On Lipschitz continuity of projections onto polyhedral moving sets

TL;DR

This paper proves the local Lipschitz continuity of projections onto parametric polyhedral sets in Hilbert spaces, under certain regularity conditions, which is important for stability analysis in optimization.

Contribution

It establishes the local Lipschitzness of projections onto parametric polyhedral sets with data depending Lipschitz continuously on parameters, under relaxed constraint qualifications.

Findings

Projections are locally Lipschitz continuous in Hilbert spaces.

Lipschitzness holds under relaxed constant rank constraint qualification.

Results apply to parametric systems with data as Lipschitz functions.

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.