Perturbation of error bounds

Abstract

Our aim in the current article is to extend the developments in Kruger, Ngai & Théra, SIAM J. Optim. 20(6), 3280-3296 (2010) and, more precisely, to characterize, in the Banach space setting, the stability of the local and global error bound property of inequalities determined by lower semicontinuous functions under data perturbations. We propose new concepts of (arbitrary, convex and linear) perturbations of the given function defining the system under consideration, which turn out to be a useful tool in our analysis. The characterizations of error bounds for families of perturbations can be interpreted as estimates of the 'radius of error bounds'. The definitions and characterizations are illustrated by examples.

Characterizations of Stability of Error Bounds for Convex Inequality Constraint Systems

“…(a). From [33,42], the condition (11) means that g is an ε-perturbation of f near x, and the condition inf h =1 f (x, h) = 0 is proved to be equivalent to the stability of this ε-perturbation of local error bounds. Further, it has been shown in Theorem 5 that the stability of such ε-perturbation is essentially equivalent to that of ε-linear perturbation.…”

“…We denote by τ min (F ) := inf{τ > 0 : (33) holds} the global error bound modulus of S F . For x ∈ bdry(S F ), system (30) is said to have a local error bound at x if there exist constants τ, δ ∈ (0, +∞) such that…”

“…In 2012, by relaxing the convexity assumption, Zheng and Wei [56] discussed the stability of error bounds for quasi-subsmooth (not necessarily convex) inequalities in a general Banach space and provided Clarke subdifferential characterizations of the stability of error bounds. In 2018, Kruger, López and Théra [33] extended the development in [35,42] and characterized the stability of error bounds for convex inequalities in the Banach space setting. From the viewpoint of infinite dimensional Banach spaces, results on the stability of error bounds in [33,35,42,56] are dual conditions, and it is a pretty natural idea to study this issue not involving the dual space since information on the dual space may be missing.…”

“…In 2018, Kruger, López and Théra [33] extended the development in [35,42] and characterized the stability of error bounds for convex inequalities in the Banach space setting. From the viewpoint of infinite dimensional Banach spaces, results on the stability of error bounds in [33,35,42,56] are dual conditions, and it is a pretty natural idea to study this issue not involving the dual space since information on the dual space may be missing. Inspired by this observation, we study characterizations of stability of local and global error bounds of a single convex inequality and semi-infinite convex constraint systems via directional derivatives.…”

Characterizations of Stability of Error Bounds for Convex Inequality Constraint Systems

“…In 2012, by relaxing the convexity assumption, Zheng and Wei [57] discussed the stability of error bounds for quasi-subsmooth (not necessarily convex) inequalities in a general Banach space and provided Clarke subdifferential characterizations of the stability of error bounds. In 2018, Kruger, López and Théra [35] extended the development in [33,42] and characterized the stability of error bounds for convex inequalities in the Banach space setting. From the viewpoint of infinite dimensional Banach spaces, results on the stability of error bounds in [33,35,42,57] are dual conditions, and it is a pretty natural idea to study this issue not involving the dual space since information on the dual space may be missing.…”