On various notions of regularity of sets in nonsmooth analysis

Bounkhel, Messaoud; Thibault, Lionel

doi:10.1016/s0362-546x(00)00183-8

Cited by 80 publications

(54 citation statements)

References 17 publications

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“…Subdifferential properties of the distance function (1.3) have been well investigated and applied in many publications; see, e.g., [2,3,5,12,13,14,15,18] and the references therein. Much less has been done for the minimal time function (1.1).…”

Section: Introductionmentioning

confidence: 99%

Limiting subgradients of minimal time functions in Banach spaces

Mordukhovich

Nam²

2009

J Glob Optim

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Abstract. The paper mostly concerns the study of generalized differential properties of the so-called minimal time functions associated, in particular, with constant dynamics and arbitrary closed target sets in control theory. Functions of this type play a significant role in many aspects of optimization, control theory: and Hamilton-Jacobi partial differential equations. We pay the main attention to computing and estimating limiting subgradients of the minimal value functions and to deriving the corresponding relations for Frechet type c-subgradients in arbitrary Banach spaces.

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Section: Introductionmentioning

confidence: 99%

Limiting subgradients of minimal time functions in Banach spaces

Mordukhovich

Nam²

2009

J Glob Optim

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“…Various properties of the normal cones (as well as of the subdifferentials of lower semicontinuous functions) can be found, e.g., in [10,2,36,12,31,6]). Similarly as for the subdifferentials, a closed set C is said to be proximally (normally,…”

Section: Preliminariesmentioning

confidence: 99%

Regularity of a Kind of Marginal Functions in Hilbert Spaces

Pereira

Goncharov

2014

Optimization in Science and Engineering

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and goncha@uevora.pt Dedicated to Panos M. Pardalos in honor of his 60th birthday Summary. We study well-posedness of some mathematical programming problem depending on a parameter that generalizes in a certain sense the metric projection onto a closed nonconvex set. We are interested in regularity of the set of minimizers as well as of the value function, which can be seen, on one hand, as the viscosity solution to a Hamilton-Jacobi equation, while, on the other, as the minimal time in some related optimal time control problem. The regularity includes both the Fréchet differentiability of the value function and the Hölder continuity of its (Fréchet) gradient.

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“…We recall (see [14]) that any one of the two above normal regularities entails the (Clarke) tangential regularity. We refer to ( [14]) for the development of a detailed comparison between the above concepts of normal and tangential regularities and various others.…”

Section: Frédéric Bernard Lionel Thibault and Nadia Zlatevamentioning

confidence: 99%

Prox-regular sets and epigraphs in uniformly convex Banach spaces: Various regularities and other properties

Bernard

Thibault

Zlateva

2011

Trans. Amer. Math. Soc.

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Abstract. We continue the study of prox-regular sets that we began in a previous work in the setting of uniformly convex Banach spaces endowed with a norm both uniformly smooth and uniformly convex (e.g., L p , W m,p spaces). We prove normal and tangential regularity properties for these sets, and in particular the equality between Mordukhovich and proximal normal cones. We also compare in this setting the proximal normal cone with different Hölderian normal cones depending on the power types s, q of moduli of smoothness and convexity of the norm. In the case of sets that are epigraphs of functions, we show that J-primal lower regular functions have prox-regular epigraphs and we compare these functions with Poliquin's primal lower nice functions depending on the power types s, q of the moduli. The preservation of prox-regularity of the intersection of finitely many sets and of the inverse image is obtained under a calmness assumption. A conical derivative formula for the metric projection mapping of prox-regular sets is also established. Among other results of the paper it is proved that the Attouch-Wets convergence preserves the uniform r-prox-regularity property and that the metric projection mapping is in some sense continuous with respect to this convergence for such sets.

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On various notions of regularity of sets in nonsmooth analysis

Cited by 80 publications

References 17 publications

Limiting subgradients of minimal time functions in Banach spaces

Limiting subgradients of minimal time functions in Banach spaces

Regularity of a Kind of Marginal Functions in Hilbert Spaces

Prox-regular sets and epigraphs in uniformly convex Banach spaces: Various regularities and other properties

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