Abstract
We consider error bound issue for conic inequalities in Hilbert spaces. In terms of proximal subdifferentials of vector-valued functions, we provide sufficient conditions for the existence of a local error bound for a conic inequality. In the Hilbert space case, our result improves and extends some existing results on local error bounds.
Highlights
Let X be a Banach space and let f : X → R := R ∪ {+∞} be a proper lower semicontinuous function
Condition (28) will be much easier to be satisfied if∂KΦ(x) is replaced by ∂KPΦ(x). Taking this fact into account, Theorem 6 provides a sharper result with∂KΦ replaced by ∂KPΦ and gives a relationship between the modulus of error bound and corresponding radius which were not mentioned in Theorem I
“Weak sharp minima, well-behaving functions and global error bounds for convex inequalities in Banach spaces,” in Proceedings of the 12th Baikal International Conference on Optimization Methods and Their Applications, pp. 272– 284, Irkutsk, Russia, 2001
Summary
Let X be a Banach space and let f : X → R := R ∪ {+∞} be a proper lower semicontinuous function. The work of Ioffe in this area will be important The author in his seminal work [10] first characterized error bound (under a different name) in terms of the subdifferentials and gave the following interesting result: if f is locally Lipschitz at a ∈ Sf and there exist η, δ ∈ (0, +∞) such that η ≤ d (0, ∂f (x)) ∀x ∈ B (a, δ) \ Sf,. For such a kind of systems, we consider their error bounds. The main aim of this paper is to extend the abovementioned Ioffe’s classic result on error bounds to conic inequality (CIE) with X, Y being any pair of Hilbert spaces in terms of proximal subdifferentials of vector-valued functions defined by proximal normal cone with a kind of variational behavior of “order two.”. Zheng and Ng’s result in the Hilbert space setting
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