Theorems on estimates in the neighborhood of a singular point of a mapping

Avakov, E. R.

doi:10.1007/bf01158081

Cited by 40 publications

(13 citation statements)

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“…The following definitions are direct extensions of those previously introduced in the twice differentiable case, see [3][4][5][6]18,1,16,19]. If F is twice Fréchet-differentiable atx, our definitions reduce to standard notions of 2-regularity theory.…”

Section: -Regularity Error Bounds For 2-regular Mappingsmentioning

confidence: 90%

Error bounds for 2-regular mappings with Lipschitzian derivatives and their applications

Izmailov¹,

Solodov²

2001

Math. Program.

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We obtain local estimates of the distance to a set defined by equality constraints under assumptions which are weaker than those previously used in the literature. Specifically, we assume that the constraints mapping has a Lipschitzian derivative, and satisfies a certain 2-regularity condition at the point under consideration. This setting directly subsumes the classical regular case and the twice differentiable 2-regular case, for which error bounds are known, but it is significantly richer than either of these two cases. When applied to a certain equation-based reformulation of the nonlinear complementarity problem, our results yield an error bound under an assumption more general than b-regularity. The latter appears to be the weakest assumption under which a local error bound for complementarity problems was previously available. We also discuss an application of our results to the convergence rate analysis of the exterior penalty method for solving irregular problems.Key words. error bound -C 1,1 -mapping -2-regularity -nonlinear complementarity problem -exterior penalty -rate of convergence Error bounds and their applicationsAmong the most important tools for theoretical and numerical treatment of nonlinear operator equations, optimization problems, variational inequalities, and other related problems, are the so-called error bounds, i.e., upper estimates of the distance to a given set in terms of some residual function. We refer the reader to [31] for a survey of error bounds and their applications. When the set is defined by functional constraints, a typical residual function is some measure of violation of constraints at the given point. When available, error bounds can often be used to obtain a constructive local description of the set under consideration. This description, in turn, plays a central role in the theory of optimality conditions. As another important application of error bounds, we mention development, implementation, and convergence rate analysis of numerical methods for solving optimization and related problems.In this paper, we consider error bounds for sets given by equality constraints. However, it is worth to point out that this setting implicitly includes sets of more general structure, namely those whose constraints can be (equivalently) reformulated into equations. One such example is solution set of the nonlinear complementarity problem, which we shall study in Sect. 3.

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Section: -Regularity Error Bounds For 2-regular Mappingsmentioning

confidence: 90%

Error bounds for 2-regular mappings with Lipschitzian derivatives and their applications

Izmailov¹,

Solodov²

2001

Math. Program.

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“…However, the inclusion T S (x) ⊃ ker F (x) is not necessarily true when (5) is violated. The first tangent cone results valid without the normality condition were obtained in [21] (under the assumption that F (x) = 0) and [7] (in the general setting; see also [3,8,15,16]). Let P be the orthogonal projector onto (im F (x)) ⊥ in R l (note that (5) is violated if and only if im F (x) is a proper subspace in R l ).…”

Section: Introductionmentioning

confidence: 99%

Tangent vectors to a zero set at abnormal points

Arutyunov

Izmailov

2004

Journal of Mathematical Analysis and Applications

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“…Such a technique, relying on the so-called 2-regularity concept, was developed in [6][7][8]20] (see also [2,[13][14][15]). In many cases it gives an opportunity to characterize the zero set near a singular point in terms of tangency, or even "up to diffeomorphism."…”

Section: Introductionmentioning

confidence: 99%

Bifurcation Theorems via Second-Order Optimality Conditions

Arutyunov¹,

Izmailov²

2001

Journal of Mathematical Analysis and Applications

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We present a new approach to bifurcation study that relies on the theory of second-order optimality conditions for abnormal constrained optimization problems developed earlier by the first author. This theory does not subsume the "primal" description of the feasible set in terms of tangent vectors or in any other way. As a result, we obtain new sufficient conditions for bifurcation, which are to some extent complementary with respect to the known bifurcation theory. 

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Theorems on estimates in the neighborhood of a singular point of a mapping

Cited by 40 publications

References 1 publication

Error bounds for 2-regular mappings with Lipschitzian derivatives and their applications

Error bounds for 2-regular mappings with Lipschitzian derivatives and their applications

Tangent vectors to a zero set at abnormal points

Bifurcation Theorems via Second-Order Optimality Conditions

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