Error Bounds of Generalized D-Gap Functions for Nonsmooth and Nonmonotone Variational Inequality Problems

Li, Guoyin; Ng, Kung Fu

doi:10.1137/070696283

Cited by 54 publications

(24 citation statements)

References 31 publications

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“…Further studies are needed to deal with this problem. Second, recently, Hölderian error bound properties, as well as Hölderian metric subregularity/regularity, has attracted the study of a large number of authors (see Frankowska and Quincampoix [22], Ioffe [28], Li and Ng [38], Wu and Ye [57], and the references given therein). Note that if we replace the function in relation (13) of Theorem 1 by with some given > 0, we obtain a sufficient condition for metric subregularity with order of the multifunction F .…”

Section: Propositionmentioning

confidence: 99%

Metric Subregularity of Multifunctions: First and Second Order Infinitesimal Characterizations

Ngai¹,

Tĩnh²

2015

Mathematics of OR

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Metric subregularity and regularity of multifunctions are fundamental notions in variational analysis and optimization. Using the concept of strong slope, in this paper we first establish a criterion for metric subregularity of multifunctions between metric spaces. Next, we use a combination of abstract coderivatives and contingent derivatives to derive verifiable first order conditions ensuring metric subregularity of multifunctions between Banach spaces. Then using second order approximations of convex multifunctions, we establish a second order condition for metric subregularity of mixed smooth-convex constraint systems, which generalizes a result established recently by Gfrerer [Gfrerer H (2011) First order and second order characterizations of metric subregularity and calmness of constraint set mapping. SIAM J. Optim. 21(4):1439-1474].

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Section: Propositionmentioning

confidence: 99%

Metric Subregularity of Multifunctions: First and Second Order Infinitesimal Characterizations

Ngai¹,

Tĩnh²

2015

Mathematics of OR

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“…Note the only attempt so far to consider the case q > 1 in [51]. Nevertheless, the Hölder or more general nonlinear estimates of metric subregularity/error bounds have proved to be important in sensitivity analysis and quantifying linear/sublinear convergence rates for the proximal point and alternating projection algorithms in optimization and variational inequalities [2,3,9,16,45,47,48,58].…”

Section: Introductionmentioning

confidence: 99%

Error Bounds and Hölder Metric Subregularity

Kruger

2015

Set-Valued Var. Anal

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The Hölder setting of the metric subregularity property of set-valued mappings between general metric or Banach/Asplund spaces is investigated in the framework of the theory of error bounds for extended real-valued functions of two variables. A classification scheme for the general Hölder metric subregularity criteria is presented. The criteria are formulated in terms of several kinds of primal and subdifferential slopes.

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“…Following [1,2,18,21,22,24], a set C ⊆ R n is said to be (i) semianalytic, if for any x ∈ R n , there exists a neighbourhood U of x such that…”

Section: Semismoothness Of the Maximum Eigenvalue Functionmentioning

confidence: 99%

Semismoothness of the maximum eigenvalue function of a symmetric tensor and its application

2013

Linear Algebra and its Applications

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In this paper, we examine the maximum eigenvalue function of an even order real symmetric tensor. By using the variational analysis techniques, we first show that the maximum eigenvalue function is a continuous and convex function on the symmetric tensor space. In particular, we obtain the convex subdifferential formula for the maximum eigenvalue function. Next, for an mth-order n-dimensional symmetric tensor A, we show that the maximum eigenvalue function is always ρth-order semismooth at A for some rational number ρ > 0. In the special case when the geometric multiplicity is one, we show that ρ can be set as 1 (2m−1) n . Sufficient condition ensuring the strong semismoothness of the maximum eigenvalue function is also provided. As an application, we propose a generalized Newton method to solve the space tensor conic linear programming problem which arises in medical imaging area. Local convergence rate of this method is established by using the semismooth property of the maximum eigenvalue function.

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Error Bounds of Generalized D-Gap Functions for Nonsmooth and Nonmonotone Variational Inequality Problems

Cited by 54 publications

References 31 publications

Metric Subregularity of Multifunctions: First and Second Order Infinitesimal Characterizations

Metric Subregularity of Multifunctions: First and Second Order Infinitesimal Characterizations

Error Bounds and Hölder Metric Subregularity

Semismoothness of the maximum eigenvalue function of a symmetric tensor and its application

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