On Lipschitz Continuity of Projections onto 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- 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.

Dynamical System Related to Primal–Dual Splitting Projection Methods

“…Then, by Proposition 1, x 1 (t) − w ≥ z − w and x 2 (t) − w ≥ z − w for t ≥ t. However, since z ∈ D and, by (8),…”

“…Assumption (A ) follows from (28) i.e., Fix T z ∈ H (x, Tx) for all x ∈ D and x ∈ Fix T ∩ D ⇐⇒ x = z. Assumption B follows from fact that for any x ∈ D ⊂ D, z ∈ C(x), hence z ∈ H ( w, P C(x) ( w)) and therefore P C(x) ( w) ∈ B( w+z 2 , w−z 2 ) = D (see (8)). Assumption (C ) follows from fact that for any…”