Lyusternik's Theorem and the Theory of Extrema

Дмитрук, А. В.; Milyutin, A. A.; Osmolovskii, Nikolai P.

doi:10.1070/rm1980v035n06abeh001973

Cited by 157 publications

(127 citation statements)

References 10 publications

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“…To ensure that D~1 G(y, x)(O) = {0}, we proceed similarly to Case 1 observing that the calculus rule in (3,6) holds by [16,Corollary 3,5] under the qualification condition (3,5) due to the SNC property of S12 at (y, x), which is obviously implied by the assumed SNC property of S1 at i, To conclude now by [ 4 Frechet-Like Normals to Set Images…”

Section: • F(:c Y)(y')mentioning

confidence: 99%

“…We refer the reader to [4,6,10,11,13,15,16,20,21] for the genesis of ideas and various approaches and for more results and discussions in this direction in finite-din1ensional and infinite-dimensional Banach spaces.…”

Section: • F(:c Y)(y')mentioning

confidence: 99%

“…The concept of metric regularity goes back to the seminal Lyusternik-Graves theorem of the classical nonlinear analysis and has been long recognized among the most fundamental tools in the modern stage of nonlinear analysis especially regarding its variational aspects; see, e.g., the books [4,14,16,17,20,21], the extended surveys [2,6,11], the papers [1,12,13,15,18,19], and the references therein for more details as well as for recent developments. Alex Ioffe is surely one of the major original contributors and nowadays leaders in this and related fields of modern set-valued and variational analysis.…”

Section: Introductionmentioning

confidence: 99%

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Metric Regularity of Mappings and Generalized Normals to Set Images

Mordukhovich

Nam²,

Wang

2009

Set-Valued Anal

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Abstract. The primary goal of this paper is to study some notions of normals to nonconvex sets in finite-dimensional and infinite-dimensional spaces and their images under single-valued and set.-valued mappings. The main motivation for our study comes from variational analysis and optimization. ·where the problems under consideration play a crucial role in many important aspects of generalized differential calculus and applications. Our major results provide precise equality formulas (sometimes just efiicient upper estimates) allowing us to compute generalized normals in various senses to direct and inverse images of nonconvex sets under single-valued and set-valued mappings between Banach spaces. The main tools of our analysis revolve around variational principles and the fundamental concept of metric regularity properly modified in this paper.

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Section: • F(:c Y)(y')mentioning

confidence: 99%

Section: • F(:c Y)(y')mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Metric Regularity of Mappings and Generalized Normals to Set Images

Mordukhovich

Nam²,

Wang

2009

Set-Valued Anal

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“…For a thorough study of covering theorems and their applications, we refer the reader to Dmitruk et al (1980). Below, we prove a covering theorem (with a quantitive estimate, see (4.2)!)…”

Section: Theorem 12 Under the Assumptions Of Theorem 11 It Holds mentioning

confidence: 99%

“…Covering can be used for deriving optimality conditions via the following considerations (in Dmitruk et al 1980, this approach was called simultaneous covering). For problem (1.3), (1.4), consider the mapping…”

Section: Theorem 12 Under the Assumptions Of Theorem 11 It Holds mentioning

confidence: 99%

The Theory of 2-Regularity for Mappings with Lipschitzian Derivatives and its Applications to Optimality Conditions

Izmailov

Solodov

2002

Mathematics of OR

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We study local structure of a nonlinear mapping near points where standard regularity and/or smoothness assumptions need not be satisfied. We introduce a new concept of 2-regularity (a certain kind of second-order regularity) for a once differentiable mapping whose derivative is Lipschitz continuous. Under this 2-regularity condition, we obtain the representation theorem and the covering theorem (i.e., stability with respect to "right-hand side" perturbations) under assumptions that are weaker than those previously employed in the literature for results of this type. These results are further used to derive a constructive description of the tangent cone to a set defined by (2-regular) equality constraints and optimality conditions for related optimization problems. The class of mappings introduced and studied in the paper appears to be a convenient tool for treating complementarity structures by means of an appropriate equation-based reformulation. Optimality conditions for mathematical programs with (equivalently reformulated) complementarity constraints are also discussed.1. Introduction. In this paper, we consider the problem of local approximation of a nonlinear mapping near a given point in the absence of standard combinations of smoothness and regularity assumptions. We further study consequences of this approximation, in particular those relevant for constructive description of tangent directions to level surfaces, and optimality conditions for problems with feasible regions defined by irregular mappings.Let X and Y be Banach spaces, and consider a pointx ∈ X, its neighborhood V in X, and a mapping F V → Y . In what follows, X Y stands for the space of continuous linear operators from X to Y . For a linear operator ∈ X Y , ker = x ∈ X x = 0 is its null space, im = y ∈ Y y = x for some x ∈ X is its image space, and * Y * → X * is the linear operator adjoint to , where X * and Y * denote the dual spaces of X and Y , respectively.It is well known that if F is (first-order) regular atx, i.e.,

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Subtraction Theorems and Approximate Openness for Multifunctions: Topological and Infinitesimal Viewpoints

Azé

Chou

Penot

1998

Journal of Mathematical Analysis and Applications

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An approximate open mapping principle for multifunctions between metric spaces with closed graphs is given. It is shown that infinitesimal assumptions involving tangent cones and approximate tangent cones ensure that openness results can be derived from the general principle. Several examples and counterexamples enlight the validity and the interplay of the results. Applications are given to the lower semicontinuity of the intersection of two multifunctions. ᮊ 1998 Academic Press 33

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Lyusternik's Theorem and the Theory of Extrema

Cited by 157 publications

References 10 publications

Metric Regularity of Mappings and Generalized Normals to Set Images

Metric Regularity of Mappings and Generalized Normals to Set Images

The Theory of 2-Regularity for Mappings with Lipschitzian Derivatives and its Applications to Optimality Conditions

Subtraction Theorems and Approximate Openness for Multifunctions: Topological and Infinitesimal Viewpoints

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