Journal of Convex Analysis, ISSN 0944-6532, 2016, Volume 23, Issue 2, pp. 511 - 530

We extend the results of Correa, Garcia and Hantoute [6], dealing with the integration of nonconvex epi-pointed functions using the Fenchel subdifferential. In...

MATHEMATICS | SUBDIFFERENTIALS

MATHEMATICS | SUBDIFFERENTIALS

Journal Article

Journal of Convex Analysis, ISSN 0944-6532, 2019, Volume 26, Issue 1, pp. 189 - 200

In recent years there has been great interest in variational analysis of a class of nonsmooth functions called the minimal time function. In this paper we...

Limiting subdifferential | Calmness | Singular subdifferential | ε-Fréchet subdifferential | Minimal time function | limiting subdifferential | MATHEMATICS | epsilon-Frechet subdifferential | singular subdifferential | calmness | SUBDIFFERENTIALS | DISTANCE FUNCTIONS

Limiting subdifferential | Calmness | Singular subdifferential | ε-Fréchet subdifferential | Minimal time function | limiting subdifferential | MATHEMATICS | epsilon-Frechet subdifferential | singular subdifferential | calmness | SUBDIFFERENTIALS | DISTANCE FUNCTIONS

Journal Article

Journal of Optimization Theory and Applications, ISSN 0022-3239, 2/2019, Volume 180, Issue 2, pp. 397 - 427

This paper aims at providing some formulae for the subdifferential and the conjungate function of the supremum function over an arbitrary family of functions....

Pointwise supremum function | Mathematics | Theory of Computation | Optimization | Calculus of Variations and Optimal Control; Optimization | varepsilon $$ ε -Subdifferential | 90C25 | Operations Research/Decision Theory | 90C34 | Applications of Mathematics | Engineering, general | Fenchel conjugate | 46N10 | Convex analysis | ε-Subdifferential | MATHEMATICS, APPLIED | SET | CALCULUS RULES | FENCHEL SUBDIFFERENTIALS | EPSILON-SUBDIFFERENTIALS | EPI-POINTED FUNCTIONS | MINIMAX | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | epsilon-Subdifferential | INTEGRATION | CONVEX | PROPER APPROXIMATE SOLUTIONS | Employee motivation | Analysis

Pointwise supremum function | Mathematics | Theory of Computation | Optimization | Calculus of Variations and Optimal Control; Optimization | varepsilon $$ ε -Subdifferential | 90C25 | Operations Research/Decision Theory | 90C34 | Applications of Mathematics | Engineering, general | Fenchel conjugate | 46N10 | Convex analysis | ε-Subdifferential | MATHEMATICS, APPLIED | SET | CALCULUS RULES | FENCHEL SUBDIFFERENTIALS | EPSILON-SUBDIFFERENTIALS | EPI-POINTED FUNCTIONS | MINIMAX | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | epsilon-Subdifferential | INTEGRATION | CONVEX | PROPER APPROXIMATE SOLUTIONS | Employee motivation | Analysis

Journal Article

Computational Optimization and Applications, ISSN 0926-6003, 06/2017, Volume 67, Issue 2, pp. 421 - 442

Computing explicitly the epsilon-subdifferential of a proper function amounts to computing the level set of a convex function namely the conjugate minus a...

Piecewise linear–quadratic functions | Visualization | ε-Subdifferentials | Convex function | Subdifferentials | Computational convex analysis (CCA) | LEVEL BUNDLE METHODS | MATHEMATICS, APPLIED | APPROXIMATE SUBDIFFERENTIALS | epsilon-Subdifferentials | CALCULUS | Piecewise linear-quadratic functions | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | MINIMIZATION | CONVEX | CONJUGATE | OPTIMIZATION | BIVARIATE FUNCTIONS

Piecewise linear–quadratic functions | Visualization | ε-Subdifferentials | Convex function | Subdifferentials | Computational convex analysis (CCA) | LEVEL BUNDLE METHODS | MATHEMATICS, APPLIED | APPROXIMATE SUBDIFFERENTIALS | epsilon-Subdifferentials | CALCULUS | Piecewise linear-quadratic functions | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | MINIMIZATION | CONVEX | CONJUGATE | OPTIMIZATION | BIVARIATE FUNCTIONS

Journal Article

Journal of Global Optimization, ISSN 0925-5001, 7/2011, Volume 50, Issue 3, pp. 485 - 502

In this paper we first provide a general formula of inclusion for the Dini-Hadamard ε-subdifferential of the difference of two functions and show that it...

Fréchet ε -subdifferential | Cone-constrained nonsmooth nonconvex optimization problems | Optimality conditions in subdifferential form | Dini-Hadamard ε -subdifferential | Optimization | Economics / Management Science | Directionally approximately starshaped functions | 49J52 | Operations Research/Decision Theory | 90C56 | Sponge | Approximately starshaped functions | 26B25 | Computer Science, general | Real Functions | Dini-Hadamard ε-subdifferential | Fréchet ε-subdifferential | MATHEMATICS, APPLIED | APPROXIMATE SUBDIFFERENTIALS | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Frechet epsilon-subdifferential | SUBGRADIENTS | CALCULUS | Cone-constrained nonsmooth nonconvex optimization problems Optimality conditions in subdifferential form | Dini-Hadamard epsilon-subdifferential | GRADIENTS | Computer science | Universities and colleges | Studies | Mathematical problems

Fréchet ε -subdifferential | Cone-constrained nonsmooth nonconvex optimization problems | Optimality conditions in subdifferential form | Dini-Hadamard ε -subdifferential | Optimization | Economics / Management Science | Directionally approximately starshaped functions | 49J52 | Operations Research/Decision Theory | 90C56 | Sponge | Approximately starshaped functions | 26B25 | Computer Science, general | Real Functions | Dini-Hadamard ε-subdifferential | Fréchet ε-subdifferential | MATHEMATICS, APPLIED | APPROXIMATE SUBDIFFERENTIALS | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Frechet epsilon-subdifferential | SUBGRADIENTS | CALCULUS | Cone-constrained nonsmooth nonconvex optimization problems Optimality conditions in subdifferential form | Dini-Hadamard epsilon-subdifferential | GRADIENTS | Computer science | Universities and colleges | Studies | Mathematical problems

Journal Article

6.
Full Text
Differential Stability of Convex Optimization Problems with Possibly Empty Solution Sets

Journal of Optimization Theory and Applications, ISSN 0022-3239, 4/2019, Volume 181, Issue 1, pp. 126 - 143

This paper studies differential stability of infinite-dimensional convex optimization problems, whose solution sets may be empty. By using suitable sum rules...

Optimal value function | Mathematics | Theory of Computation | Parametric convex programming | Optimization | 90C31 | varepsilon $$ ε -Subdifferentials | varepsilon $$ ε -Normal directions | 49J53 | Calculus of Variations and Optimal Control; Optimization | 90C25 | Operations Research/Decision Theory | Conjugate function | 49Q12 | Applications of Mathematics | Engineering, general | ε-Normal directions | ε-Subdifferentials | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | SUBGRADIENTS | epsilon-Subdifferentials | CALCULUS | epsilon-Normal directions | Computational geometry | Convexity | Sum rules | Dimensional stability | Convex analysis

Optimal value function | Mathematics | Theory of Computation | Parametric convex programming | Optimization | 90C31 | varepsilon $$ ε -Subdifferentials | varepsilon $$ ε -Normal directions | 49J53 | Calculus of Variations and Optimal Control; Optimization | 90C25 | Operations Research/Decision Theory | Conjugate function | 49Q12 | Applications of Mathematics | Engineering, general | ε-Normal directions | ε-Subdifferentials | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | SUBGRADIENTS | epsilon-Subdifferentials | CALCULUS | epsilon-Normal directions | Computational geometry | Convexity | Sum rules | Dimensional stability | Convex analysis

Journal Article

SIAM JOURNAL ON OPTIMIZATION, ISSN 1052-6234, 2019, Volume 29, Issue 2, pp. 1714 - 1743

This work provides calculus rules for the Frechet and Mordukhovich subdifferentials of the pointwise supremum given by an arbitrary family of lower...

MATHEMATICS, APPLIED | APPROXIMATE SUBDIFFERENTIALS | INFINITE SYSTEMS | RICH FAMILIES | supremum functions | REPRESENTATION | variational analysis and optimization | calculus rules | SEMIINFINITE | FINITE | SETS | TANGENTIAL EXTREMAL PRINCIPLES | CONSTRAINED OPTIMIZATION | subdifferentials

MATHEMATICS, APPLIED | APPROXIMATE SUBDIFFERENTIALS | INFINITE SYSTEMS | RICH FAMILIES | supremum functions | REPRESENTATION | variational analysis and optimization | calculus rules | SEMIINFINITE | FINITE | SETS | TANGENTIAL EXTREMAL PRINCIPLES | CONSTRAINED OPTIMIZATION | subdifferentials

Journal Article

Journal of the London Mathematical Society, ISSN 0024-6107, 06/2011, Volume 83, Issue 3, pp. 637 - 658

Continuity of set‐valued maps is hereby revisited: after recalling some basic concepts of variational analysis and a short description of the state‐of‐the‐art,...

MATHEMATICS | OPTIMIZATION | SUBDIFFERENTIALS | MULTIFUNCTIONS

MATHEMATICS | OPTIMIZATION | SUBDIFFERENTIALS | MULTIFUNCTIONS

Journal Article

Set-Valued and Variational Analysis, ISSN 1877-0533, 12/2015, Volume 23, Issue 4, pp. 643 - 665

We introduce a subfamily of additive enlargements of a maximally monotone operator. Our definition is inspired by the early work of Simon Fitzpatrick. These...

Fitzpatrick function | Subdifferential operator | Enlargement of an operator | Brøndsted- Rockafellar enlargements | Mathematics | ε -subdifferential mapping | Convex lower semicontinuous function | Additive enlargements | Geometry | Maximally monotone operator | 90C30 | 90C46 | 49J52 | 90C25 | Analysis | Brøndsted- Rockafellar property | Fenchel-Young function | 48N15 | ε-subdifferential mapping | MATHEMATICS, APPLIED | REPRESENTATIONS | BUNDLE METHODS | PROOF | EXTENSION | EPSILON-ENLARGEMENTS | CONVEX-FUNCTIONS | FAMILY | Brondsted-Rockafellar property | epsilon-subdifferential mapping | Brondsted-Rockafellar enlargements | CONVERGENCE | Optimization and Control

Fitzpatrick function | Subdifferential operator | Enlargement of an operator | Brøndsted- Rockafellar enlargements | Mathematics | ε -subdifferential mapping | Convex lower semicontinuous function | Additive enlargements | Geometry | Maximally monotone operator | 90C30 | 90C46 | 49J52 | 90C25 | Analysis | Brøndsted- Rockafellar property | Fenchel-Young function | 48N15 | ε-subdifferential mapping | MATHEMATICS, APPLIED | REPRESENTATIONS | BUNDLE METHODS | PROOF | EXTENSION | EPSILON-ENLARGEMENTS | CONVEX-FUNCTIONS | FAMILY | Brondsted-Rockafellar property | epsilon-subdifferential mapping | Brondsted-Rockafellar enlargements | CONVERGENCE | Optimization and Control

Journal Article

10.
Full Text
The Fréchet and limiting subdifferentials of integral functionals on the spaces L1 (Ω, E)

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 12/2009, Volume 360, Issue 2, pp. 704 - 710

A new approach to computing the Fréchet subdifferential and the limiting subdifferential of integral functionals is proposed. Thanks to this way, we obtain...

spaces | Fenchel subdifferential | Integral functional | Limiting subdifferential | Fréchet subdifferential

spaces | Fenchel subdifferential | Integral functional | Limiting subdifferential | Fréchet subdifferential

Journal Article

Numerical Functional Analysis and Optimization, ISSN 0163-0563, 12/2018, Volume 39, Issue 16, pp. 1833 - 1854

Necessary optimality conditions for local Henig efficient and superefficient solutions of vector equilibrium problems involving equality, inequality, and set...

Clarke subdifferential | Michel-Penot subdifferential | vector equilibrium problems | vector optimization problems | vector variational inequalities | local superefficient solutions | local Henig efficient solutions | Michel–Penot subdifferential | MATHEMATICS, APPLIED | CALCULUS | Banach space | Convexity | Nonlinear programming

Clarke subdifferential | Michel-Penot subdifferential | vector equilibrium problems | vector optimization problems | vector variational inequalities | local superefficient solutions | local Henig efficient solutions | Michel–Penot subdifferential | MATHEMATICS, APPLIED | CALCULUS | Banach space | Convexity | Nonlinear programming

Journal Article

Advances in Nonlinear Analysis, ISSN 2191-9496, 02/2012, Volume 1, Issue 1, pp. 47 - 120

The theory presented in the paper consists of two parts. The first is devoted to basic concepts and principles such as the very concept of a subdifferential,...

metric modification | subdifferential | trustworthiness | subdifferential calculus | normal cone | separable reduction | separable space | limiting subdifferentials | coderivative | Banach space | Separable space | Subdifferential calculus | Trustworthiness | Normal cone | Coderivative | Limiting subdifferentials | Metric modification | G-subdifferential | Separable reduction | MATHEMATICS | MATHEMATICS, APPLIED

metric modification | subdifferential | trustworthiness | subdifferential calculus | normal cone | separable reduction | separable space | limiting subdifferentials | coderivative | Banach space | Separable space | Subdifferential calculus | Trustworthiness | Normal cone | Coderivative | Limiting subdifferentials | Metric modification | G-subdifferential | Separable reduction | MATHEMATICS | MATHEMATICS, APPLIED

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 01/2020, Volume 481, Issue 1, p. 123445

This is the first of two closely related papers on transversality. Here we introduce the notion of tangential transversality of two closed subsets of a Banach...

Lagrange multiplier rule | Tangential transversality | Nonseparation of sets | Intersection properties | MATHEMATICS | MATHEMATICS, APPLIED | APPROXIMATE SUBDIFFERENTIALS | Mathematics - Optimization and Control

Lagrange multiplier rule | Tangential transversality | Nonseparation of sets | Intersection properties | MATHEMATICS | MATHEMATICS, APPLIED | APPROXIMATE SUBDIFFERENTIALS | Mathematics - Optimization and Control

Journal Article

Journal of Functional Analysis, ISSN 0022-1236, 02/2016, Volume 270, Issue 4, pp. 1361 - 1378

Asplund property of a Banach space X is characterized by the existence of a rich family, in the product X×X⁎, consisting of some carefully chosen separable...

Rich family | Fréchet subdifferential | Asplund space | Separable reduction | MATHEMATICS | Frechet subdifferential

Rich family | Fréchet subdifferential | Asplund space | Separable reduction | MATHEMATICS | Frechet subdifferential

Journal Article

SIAM Journal on Optimization, ISSN 1052-6234, 2018, Volume 28, Issue 1, pp. 30 - 44

We continue the investigation of a general minimal time problem with a convex constant dynamics and a lower semicontinuous extended real-valued target function...

Infimal convolution | Fréchet subdifferential | Minimal time projection | Legendre-Fenchel conjugate | Minimal time function | SPACE | MATHEMATICS, APPLIED | Frechet subdifferential | minimal time projection | FORMULA | minimal time function | infimal convolution | Mathematics

Infimal convolution | Fréchet subdifferential | Minimal time projection | Legendre-Fenchel conjugate | Minimal time function | SPACE | MATHEMATICS, APPLIED | Frechet subdifferential | minimal time projection | FORMULA | minimal time function | infimal convolution | Mathematics

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 12/2017, Volume 456, Issue 2, pp. 1101 - 1122

We investigate several possibilities of obtaining a Łojasiewicz inequality for definable multifunctions and give some examples of applications thereof. In...

Tame geometry | Multifunctions | Subdifferentials | Łojasiewicz inequality | MATHEMATICS | MATHEMATICS, APPLIED | ANALYTIC SETS | DIFFERENTIABILITY | Lojasiewicz inequality | KURATOWSKI CONVERGENCE

Tame geometry | Multifunctions | Subdifferentials | Łojasiewicz inequality | MATHEMATICS | MATHEMATICS, APPLIED | ANALYTIC SETS | DIFFERENTIABILITY | Lojasiewicz inequality | KURATOWSKI CONVERGENCE

Journal Article

Journal of Optimization Theory and Applications, ISSN 0022-3239, 03/2019, Volume 180, Issue 3, pp. 775 - 786

Two criteria for the robust quasiconvexity of lower semicontinuous functions are established in terms of Frechet subdifferentials in Asplund spaces. The first...

Quasiconvexity | Quasimonotone | Approximate mean value theorem | Robust quasiconvexity | Fréchet subdifferential | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Frechet subdifferential | MONOTONICITY | Mathematical analysis | Optimization | Criteria

Quasiconvexity | Quasimonotone | Approximate mean value theorem | Robust quasiconvexity | Fréchet subdifferential | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Frechet subdifferential | MONOTONICITY | Mathematical analysis | Optimization | Criteria

Journal Article

Nonlinear Analysis, ISSN 0362-546X, 1999, Volume 38, Issue 6, pp. 687 - 773

Nonsmooth analysis had its origins in the early 1970s when control theorists and nonlinear programmers attempted to deal with necessary optimality conditions...

Mean value theorems | Mean value inequalities | Subdifferential calculus | Smooth spaces | Hamilton–Jacobi equations | Open mapping | Viscosity subdifferential | Viscosity solutions | Proximal subdifferential | Sensitivity | Geometric subdifferential | Generalized gradients | Constrained optimization problems | Limiting coderivatives | Metric regularity | Limiting subdifferentials | Implicit function theorems | Coderivative calculus | MATHEMATICS, APPLIED | proximal subdifferential | geometric subdifferential | smooth spaces | limiting subdifferentials | implicit function theorems | SMOOTH VARIATIONAL PRINCIPLE | MATHEMATICS | generalized gradients | coderivative calculus | Hamilton-Jacobi equations | LOWER SEMICONTINUOUS FUNCTIONS | APPROXIMATE SUBDIFFERENTIALS | NONCONVEX DIFFERENTIAL-INCLUSIONS | MEAN-VALUE THEOREM | INFINITE DIMENSIONS | mean value theorems | limiting coderivatives | subdifferential calculus | UNBOUNDED LINEAR TERMS | metric regularity | BANACH-SPACES | viscosity subdifferential | viscosity solutions | sensitivity | mean value inequalities | constrained optimization problems | open mapping

Mean value theorems | Mean value inequalities | Subdifferential calculus | Smooth spaces | Hamilton–Jacobi equations | Open mapping | Viscosity subdifferential | Viscosity solutions | Proximal subdifferential | Sensitivity | Geometric subdifferential | Generalized gradients | Constrained optimization problems | Limiting coderivatives | Metric regularity | Limiting subdifferentials | Implicit function theorems | Coderivative calculus | MATHEMATICS, APPLIED | proximal subdifferential | geometric subdifferential | smooth spaces | limiting subdifferentials | implicit function theorems | SMOOTH VARIATIONAL PRINCIPLE | MATHEMATICS | generalized gradients | coderivative calculus | Hamilton-Jacobi equations | LOWER SEMICONTINUOUS FUNCTIONS | APPROXIMATE SUBDIFFERENTIALS | NONCONVEX DIFFERENTIAL-INCLUSIONS | MEAN-VALUE THEOREM | INFINITE DIMENSIONS | mean value theorems | limiting coderivatives | subdifferential calculus | UNBOUNDED LINEAR TERMS | metric regularity | BANACH-SPACES | viscosity subdifferential | viscosity solutions | sensitivity | mean value inequalities | constrained optimization problems | open mapping

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

Annals of Operations Research, ISSN 0254-5330, 10/2018, Volume 269, Issue 1, pp. 727 - 751

The aim of this work is twofold. First, we establish sum rules for the directionally coderivatives of multifunctions and intersection rules for the...

Directional subdifferential | Directional coderivative | Business and Management | Set-valued optimization | Operations Research/Decision Theory | Optimality condition | Theory of Computation | Combinatorics | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | MULTIOBJECTIVE OPTIMIZATION | SUBREGULARITY | EQUILIBRIUM CONSTRAINTS | NONSMOOTH MATHEMATICAL PROGRAMS | SUBDIFFERENTIALS | OPTIMALITY CONDITIONS | Functions | Management science | Research | Functional equations | Mathematical research | Mathematical optimization | Studies | Operations research | Sum rules | Cones

Directional subdifferential | Directional coderivative | Business and Management | Set-valued optimization | Operations Research/Decision Theory | Optimality condition | Theory of Computation | Combinatorics | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | MULTIOBJECTIVE OPTIMIZATION | SUBREGULARITY | EQUILIBRIUM CONSTRAINTS | NONSMOOTH MATHEMATICAL PROGRAMS | SUBDIFFERENTIALS | OPTIMALITY CONDITIONS | Functions | Management science | Research | Functional equations | Mathematical research | Mathematical optimization | Studies | Operations research | Sum rules | Cones

Journal Article

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