Error Bounds for Approximate Solutions of Abstract Inequality Systems and Infinite Systems of Inequalities on Banach Spa
- PDF / 478,355 Bytes
- 21 Pages / 439.642 x 666.49 pts Page_size
- 13 Downloads / 187 Views
Error Bounds for Approximate Solutions of Abstract Inequality Systems and Infinite Systems of Inequalities on Banach Spaces Jinhua Wang1 · Mingwu Ye2 · Sy-Ming Guu3,4 · Chong Li5 Received: 14 February 2019 / Accepted: 14 August 2020 / © Springer Nature B.V. 2020
Abstract Using the result of the error estimate of the simple extended Newton method established in the present paper for solving abstract inequality systems, we study the error bound property of approximate solutions of abstract inequality systems on Banach spaces with the involved function F being Fr´echet differentiable and its derivative F satisfying the centerLipschitz condition (not necessarily the Lipschitz condition) around a point x0 . Under some mild conditions, we establish results on the existence of the solutions, and the error bound properties for approximate solutions of abstract inequality systems. Applications of these results to finite/infinite systems of inequalities/equalities on Banach spaces are presented and the error bound properties of approximate solutions of finite/infinite systems of inequalities/equalities are also established. Our results extend the corresponding results in [3, 18, 19]. Keywords Abstract inequality systems · Infinite systems of inequalities · Error bound · weak-Robinson condition Mathematics Subject Classification (2010) 47J99 · 90C48 · 90C31
1 Introduction Let F : ⊆ X → Y be a continuously Fr´echet differentiable function with its Fr´echet derivative denoted by F , where X and Y are Banach spaces, and ⊆ Y is an open subset.
Jinhua Wang’s work was supported in part by the National Natural Science Foundation of China (grant 11771397). Mingwu Ye’s work was supported in part by Program Foundation for Talents of Guizhou University (grant 2017(60)). Sy-Ming Guu’s work was supported in part by MOST 106-2221-E-182-038-MY2 and BMRPD17. Chong Li’s work was supported in part by the National Natural Science Foundation of China (grant 11971429) and Zhejiang Provincial Natural Science Foundation of China (grant LY18A010004). Chong Li
[email protected]
Extended author information available on the last page of the article.
J. Wang et al.
Let W be a nonempty subset of Y . Recall that the partial order (or preorder) ≥W associated with W is defined by y1 ≥W y2 ⇐⇒ y1 − y2 ∈ W
for any y1 , y2 ∈ Y .
Let K ⊂ Y be a nonempty convex subset satisfying that λz ∈ K
for any z ∈ K and λ > 0
(1.1)
(we also call a convex subset K satisfying (1.1) a “convex cone”, and note that the “convex cone” K is not necessary closed and 0 ∈ / K is allowed), and let C be the closure of K, that is, C = K. We consider the following two abstract conic inequality systems: F (x) ≥C 0
(1.2)
F (x) ≥K 0.
(1.3)
and The solution sets of (1.2) and (1.3) are denoted by Scl and S, respectively, that is Scl := {x ∈ X| F (x) ∈ C}
and S := {x ∈ X| F (x) ∈ K}.
We are especially interested in the following classical finite/infinite systems of inequalities and inequality/strict-inequalities, which are respectively two special cases of (1.3): fi (x) ≥
Data Loading...
