Constrained Lipschitzian Error Bounds and Noncritical Solutions of Constrained Equations
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Constrained Lipschitzian Error Bounds and Noncritical Solutions of Constrained Equations A. Fischer1
· A. F. Izmailov2
· M. Jelitte1
Received: 9 March 2020 / Accepted: 20 October 2020 / © The Author(s) 2020
Abstract For many years, local Lipschitzian error bounds for systems of equations have been successfully used for the design and analysis of Newton-type methods. There are characterizations of those error bounds by means of first-order derivatives like a recent result by Izmailov, Kurennoy, and Solodov on critical solutions of nonlinear equations. We aim at extending this result in two directions which shall enable, to some extent, to include additional constraints and to consider mappings with reduced smoothness requirements. This leads to new necessary as well as sufficient conditions for the existence of error bounds. Keywords Constrained equation · constrained error bound · critical and noncritical solutions · (piecewise) semidifferentiable functions Mathematics Subject Classification (2010) 49J52 · 49J53 · 90C33
The work was funded by the Volkswagen Foundation, by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 409756759, and by the Russian Foundation for Basic Research Grants 19-51-12003 NNIO a and 20-01-00106. Dedicated to Professor R. Tyrrell Rockafellar on the occasion of his 85th birthday. M. Jelitte
[email protected] A. Fischer [email protected] A. F. Izmailov [email protected] 1
Faculty of Mathematics, Technische Universit¨at Dresden, 01062 Dresden, Germany
2
VMK Faculty, OR Department, Lomonosov Moscow State University (MSU), Uchebniy Korpus 2, Leninskiye Gory, 119991 Moscow, Russia
A. Fischer et al.
1 Introduction The present article focuses on the characterization of constrained local Lipschitzian error bounds. More in detail, let a set ⊂ Rn and a function f : Rn → [0, ∞] be given such that U := f −1 (0) ∩
(1)
is the nonempty solution set of the constrained system f (u) = 0 s.t.
u ∈ .
We say that f provides a local -error bound at c, ε > 0 such that
u∗
∈ U for U , if there are constants
cdist [u, U ] ≤ f (u) for all u ∈ ∩ (u∗ + εB), where the distance of u ∈
Rn
to a nonempty set W ⊂
Rn
(2)
(3)
is given by
dist [u, W ] := inf{ u − w | w ∈ W }, while · stands for the Euclidean norm, and B := {u ∈ Rn | u ≤ 1} denotes the unit ball. Obviously, the inequality in (3) is restricted to some local neighborhood of u∗ intersected with the constraint set , so that this property is also called constrained local Lipschitzian error bound. For brevity, we will omit the term “Lipschitzian” throughout. In the special case when = Rn , the property in question will be simply called local error bound. The function f may come from different applications. For example, f (u) := F (u)
(4)
is often considered for some given mapping F : Rn → Rm . Another application can be found in constrained minimization problems f0 (u) → min
s.t. u ∈
by setting f (u) := f0 (u) − f∗ , where f∗ ∈ R denotes the minimal value of f on (if it exists), see
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