On uniform regularity and strong regularity. (4th March 2019)

• Record Type: Journal Article
• Authors: Cibulka, R.; Preininger, J.; Roubal, T.
• Is Part Of: Optimization. Volume 68:Number 2/3(2019)
• Page Start: 549
• Page End: 577
• Publication Date: 2019-03-04
• Subjects: Control system -- uniform metric regularity -- uniform strong metric regularity -- discrete approximation -- path-following; 49k40 -- 49J40 -- 49J53 -- 90c31; Mathematical optimization -- Periodicals; 519.7
• DOI: 10.1080/02331934.2018.1547383
• Languages: English
• ISSNs: 0233-1934

Abstract

ABSTRACT: We investigate uniform versions of (metric) regularity and strong (metric) regularity on compact subsets of Banach spaces, in particular, along continuous paths. These two properties turn out to play a key role in analyzing path-following schemes for tracking a solution trajectory of a parametric generalized equation or, more generally, of a differential generalized equation (DGE). The latter model allows us to describe in a unified way several problems in control and optimization such as differential variational inequalities and control systems with state constraints. We study two inexact path-following methods for DGEs having the order of the grid errorO ( h ) andO ( h 2 ), respectively. We provide numerical experiments, comparing the schemes derived, for simple problems arising in physics. Finally, we study metric regularity of mappings associated with a particular case of the DGE arising in control theory. We establish the relationship between the pointwise version of this property and its counterpart in function spaces.