and Applications, ISSN 0926-6003, 9/2012, Volume 53, Issue 1, pp. 45 - 89

We present an inexact spectral bundle method for solving convex quadratic semidefinite optimization problems...

Semidefinite programming | Operations Research/Decision Theory | Convex and Discrete Geometry | Nonsmooth optimization methods | Inexact spectral bundle method | Mathematics | Operations Research, Management Science | Statistics, general | Approximate subgradients | Optimization | Eigenvalue minimization problem | EIGENVALUE | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | PATH-FOLLOWING ALGORITHM | DIRECTION | INTERIOR-POINT METHODS | NEAREST CORRELATION MATRIX | Studies | Computational mathematics | Eigen values | Bundling | Computation | Mathematical models | Spectra | Computational efficiency | Iterative methods | Convergence

Semidefinite programming | Operations Research/Decision Theory | Convex and Discrete Geometry | Nonsmooth optimization methods | Inexact spectral bundle method | Mathematics | Operations Research, Management Science | Statistics, general | Approximate subgradients | Optimization | Eigenvalue minimization problem | EIGENVALUE | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | PATH-FOLLOWING ALGORITHM | DIRECTION | INTERIOR-POINT METHODS | NEAREST CORRELATION MATRIX | Studies | Computational mathematics | Eigen values | Bundling | Computation | Mathematical models | Spectra | Computational efficiency | Iterative methods | Convergence

Journal Article

SIAM Journal on Optimization, ISSN 1052-6234, 2007, Volume 18, Issue 2, pp. 613 - 642

...). C, where F is a nonlinear Frechet differentiable function from R-v to R-m. When C is the set of minimum points of a convex real- valued function h on Rm and F...

Convex composite optimization | The Gauss-Newton method | Majorizing function | Convergence | majorizing function | MATHEMATICS, APPLIED | convex composite optimization | MULTIFUNCTIONS | convergence | THEOREM | STABILITY | SYSTEMS | ERROR-BOUNDS | INVERSE | the Gauss-Newton method

Convex composite optimization | The Gauss-Newton method | Majorizing function | Convergence | majorizing function | MATHEMATICS, APPLIED | convex composite optimization | MULTIFUNCTIONS | convergence | THEOREM | STABILITY | SYSTEMS | ERROR-BOUNDS | INVERSE | the Gauss-Newton method

Journal Article

Journal of Optimization Theory and Applications, ISSN 0022-3239, 7/2018, Volume 178, Issue 1, pp. 36 - 55

.... It is shown that infinite-dimensional quadratic programming problems and infinite-dimensional linear fractional vector optimization problems can be studied by using affine variational inequalities...

Infinite-dimensional quadratic programming | 90C29 | Mathematics | Theory of Computation | Generalized polyhedral convex set | Optimization | Infinite-dimensional affine variational inequality | 90C20 | Calculus of Variations and Optimal Control; Optimization | 49K40 | Operations Research/Decision Theory | 49J40 | Infinite-dimensional linear fractional vector optimization | 49J50 | Solution set | Applications of Mathematics | Engineering, general | MATHEMATICS, APPLIED | STABILITY | COMPONENTS | SOLUTION MAP | CONTINUITY | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | NORMAL CONE MAPPINGS | SOLUTION SETS | Lagrange multiplier | Quadratic programming | Lagrange multipliers | Estimates | Inequalities | Mathematical programming

Infinite-dimensional quadratic programming | 90C29 | Mathematics | Theory of Computation | Generalized polyhedral convex set | Optimization | Infinite-dimensional affine variational inequality | 90C20 | Calculus of Variations and Optimal Control; Optimization | 49K40 | Operations Research/Decision Theory | 49J40 | Infinite-dimensional linear fractional vector optimization | 49J50 | Solution set | Applications of Mathematics | Engineering, general | MATHEMATICS, APPLIED | STABILITY | COMPONENTS | SOLUTION MAP | CONTINUITY | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | NORMAL CONE MAPPINGS | SOLUTION SETS | Lagrange multiplier | Quadratic programming | Lagrange multipliers | Estimates | Inequalities | Mathematical programming

Journal Article

8.
Full Text
Nonlinear Chance Constrained Problems: Optimality Conditions, Regularization and Solvers

Journal of Optimization Theory and Applications, ISSN 0022-3239, 8/2016, Volume 170, Issue 2, pp. 419 - 436

We deal with chance constrained problems with differentiable nonlinear random functions and discrete distribution...

Mathematics | Theory of Computation | Free MATLAB codes | 90C15 | 90C26 | Optimization | Optimality conditions | Algorithms | Calculus of Variations and Optimal Control; Optimization | 49M05 | Applications of Mathematics | Engineering, general | Operation Research/Decision Theory | Chance constrained programming | Regularization | MATHEMATICS, APPLIED | OPTIMIZATION PROBLEMS | CONVEX APPROXIMATIONS | DISTRIBUTIONS | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | QUALIFICATION | DIFFERENCE | VALUE-AT-RISK | MATHEMATICAL PROGRAMS | PROBABILISTIC CONSTRAINTS | Studies | Regularization methods | Nonlinear programming | Mathematical analysis | Constraints | Solvers | Nonlinearity | Feasibility | Mathematical models

Mathematics | Theory of Computation | Free MATLAB codes | 90C15 | 90C26 | Optimization | Optimality conditions | Algorithms | Calculus of Variations and Optimal Control; Optimization | 49M05 | Applications of Mathematics | Engineering, general | Operation Research/Decision Theory | Chance constrained programming | Regularization | MATHEMATICS, APPLIED | OPTIMIZATION PROBLEMS | CONVEX APPROXIMATIONS | DISTRIBUTIONS | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | QUALIFICATION | DIFFERENCE | VALUE-AT-RISK | MATHEMATICAL PROGRAMS | PROBABILISTIC CONSTRAINTS | Studies | Regularization methods | Nonlinear programming | Mathematical analysis | Constraints | Solvers | Nonlinearity | Feasibility | Mathematical models

Journal Article

Computational optimization and applications, ISSN 1573-2894, 2018, Volume 72, Issue 3, pp. 769 - 795

We developed a long-step path-following algorithm for a class of symmetric programming problems with nonlinear convex objective functions...

Symmetric programming | Self-concordance | Von Neumann entropy | Mathematics | Statistics, general | Interior-point methods | Nonlinear objective functions | Optimization | Convex optimization | Operations Research/Decision Theory | Convex and Discrete Geometry | Operations Research, Management Science | Matrix monotonicity | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Algorithms | Economic models | Pareto optimum | Upper bounds

Symmetric programming | Self-concordance | Von Neumann entropy | Mathematics | Statistics, general | Interior-point methods | Nonlinear objective functions | Optimization | Convex optimization | Operations Research/Decision Theory | Convex and Discrete Geometry | Operations Research, Management Science | Matrix monotonicity | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Algorithms | Economic models | Pareto optimum | Upper bounds

Journal Article

Journal of optimization theory and applications, ISSN 1573-2878, 2018, Volume 179, Issue 1, pp. 103 - 126

Difference-of-Convex programming and related algorithms, which constitute the backbone of nonconvex programming and global optimization, were introduced in 1985 by Pham Dinh Tao and have been...

Subdifferential | Subanalyticity | Mathematics | Theory of Computation | 90C26 | Optimization | Difference-of-Convex algorithm | 90C30 | Calculus of Variations and Optimal Control; Optimization | Operations Research/Decision Theory | Convergence rate | Lojasiewicz exponent | Applications of Mathematics | Engineering, general | 47N10 | Difference-of-Convex programming | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | TRUST-REGION SUBPROBLEM | DESCENT METHODS | Computer science | Analysis | Algorithms | Computational geometry | Global optimization | Critical point | Convexity | Nonlinear programming | Continuity (mathematics) | Convergence | Mathematical programming

Subdifferential | Subanalyticity | Mathematics | Theory of Computation | 90C26 | Optimization | Difference-of-Convex algorithm | 90C30 | Calculus of Variations and Optimal Control; Optimization | Operations Research/Decision Theory | Convergence rate | Lojasiewicz exponent | Applications of Mathematics | Engineering, general | 47N10 | Difference-of-Convex programming | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | TRUST-REGION SUBPROBLEM | DESCENT METHODS | Computer science | Analysis | Algorithms | Computational geometry | Global optimization | Critical point | Convexity | Nonlinear programming | Continuity (mathematics) | Convergence | Mathematical programming

Journal Article

Journal of Global Optimization, ISSN 0925-5001, 12/2007, Volume 39, Issue 4, pp. 555 - 575

We consider two classes of proximal-like algorithms for minimizing a proper lower semicontinuous quasi-convex function f(x...

Proximal-like method | Quasi-convex programming | Operations Research/Decision Theory | Computer Science, general | Optimization | Entropy-like distance | Economics / Management Science | Real Functions | quasi-convex programming | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | MINIMIZATION | proximal-like method | entropy-like distance | CONVERGENCE | POINT ALGORITHM | EXPONENTIAL MULTIPLIER METHOD | Algorithms | Mathematical programming

Proximal-like method | Quasi-convex programming | Operations Research/Decision Theory | Computer Science, general | Optimization | Entropy-like distance | Economics / Management Science | Real Functions | quasi-convex programming | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | MINIMIZATION | proximal-like method | entropy-like distance | CONVERGENCE | POINT ALGORITHM | EXPONENTIAL MULTIPLIER METHOD | Algorithms | Mathematical programming

Journal Article

Mathematical programming, ISSN 1436-4646, 2019, pp. 1 - 24

.... We extend the partition of the identity into a sum of firmly nonexpansive mappings and Moreau’s decomposition of the quadratic function into envelopes and proximal mappings into the multi-marginal settings...

Operators (mathematics) | Quadratic equations | Transport theory | Convex analysis

Operators (mathematics) | Quadratic equations | Transport theory | Convex analysis

Journal Article

Computational Optimization and Applications, ISSN 0926-6003, 5/2019, Volume 73, Issue 1, pp. 129 - 158

.... Furthermore, the generated sequence is globally convergent to a stationary point if the objective function satisfies the Kurdyka–Łojasiewicz property...

Nonsmooth | Global convergence | Kurdyka–Łojasiewicz property | Operations Research/Decision Theory | Convex and Discrete Geometry | Inertial proximal gradient method | Mathematics | Operations Research, Management Science | Statistics, general | Nonconvex | Optimization | MATHEMATICS, APPLIED | Kurdyka-ojasiewicz property | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | CONVERGENCE | Methods | Business schools | Nonlinear programming | Quadratic programming | Convergence

Nonsmooth | Global convergence | Kurdyka–Łojasiewicz property | Operations Research/Decision Theory | Convex and Discrete Geometry | Inertial proximal gradient method | Mathematics | Operations Research, Management Science | Statistics, general | Nonconvex | Optimization | MATHEMATICS, APPLIED | Kurdyka-ojasiewicz property | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | CONVERGENCE | Methods | Business schools | Nonlinear programming | Quadratic programming | Convergence

Journal Article

Optimal Control Applications and Methods, ISSN 0143-2087, 09/2016, Volume 37, Issue 5, pp. 1035 - 1055

.... We study control systems with a priori given time‐driven switching mechanism in the presence of a quadratic cost functional...

nonlinear systems | optimal control | fixed‐levels control | constrained LQ optimization | fixed-levels control | MATHEMATICS, APPLIED | LINEAR SWITCHED CONTROL | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | FEEDBACK STABILIZATION | CONVEX | RELAXATIONS | OPTIMIZATION | AUTOMATION & CONTROL SYSTEMS | Control systems | Nonlinear dynamics | Approximation | Mathematical analysis | Optimal control | Constrictions | Dynamical systems | Optimization

nonlinear systems | optimal control | fixed‐levels control | constrained LQ optimization | fixed-levels control | MATHEMATICS, APPLIED | LINEAR SWITCHED CONTROL | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | FEEDBACK STABILIZATION | CONVEX | RELAXATIONS | OPTIMIZATION | AUTOMATION & CONTROL SYSTEMS | Control systems | Nonlinear dynamics | Approximation | Mathematical analysis | Optimal control | Constrictions | Dynamical systems | Optimization

Journal Article

Journal of Global Optimization, ISSN 0925-5001, 8/2013, Volume 56, Issue 4, pp. 1563 - 1589

In this paper, we introduce an iterative algorithm for finding a common element of the set of solutions of a system of mixed equilibrium problems, the set of...

Infinite family of strictly pseudo-contractive mappings | Variational inclusion | Nonexpansive semigroups | Optimization | Nonexpansive mapping | Economics / Management Science | Metric projection | System of mixed equilibrium problem | 47H10 | 47J20 | 49J30 | Operations Research/Decision Theory | 49J40 | 90C99 | η -strongly convex functions | 47H09 | 49M05 | Computer Science, general | Optimization problems | 47H17 | Real Functions | η-strongly convex functions | HILBERT-SPACES | MATHEMATICS, APPLIED | INEQUALITIES | MIXED EQUILIBRIUM PROBLEMS | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | QUADRATIC OPTIMIZATION | BANACH-SPACES | SETS | VISCOSITY APPROXIMATION METHODS | eta-strongly convex functions | STRONG-CONVERGENCE | Algorithms | Studies | Mapping | Iterative methods | Analysis | Fixed points (mathematics) | Group theory | Nonlinearity | Mathematical models | Iterative algorithms | Inverse | Convergence

Infinite family of strictly pseudo-contractive mappings | Variational inclusion | Nonexpansive semigroups | Optimization | Nonexpansive mapping | Economics / Management Science | Metric projection | System of mixed equilibrium problem | 47H10 | 47J20 | 49J30 | Operations Research/Decision Theory | 49J40 | 90C99 | η -strongly convex functions | 47H09 | 49M05 | Computer Science, general | Optimization problems | 47H17 | Real Functions | η-strongly convex functions | HILBERT-SPACES | MATHEMATICS, APPLIED | INEQUALITIES | MIXED EQUILIBRIUM PROBLEMS | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | QUADRATIC OPTIMIZATION | BANACH-SPACES | SETS | VISCOSITY APPROXIMATION METHODS | eta-strongly convex functions | STRONG-CONVERGENCE | Algorithms | Studies | Mapping | Iterative methods | Analysis | Fixed points (mathematics) | Group theory | Nonlinearity | Mathematical models | Iterative algorithms | Inverse | Convergence

Journal Article

Optimization, ISSN 1029-4945, 2018, Volume 68, Issue 1, pp. 279 - 348

The Bregman divergence (Bregman distance, Bregman measure of distance) is a certain useful substitute for a distance, obtained from a well-chosen function...

gauge | negative Boltzmann-Gibbs-Shannon entropy | uniformly convex | negative iterated log entropy | Bregman divergence | strongly convex | negative Havrda-Charvát-Tsallis entropy | negative Burg entropy | relative uniform convexity | Bregman function | MATHEMATICS, APPLIED | MAXIMUM-ENTROPY | DISTANCE | CONVEXITY | VARIATIONAL INEQUALITY PROBLEM | PROXIMAL POINT ALGORITHM | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | MIRROR DESCENT | CONVERGENCE | OPTIMIZATION | negative Havrda-Charvat-Tsallis entropy | OPERATORS | Operations research | Properties (attributes) | Divergence | Entropy (Information theory) | Machine learning | Nonlinear analysis | Entropy | Convexity | Banach space | Information theory

gauge | negative Boltzmann-Gibbs-Shannon entropy | uniformly convex | negative iterated log entropy | Bregman divergence | strongly convex | negative Havrda-Charvát-Tsallis entropy | negative Burg entropy | relative uniform convexity | Bregman function | MATHEMATICS, APPLIED | MAXIMUM-ENTROPY | DISTANCE | CONVEXITY | VARIATIONAL INEQUALITY PROBLEM | PROXIMAL POINT ALGORITHM | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | MIRROR DESCENT | CONVERGENCE | OPTIMIZATION | negative Havrda-Charvat-Tsallis entropy | OPERATORS | Operations research | Properties (attributes) | Divergence | Entropy (Information theory) | Machine learning | Nonlinear analysis | Entropy | Convexity | Banach space | Information theory

Journal Article

Optimization, ISSN 0233-1934, 10/2017, Volume 66, Issue 10, pp. 1577 - 1622

In this article, we develop a general theory of exact parametric penalty functions for constrained optimization problems...

localization principle | Penalty function | rate of steepest descent | smoothing approximation | exact penalization | zero duality gap | exact penalty method | INEQUALITY CONSTRAINED OPTIMIZATION | IMAGE SPACE | EXTREMUM | MATHEMATICS, APPLIED | ALGORITHM | NONLINEAR SEPARATION | METRIC REGULARITY | PROGRAMMING-PROBLEMS | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | CONVEX | BARRIER METHODS | Inventory control | Nonlinear programming | Smoothing | Optimization | Mathematics - Optimization and Control

localization principle | Penalty function | rate of steepest descent | smoothing approximation | exact penalization | zero duality gap | exact penalty method | INEQUALITY CONSTRAINED OPTIMIZATION | IMAGE SPACE | EXTREMUM | MATHEMATICS, APPLIED | ALGORITHM | NONLINEAR SEPARATION | METRIC REGULARITY | PROGRAMMING-PROBLEMS | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | CONVEX | BARRIER METHODS | Inventory control | Nonlinear programming | Smoothing | Optimization | Mathematics - Optimization and Control

Journal Article