On Lipschitz-Like Property for Polyhedral Moving Sets

Abstract

In Hilbert space setting we prove local lipchitzness of projections onto parametric polyhedral sets represented as solutions to systems of inequalities and equations with parameters appearing both in left-hand-sides and right-hand-sides of the constraints. In deriving main results we assume that data are locally Lipschitz functions of parameter and the relaxed constant rank constraint qualification condition is satisfied.2010 Mathematics Subject Classification. 47N10,49J52,49J53,49K40,90C31.

On Lipschitz-like continuity of a class of set-valued mappings

“…We generalize results from [25] and [3]. In [25] the respective results are obtained under stronger assumptions of functions h i , i ∈ I 0 ∪ I, while in [3] the Lipschitz-likeness of F is obtained for h i (p, x) = x | g i (p) − f i (p), i ∈ I 0 ∪ I, where f i : G → R, g i : G → H, i ∈ I 0 ∪ I, are locally Lipschitz functions. We also correct the mistake in the proof of Lemma 3 of [25].…”

“…Let us note that for some particular functions h i (·, ·), i ∈ I ∪ I 0 Theorem 5.3 has been already proved in [3].…”

“…In [3] we discussed in details the proof of Lemma 3 of [25] for some special functions h i , i = 1, . .…”

On Lipschitz Continuity of Projections onto Polyhedral Moving Sets

“…Essential part of our considerations is based on the relaxed constant rank constraint qualification (RCRCQ) introduced in [19] and investigated in [3,17,20]. According to our knowledge, no result is known in the literature, in which this particular constraint qualification condition is used in the context of stability of solutions to parametric problems (Par) with I 1 = ∅.…”

“…We take this fact into account by introducing the concept of equivalent representation (Definition 4) and the concept of equivalent stable representation (Definition 5). In Theorem 5 we show that the under assumption (H1) the existence of a suitable equivalent representation is necessary for the continuity of projections onto sets C( p), p ∈ D, given by (3).…”