Error bounds for systems of lower semicontinuous functions in Asplund spaces

Ngai, Huynh Van; Théra, Michel

doi:10.1007/s10107-007-0121-9

Cited by 67 publications

(9 citation statements)

References 46 publications

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“…As we detail in Section 2, the expression for H(A) in Proposition 1 can readily be seen to be at least as sharp as some bounds on H(A) derived by Güler et al [16] and Burke and Tseng [8]. We also note that weaker versions of Theorem 1 can be obtained from results on error bounds in Asplund spaces as those developed in the article by Van Ngai and Théra [42]. Our goal to devise algorithms to compute Hoffman constants is in the spirit of and draws on the work by Freund and Vera [11,12] to compute the distance to ill-posedness of a system of linear constraints.…”

Section: Introductionmentioning

confidence: 55%

New characterizations of Hoffman constants for systems of linear constraints

2020

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We give a characterization of the Hoffman constant of a system of linear constraints in R n relative to a reference polyhedron R ⊆ R n . The reference polyhedron R represents constraints that are easy to satisfy such as box constraints. In the special case R = R n , we obtain a novel characterization of the classical Hoffman constant.More precisely, suppose R ⊆ R n is a reference polyhedron, A ∈ R m×n , and A(R) :Our characterization is stated in terms of the largest of a canonical collection of easily computable Hoffman constants. Our characterization in turn suggests new algorithmic procedures to compute Hoffman constants.Furthermore, observe that if J ∈ S(A) andṽ ∈ ext{v ∈ R J + , A T J v * ≤ 1} then A J ′ must have full row rank for J ′ := {i :ṽ i > 0} ⊆ J. Therefore Proposition 2 also implies thatwhich is precisely the characterization (4) of H(A). In addition, Proposition 2 implies the following bound on H(A) in terms of the χ(A) condition measure [41,44,46] established in [50] for the special case when A ∈ R m×n is full column rank and both R m and R n are endowed with Euclidean norms:|J |=n A J non-singular max{ v : v ∈ R J , A T J v ≤ 1} = max J ⊆[m],|J |=n A J non-singular A −1 J = χ(A). The last step follows from [50, Prop. 3.7].

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Section: Introductionmentioning

confidence: 55%

New characterizations of Hoffman constants for systems of linear constraints

2020

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“…As we mentioned before, the converse of the latter corollary does not always hold. The results of Corollary 8, Corollary 11 have been appeared in numerous works, for instance, see [30,42,62,68,63,64]. This result gives an useful tools for establishing the quantitative error bounds, see [47,46].…”

Section: This Is the Content Of [60 Theorem 25] A Local Version Of Th...mentioning

confidence: 86%

A stroll in the jungle of error bounds

Nguyen¹

2017

Preprint

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The aim of this paper is to give a short overview on error bounds and to provide the first bricks of a unified theory. Inspired by the works of [8,15,13,16, 10], we show indeed the centrality of the Lojasiewicz gradient inequality. For this, we review some necessary and sufficient conditions for global/local error bounds, both in the convex and nonconvex case. We also recall some results on quantitative error bounds which play a major role in convergence rate analysis and complexity theory of many optimization methods.

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“…For a proper lower semicontinuous function : → R, we denote by dom( ) and epi( ) the domain and the epigraph of , respectively; that is, dom ( ) := { ∈ : ( ) < +∞} , epi ( ) := {( , ) ∈ × : ( ) ≤ } . (6) Throughout the paper, the symbol → always denotes the convergence relative to the distance (⋅, ⋅) induced by the norm while the arrow * → signifies the weak * convergence in the dual space * .…”

Section: Preliminariesmentioning

confidence: 99%

“…In the variational analysis literature, sensitivity analysis of mathematical programming and convergence analysis of some algorithms of optimization problems are deeply tied to the notion of error bound. Since Hoffman's pioneering work [1], the study of error bounds has received extensive attention in the mathematical programming literature (for details, see [2][3][4][5][6][7][8][9] and references therein). For us, the work of Ioffe in this area will be particularly important.…”

Section: Introductionmentioning

confidence: 99%