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On the strong metric subregularity in mathematical programming

Osmolovskii Nikolai P., Veliov Vladimir M.

Control and Cybernetics

2021

Vol. 50, No. 4

457--471

This note presents sufficient conditions for the property of strong metric subregularity (SMSr) of the system of first order optimality conditions for a mathematical programming problem in a Banach space (the Karush-Kuhn-Tucker conditions). The constraints of the problem consist of equations in a Banach space setting and a finite number of inequalities. The conditions, under which SMSr is proven, assume that the data are twice continuously Fréchet differentiable, the strict Mangasarian-Fromovitz constraint qualification is satisfied, and the second-order sufficient optimality condition holds. The obtained result extends the one known for finite-dimensional problems. Although the applicability of the result is limited to the Banach space setting (due to the twice Fréchet differentiability assumptions and the finite number of inequality constraints), the paper can be valuable due to the self-contained exposition, and provides a ground for extensions. One possible extension was recently implemented in Osmolovskii and Veliov (2021).

Metric regularity under approximations

Dontchev A. L., Veliov V. M.

Control and Cybernetics

2009

Vol. 38, no 4B

1283-1303

In this paper we show that metric regularity and strong metric regularity of a set-valued mapping imply convergence of inexact iterative methods for solving a generalized equation associated with this mapping. To accomplish this, we first focus on the question how these properties are preserved under changes of the mapping and the reference point. As an application, we consider discrete approximations in optimal control.

On regularity estimates for mappings between embedded manifolds

Ioffe A.

Control and Cybernetics

2007

Vol. 36, no 3

659-668

We describe a class of metric spaces such that for set-valued mappings into such spaces it is possible to give a precise expression of regularity moduli in terms of slopes of DeGiorgi-Marino-Tosques. We also show that smooth manifolds in Banach spaces endowed with the induced metric belong to this class.

On robustness of the regularity property of maps

Ioffe A. D.

Control and Cybernetics

2003

Vol. 32, no 3

543-554

The problem considered in the paper can be described as follows. We are given a continuous mapping from one metric space into another which is regular (in the sense of metric regularity or, equivalently, controllability at a linear rate) near a certain point. How small may be an additive perturbation of the mapping which destroys regularity? The paper contains a new proof of a recent theorem of Dontchev-Lewis-Rockafellar for linear perturbations of maps between finite-dimensional Banach spaces and an exact estimate for Lipschitz perturbations of maps between complete metric spaces.