All ressources related to this research area.
Variational Nonsmooth Analysis
Some of my research papers on Nonsmooth Variational Analysis develop various basic properties of prox-regular sets and functions, subsmooth sets, primal lower regular functions,...
Related book(s)
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Prox-Regular Sets and Applications
Editor: International Press
Publication date: 2010
Related research areas: Modern Convex Analysis, Set-Valued Analysis, Variational Nonsmooth Analysis
The above title corresponds to a chapter, pages 99-182, of ''Handbook of Nonconvex Analysis and Applications'' Int. Press, Somerville, MA, 2010, Editor
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Unilateral Variational Analysis in Banach Spaces. Part I: General Theory
Authors: Thibault
Editor: World Scientific
Publication date: 2023
Related research areas: Functional analysis, Geometry of Banach spaces, Modern Convex Analysis, Variational Nonsmooth Analysis
The monograph provides a detailed and comprehensive presentation of the rich and beautiful theory of unilateral variational analysis in infinite dimensions. It is divided into two volumes named Part I and Part II. Starting with the convergence of sets and the semilimits and semicontinuities of multimappings, the first volume develops the theories of tangent cones, of subdifferentials, of convexity and duality in locally convex spaces, of extended mean value inequalities in absence of differentiability, of metric regularity, of constrained optimization problems. The second volume is devoted to special classes of non-smooth functions and sets. It expands the theory of subsmooth functions and sets, of semiconvex functions and multimappings, of primal lower regular functions, of singularities of non-smooth mappings, of prox-regular functions and sets in general spaces, of differentiability of projection mapping and others for prox-regular sets. Both volumes I and II contain, for each chapter, extensive comments covering related developments and historical comments. Connected area fields of the material are: optimization, optimal control, variational inequalities, differential inclusions, mechanics, economics. The book is intended for PhD students, researchers, and practitioners using unilateral variational analysis tools.
cover
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Unilateral Variational Analysis in Banach Spaces. Part II: Special Classes of Functions and Sets
Authors: Thibault
Editor: World Scientific
Publication date: 2023
Related research areas: Functional analysis, Geometry of Banach spaces, Modern Convex Analysis, Variational Nonsmooth Analysis
The monograph provides a detailed and comprehensive presentation of the rich and beautiful theory of unilateral variational analysis in infinite dimensions. It is divided into two volumes named Part I and Part II. Starting with the convergence of sets and the semilimits and semicontinuities of multimappings, the first volume develops the theories of tangent cones, of subdifferentials, of convexity and duality in locally convex spaces, of extended mean value inequalities in absence of differentiability, of metric regularity, of constrained optimization problems. The second volume is devoted to special classes of non-smooth functions and sets. It expands the theory of subsmooth functions and sets, of semiconvex functions and multimappings, of primal lower regular functions, of singularities of non-smooth mappings, of prox-regular functions and sets in general spaces, of differentiability of projection mapping and others for prox-regular sets. Both volumes I and II contain, for each chapter, extensive comments covering related developments and historical comments. Connected area fields of the material are: optimization, optimal control, variational inequalities, differential inclusions, mechanics, economics. The book is intended for PhD students, researchers, and practitioners using unilateral variational analysis tools.
cover
Related article(s)
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C1,.-regularity and Lipschitz-like properties of subdifferential.
Author(s):Jourani, Thibault, Zagrodny
Related research area(s): Modern Convex Analysis, Variational Nonsmooth Analysis
Journal: Proceedings of London Mathematical Society
Reference: 105 (2012), 189--223.
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A general metric regularity in Asplund spaces
Author(s):Amahroq, Jourani, Thibault
Related research area(s): Variational Nonsmooth Analysis
Journal: Numerical Functional Analysis and Optimization
Reference: 19 (1998), 215--226.
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A note on Fréchet and approximate subdifferentials of composite functions
Related research area(s): Variational Nonsmooth Analysis
Journal: Bulletin of Australian Mathematical Society
Reference: 49 (1994), 111--115.
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A note on the Zagrodny mean value theorem
Author(s):Thibault
Related research area(s): Variational Nonsmooth Analysis
Journal: Optimization
Reference: 35 (1995), 127--130.
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Approximate and metric regularity in mathematical programming in Banach space
Related research area(s): Optimization, Variational Nonsmooth Analysis
Journal: Mathematics of Operations research
Reference: 18 (1993), 390--401.
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BV prox-regular sweeping process with bounded truncation
Related research area(s): Differential Inclusions, Variational Nonsmooth Analysis
Journal: Optimization
Reference: 69 (2020), 1391--1437.
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Calcul sous-différentiel et calcul des variations en dimension infinie
Author(s):Thibault
Related research area(s): Optimal Control , Variational Nonsmooth Analysis
Journal: Bulletin de la Société Mathématique de France
Reference: 60 (1979), 161--175.
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Chain rules for coderivatives of multivalued mappings in Banach spaces
Related research area(s): Set-Valued Analysis, Variational Nonsmooth Analysis
Journal: Proceedings of American Mathematical Society
Reference: 126 (1998), 1479--1485.
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Characterization of lower semicontinuous convex functions
Author(s):Correa, Jofré, Thibault
Related research area(s): Modern Convex Analysis, Variational Nonsmooth Analysis
Journal: Proceedings of American Mathematical Society
Reference: 116 (1992), 67--72.
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Characterizations of prox-regular sets in uniformly convex Banach spaces
Author(s):Bernard, Thibault, Zlateva
Related research area(s): Geometry of Banach spaces, Modern Convex Analysis, Variational Nonsmooth Analysis
Journal: Journal of Convex Analysis
Reference: 13 (2006), 525--559.
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Coderivative of multivalued mappings, locally compact cones and metric regularity
Related research area(s): Variational Nonsmooth Analysis
Journal: Nonlinear Analysis
Reference: 35 (1999), 925--945.
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Conditions nécessaires pour des problèmes de contrôle optimal définis par des fonctions compactement lipschitziennes
Author(s):Thibault
Related research area(s): Variational Nonsmooth Analysis
Journal: Comptes Rendus de l'Académie des Sciences de Paris
Reference: 288 (1979), 133--136.
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Convergence of subdifferentials of convexly composite functions
Author(s):Combari, Poliquin, Thibault
Related research area(s): Modern Convex Analysis, Variational Nonsmooth Analysis
Journal: Canadian Journal of Mathematics
Reference: 51 (1999), 250--265.
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D-representation of subdifferentials of directionally Lipschitz functions
Related research area(s): Modern Convex Analysis, Variational Nonsmooth Analysis
Journal: Proceedings of American Mathematical Society
Reference: 110 (1990), 117--123.
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Differential properties of the Moreau envelope
Author(s):Jourani, Thibault, Zagrodny
Related research area(s): Variational Nonsmooth Analysis
Journal: Journal of Functional Analysis
Reference: 266 (2014), 1185--1237.
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Directionally pseudo-Lipschitz set-valued mappings
Related research area(s): Variational Nonsmooth Analysis
Journal: Journal of Mathematical Analysis and Applications
Reference: 266 (2002), 269--287.
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Discontinuous sweeping process with prox-regular sets
Author(s):Adly, Nacry, Thibault
Related research area(s): Variational Nonsmooth Analysis
Journal: ESAIM Control Optim. Calc. Var.
Reference: 23 (2017), 1293--1329.
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Distance function associated to a prox-regular set
Related research area(s): Variational Nonsmooth Analysis
Journal: Set-Valued and Variational Analysis
Reference: 30 (2022), 731--750.
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Enlarged inclusions of subdifferentials
Related research area(s): Set-Valued Analysis, Variational Nonsmooth Analysis
Journal: Canadian Mathematical Bulletin
Reference: 48 (2005), 283--301.
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Envelopes for sets and functions: regularization and generalized conjugacy
Author(s):Cabot, Jourani, Thibault
Related research area(s): Modern Convex Analysis, Variational Nonsmooth Analysis
Journal: Mathematika
Reference: 63 (2017), 383--432.
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Epi-différentiels de fonctions vectorielles
Author(s):Thibault
Related research area(s): Variational Nonsmooth Analysis
Journal: Comptes Rendus de l'Académie des Sciences de Paris
Reference: 290 (1980), 87--90.
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Epiconvergence of convexly composite functions in Banach spaces
Related research area(s): Modern Convex Analysis, Variational Nonsmooth Analysis
Journal: SIAM Journal on Optimization
Reference: 13 (2003), 986--1003.
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Espérances conditionnelles d'intégrandes lipschitziens et d'intégrandes semi-continus inférieurement
Author(s):Thibault
Related research area(s): Variational Nonsmooth Analysis
Journal: Comptes Rendus de l'Académie des Sciences de Paris
Reference: 291 (1980), 551--554.
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Evolution differential inclusions associated to primla lower regular functions
Related research area(s): Differential Inclusions, Set-Valued Analysis, Variational Nonsmooth Analysis
Journal: Journal of Nonlinear and Convex Analysis
Reference: 20 (2019), 1949--1979.
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Evolution problems associated with primal lower nice functions
Related research area(s): Set-Valued Analysis, Variational Nonsmooth Analysis
Journal: Journal of Convex Analysis
Reference: 13 (2006), 385--421.
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Existence et caractérisation de différentielles généralisées d'applications localement lipschiziennes d'un espace de Banach séparable dans un espace de Banach réflexif
Author(s):Hiriart-Urruty, Thibault
Related research area(s): Variational Nonsmooth Analysis
Journal: Comptes Rendus de l'Académie des Sciences de Paris
Reference: 290 (1980), 1091--1094.
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Extensions of subdifferential calculus rules in Banach spaces
Related research area(s): Variational Nonsmooth Analysis
Journal: Canadian Journal of Mathematics
Reference: 48 (1996), 183--201.
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Fonctions compactement lipschitziennes et programmation mathématique
Author(s):Thibault
Related research area(s): Variational Nonsmooth Analysis
Journal: Comptes Rendus de l'Académie des Sciences de Paris
Reference: 287 (1978), 213--216.
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Inclusion and integration of subdifferentials
Related research area(s): Variational Nonsmooth Analysis
Journal: Journal of Nonlinear and Convex Analysis
Reference: 3 (2002), 411--434.
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Infimal convolution and optimal time control problem I: Frechet and proximal subdifferential
Related research area(s): Optimization, Variational Nonsmooth Analysis
Journal: Set-Valued and Variational Analysis
Reference: 26 (2018), 581--606.
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Infimal convolution and optimal time control problem II: limiting subdifferential
Related research area(s): Optimization, Variational Nonsmooth Analysis
Journal: Set-Valued and Variational Analysis
Reference: 25 (2017), 517--542.
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Infimal convolution and optimal time control problem III: minimal time projection set
Related research area(s): Optimization, Variational Nonsmooth Analysis
Journal: SIAM Journal on Optimization
Reference: 28 (2018), 30--44.
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Integrability of subdifferentials of certain bivariate functions
Related research area(s): Modern Convex Analysis, Variational Nonsmooth Analysis
Journal: Nonlinear Analysis
Reference: 54 (2003), 1251--1269.
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Integrability of subdifferentials of directionally Lipschitz functions
Related research area(s): Variational Nonsmooth Analysis
Journal: Proceedings of American Mathematical Society
Reference: 133 (2005), 2939--2948.
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Integration of primal lower nice functions in Hilbert space
Author(s):Bernard, Thibault, Zagrodny
Related research area(s): Variational Nonsmooth Analysis
Journal: Journal of Optimization Theory and Applications
Reference: 124 (2005), 561--579.
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Integration of subdifferentials oof lower semicontinuous functions on Banach spaces
Related research area(s): Variational Nonsmooth Analysis
Journal: Journal of Mathematical Analysis and Applications
Reference: 189 (1995), 33--58.
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Local differentiability of distance functions
Author(s):Poliquin, Rockafellar, Thibault
Related research area(s): Variational Nonsmooth Analysis
Journal: Transactions of American Mathematical Society
Reference: 352 (2000), 5231--5249.
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Metric projection and compatibly parameterized families of prox-regular sets in Hilbert space
Related research area(s): Variational Nonsmooth Analysis
Journal: Nonlinear Analysis
Reference: 75 (2012), 1547--1562.
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Metric regularity and subdifferential calculus in Banach space
Related research area(s): Variational Nonsmooth Analysis
Journal: Set-Valued Analysis
Reference: 3 (1995), 87--100.
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Metric regularity for strongly compactly Lipschitzian mappings
Related research area(s): Optimization, Variational Nonsmooth Analysis
Journal: Nonlinear Analysis
Reference: 24 (1995), 229--240.
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Moreau envelopes of s-lower regular functions
Related research area(s): Variational Nonsmooth Analysis
Journal: Nonlinear Analysis
Reference: 127 (2015), 157--181.
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Moreau sweeping process with bounded truncated retraction
Author(s):Thibault
Related research area(s): Differential Inclusions, Variational Nonsmooth Analysis
Journal: Journal of Convex Analysis
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New metric properties for prox-regular sets
Author(s):Adly, Nacry, Thibault
Related research area(s): Variational Nonsmooth Analysis
Journal: Mathematical Programming
Reference: 189 (2021), 7--36.
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Nonconvex sweeping process and prox-regularity in Hilbert space
Related research area(s): Differential Inclusions, Variational Nonsmooth Analysis
Journal: Journal of Nonlinear and Convex Analysis
Reference: 6 (2005), 359--374.
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On characterizations of submanifolds via smoothness of the distance function in Hilbert spaces
Related research area(s): Differential geometry , Variational Nonsmooth Analysis
Journal: Journal of Optimization Theory and Applications
Reference: 182 (2019), 189--210.
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On generalized differentials and subdifferentials of Lipschitz vector-valued functions
Author(s):Thibault
Related research area(s): Variational Nonsmooth Analysis
Journal: Nonlinear Analysis
Reference: 6 (1982), 1037--1053.
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On proto-differntaibility and strict proto-differentiability of multifunctions of feasible points in perturbed optimization problems
Related research area(s): Variational Nonsmooth Analysis
Journal: Numerical Functional Analysis and Optimization
Reference: 16 (1995), 1293--1307.
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On smoothness of solutions to projected differential equations
Author(s):Salas, Thibault, Vilches
Related research area(s): Differential Inclusions, Variational Nonsmooth Analysis
Journal: Discrete and continuous dynamical systems- Series A
Reference: 39 (2019), 2255--2283.
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On subdifferentials of optimal value functions
Author(s):Thibault
Related research area(s): Optimization, Variational Nonsmooth Analysis
Journal: SIAM Journal on Control and Optimization
Reference: 29 (1991), 1019--1036.
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On the discretization of truncated integro-differential sweeping process and optimal control
Author(s):Bouach, Haddad, Thibault
Related research area(s): Differential Inclusions, Optimal Control , Variational Nonsmooth Analysis
Journal: Journal of Optimization Theory and Applications
Reference: 193 (2022), 785--830.
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On various notions of regularity of sets in nonsmooth analysis
Related research area(s): Variational Nonsmooth Analysis
Journal: Nonlinear Analysis
Reference: 48 (2002), 223--246.
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Partial extensions of Attouch's theorem with applications to proto-derivatives of subgradient mappings
Author(s):Levy, Poliquin, Thibault
Related research area(s): Set-Valued Analysis, Variational Nonsmooth Analysis
Journal: Transactions of American Mathematical Society
Reference: 347 (1995), 1269--1294.
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Perturbed sweeping process with a subsmooth set depending on the state
Author(s):Haddad, Noel, Thibault
Related research area(s): Differential Inclusions, Variational Nonsmooth Analysis
Journal: Linear and Nonlinear Analysis
Reference: 2 (2016), 155--174.
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Preservation of uniform prox-regularity and application to constrained optimization
Author(s):Adly, Nacry, Thibault
Related research area(s): Optimization, Variational Nonsmooth Analysis
Journal: SIAM Journal on Optimization
Reference: 26 (2016), 448--473.
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Primal-lower-nice property of value functions in optimization and control problems
Related research area(s): Optimization, Variational Nonsmooth Analysis
Journal: Set-Valued and Variational Analysis
Reference: 18 (2010), 569--600.
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Problèmes de Bolza dans un espace de Banach séparable
Author(s):Thibault
Related research area(s): Variational Nonsmooth Analysis
Journal: Comptes Rendus de l'Académie des Sciences de Paris
Reference: 282 (1976), 1303--1306.
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Prox-regular functions in Hilbert space
Related research area(s): Variational Nonsmooth Analysis
Journal: Journal of Mathematical Analysis and Applications
Reference: 303 (2005), 1--14.
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Prox-regular sets and epigraphs in uniformly convex Banach spaces: various regularities and properties
Author(s):Bernard, Thibault, Zlateva
Related research area(s): Functional analysis, Geometry of Banach spaces, Variational Nonsmooth Analysis
Journal: Transactions of American Mathematical Society
Reference: 363 (2011), 2211--2247.
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Prox-regular sets and Legendre-Fenchel transform related to separation properties
Author(s):Adly, Nacry, Thibault
Related research area(s): Modern Convex Analysis, Variational Nonsmooth Analysis
Journal: Optimization
Reference: 71 (2022), 2097--2129.
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Prox-regularity approach to generalized equations and image projection
Author(s):Adly, Nacry, Thibault
Related research area(s): Set-Valued Analysis, Variational Nonsmooth Analysis
Journal: ESAIM Control Optim. Calc. Var.
Reference: 24 (2018), 677--708.
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Prox-regularity of functions and sets in Banach spaces
Related research area(s): Variational Nonsmooth Analysis
Journal: Set-Valued Analysis
Reference: 12 (2004), 25--47.
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Proximal and Fréchet normal formulae for some small normal cones
Related research area(s): Variational Nonsmooth Analysis
Journal: Nonlinear Analysis
Reference: 19 (1992), 599--612.
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Qualification conditions for calculus rules of coderivatives of multivalued mappings
Related research area(s): Set-Valued Analysis, Variational Nonsmooth Analysis
Journal: Journal of Mathematical Analysis and Applications
Reference: 218 (1998), 66--81.
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Quantitative characterizations of nonconvex bodies with smooth boundaries in Hilbert spaces va the metric projection
Related research area(s): Variational Nonsmooth Analysis
Journal: Journal of Mathematical Analysis and Applications
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Quasi-interiorly e-tangent cones to multifunctions
Author(s):El Abdouni, Thibault
Related research area(s): Set-Valued Analysis, Variational Nonsmooth Analysis
Journal: Numerical Functional Analysis and Optimization
Reference: 10 (1989), 619--641.
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Quelques propriétés des sous-différentiels de fonctions réelles localement lipschitziennes définies sur un espace de Banach séparable
Author(s):Thibault
Related research area(s): Variational Nonsmooth Analysis
Journal: Comptes Rendus de l'Académie des Sciences de Paris
Reference: 282 (1976), 507--510.
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Reduction of state dependent sweeping process to unsconstrained differential inclusion
Author(s):Haddad, Kecis, Thibault
Related research area(s): Differential Inclusions, Set-Valued Analysis, Variational Nonsmooth Analysis
Journal: Journal of Global Optimization
Reference: 62 (2015), 167--182.
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Regularization of dynamical systems associated with prox-regular moving sets
Related research area(s): Differential Inclusions, Set-Valued Analysis, Variational Nonsmooth Analysis
Journal: Journal of Nonlinear and Convex Analysis
Reference: 15 (2014), 647--663.
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Scalarization of tangential regularity of set-valued mappings
Author(s):Thibault
Related research area(s): Set-Valued Analysis, Variational Nonsmooth Analysis
Journal: Set-Valued Analysis
Reference: 7 (1999), 33--53.
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Smoothness of the metric projection onto nonconvex bodies in Hilbert spaces
Author(s):Correa, Salas, Thibault
Related research area(s): Variational Nonsmooth Analysis
Journal: Journal of Mathematical Analysis and Applications
Reference: 457 (2018), 1307--1332.
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Subdifferential analysis of bivariate separately regular functions
Related research area(s): Modern Convex Analysis, Variational Nonsmooth Analysis
Journal: Journal of Mathematical Analysis and Applications
Reference: 148 (1990), 157--174.
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Subdifferential characterization of s-lower regular functions
Related research area(s): Variational Nonsmooth Analysis
Journal: Applicable Analysis
Reference: 94 (2015), 85--98.
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Subdifferential determination of essentially directionally smooth functions in Banach space
Related research area(s): Functional analysis, Variational Nonsmooth Analysis
Journal: SIAM Journal on Optimization
Reference: 20 (2010), 2300--2326.
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Subdifferential monotonicity as characterization of convex functions
Author(s):Correa, Jofré, Thibault
Related research area(s): Modern Convex Analysis, Variational Nonsmooth Analysis
Journal: Numerical Functional Analysis and Optimization
Reference: 15 (1994), 531--535.
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Subdifferential regularity of directionally Lipschitzian functions
Related research area(s): Variational Nonsmooth Analysis
Journal: Canadian Mathematical Bulletin
Reference: 43 (2000), 25--36.
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Subdifferential representation formula annd subdifferential criteria for the behavior of nonsmooth functions
Author(s):Correa, Gajardo, Thibault
Related research area(s): Variational Nonsmooth Analysis
Journal: Nonlinear Analysis
Reference: 65 (2006), 864--891.
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Subdifferentials of compactly Lipschitzian vector-valued functions
Author(s):Thibault
Related research area(s): Variational Nonsmooth Analysis
Journal: Annali di Matematica Pura ed Applicata
Reference: 125 (1980), 157--192.
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Subdifferentials of nonconvex vector-valued functions
Author(s):Thibault
Related research area(s): Variational Nonsmooth Analysis
Journal: Journal of Mathematical Analysis and Applications
Reference: 86 (1982), 319--344.
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Sublevel representation of epi-Lipschitz sets and other properties
Related research area(s): Optimization, Set-Valued Analysis, Variational Nonsmooth Analysis
Journal: Mathematical Programming
Reference: 168 (2018), 555--569.
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Subsmooth sets: Functional characterizations and related topics
Author(s):Aussel, Daniilidis, Thibault
Related research area(s): Variational Nonsmooth Analysis
Journal: Transactions of American Mathematical Society
Reference: 357 (2005), 1275--1301.
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Sweeping process with regular and nonregular sets
Author(s):Thibault
Related research area(s): Differential Inclusions, Variational Nonsmooth Analysis
Journal: Journal of Differential Equations
Reference: 54 (2003), 1251--1269.
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Tangent cones and quasi-interiorly tangent cones to multifunctions
Author(s):Thibault
Related research area(s): Variational Nonsmooth Analysis
Journal: Transactions of American Mathematical Society
Reference: 277 (1983), 601--621.
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The approximate subdifferential of composite functions
Related research area(s): Set-Valued Analysis, Variational Nonsmooth Analysis
Journal: Bulletin of Australian Mathematical Society
Reference: 47 (1993), 443--455.
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The attainment set of the phi-envelope and genericity properties
Author(s):Cabot, Jourani, Thibault, Zagrodny
Related research area(s): Geometry of Banach spaces, Modern Convex Analysis, Variational Nonsmooth Analysis
Journal: Mathematica Scandinavica
Reference: 124 (2019), 203--246.
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The NSLUC property and Klee envelope
Author(s):Jourani, Thibault, Zagrodny
Related research area(s): Functional analysis, Geometry of Banach spaces, Variational Nonsmooth Analysis
Journal: Mathematische Annalen
Reference: 365 (2016), 923--967.
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The use of metric graphical regularity in approximate subdifferential calculus rules in finite dimensions
Related research area(s): Variational Nonsmooth Analysis
Journal: Optimization
Reference: 21 (1990), 509--519.
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Uniformly prox-regularity of functions and epigraphs in Hilbert spaces
Related research area(s): Variational Nonsmooth Analysis
Journal: Nonlinear Analysis
Reference: 60 (2005), 187--207.
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Various Lipschitz-like properties for functions and sets I: Directional derivative and tangential characterizations
Author(s):Correa, Gajardo, Thibault
Related research area(s): Modern Convex Analysis, Variational Nonsmooth Analysis
Journal: SIAM Journal on Optimization
Reference: 20 (2010), 1766--1785.
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Various Lipschitz-like properties for functions and sets II: subdifferential and normal characterizations
Author(s):Correa, Gajardo, Thibault
Related research area(s): Set-Valued Analysis, Variational Nonsmooth Analysis
Journal: Journal of Convex Analysis
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Verifiable conditions for openness and regularity of multivalued mappings in Banch spaces
Related research area(s): Set-Valued Analysis, Variational Nonsmooth Analysis
Journal: Transactions of American Mathematical Society
Reference: 347 (1995), 1255--1268.
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Well-posedness and subdifferentials of optimal value and infimal convolution
Related research area(s): Optimization, Variational Nonsmooth Analysis
Journal: Set-Valued and Variational Analysis
Reference: 27 (2019), 841--861.