## Metric regularity – a survey. I: Theory.(English)Zbl 1369.49001

The paper surveys recent developments in the metric regularity theory. This is one of the central chapters of variational analysis which is concerned with extensions of results related to regularity properties of linear and smooth nonlinear operators in Banach spaces to broader classes of mappings (from non-differentiable Lipschitz mappings in Banach spaces and up to set-valued mappings in metric spaces) that typically appear in some modern applications, in particular in optimization and optimal control. A number of remarkable and important results were obtained in this direction along with numerous applications to analysis and optimization theory.
The first part of the survey (theory) contains an introduction and six sections. In the introduction, the author briefly describes key ideas and features of the theory as well as the structure of the survey. The first section Classical theory: five great theorems gives a short account of the classical results behind the metric regularity theory. These are the Banach-Schauder open mapping theorem, theorems of Lyusternik and Graves on regular points, inverse and implicit function theorems and the Sard theorem.
Section 2 Metric theory. Definitions and equivalences contains the definitions of local and nonlocal versions of the basic regularity properties: linear openness (or covering), metric regularity and a certain Lipschitz-type property of the inverse map for set-valued mappings in the category of metric spaces. Here are brief explanations of the local versions of the first two concepts . Let $$F$$ be a set-valued mapping $$F$$ from $$X$$. Linear openness near a point $$(x_0,y_0)\in\text{Graph} F$$ means that there are $$r>0$$ and a neighborhood $$V$$ of $$(x_0,y_0)$$ such that for any $$(x,y)\in V$$ the inequality $$d(y,F(x))<rt$$ implies that $$y\in F(u)$$ for some $$u$$ satisfying $$d(u,x)\leq t$$. $$F$$ is metrically regular near $$(x_0,y_0)$$ if, for some $$k$$, the inequality $$d(x,F^{-1}(y)) \leq kd(y,F(x))$$ holds for all $$(x,y)$$ in a neighborhood of $$(x_0,y_0)$$. The key fact is that the properties are equivalent and moreover, the upper bound of $$r$$ in the definition of linear openness, called the rate of surjection of $$F$$ near $$(x_0,y_0)$$, and the lower bound of $$k$$ in the definition of metric regularity, called the rate of metric regularity, are reciprocal numbers. This result determines the quantitative character of the theory emphasized throughout the article.
The third section Metric theory. Regularity criteria contains a series of formulas for the regularity rates (regularity criteria) either in terms of some distance functions associated with the mapping or in terms of slopes of the functions. Slopes, introduced by Marino, Tosques and DeGiorgi in 1980, are probably the main infinitesimal instrument in modern metric analysis. The section also contains discussions of weaker versions of the regularity properties such as subregularity, calmness, controllability and linear recession.
In Section 4 Metric theory. Perturbation and stability, the reader will find a number of fundamental results relating to dependence of mappings on parameters. The first group of the results contains estimates for regularity rates of a mapping subject to Lipschitz perturbations of various sort. The results of the second group are extensions of the implicit function theorem to (set-valued) mappings in metric spaces and contains estimates for the Lipschitz-type behavior of the solution mapping. Special attention is given to the property of strong regularity that guarantees single-valuedness of the solution mapping.
Section 5 Banach space theory is concerned with regularity properties in the main special case of mappings in Banach spaces. In this case, a number of approximation mechanisms are available, including various kinds of subderivatives and subdifferentials, tangent and normal cones, bunches of linear operators etc.. This leads to a rich variety of additional characterizations of regularity and stability results for regularity rates.
The last Section 6 Finite dimensional theory deals with mappings between finite dimensional spaces. Along with results of general nature, which look especially elegant and precise in this case, special attention is paid to characterization of the subregularity property and to extensions of the classical transversality concept to arbitrary sets and set-valued mappings.
This part of the survey is completed by the references (98 items).
The survey may be interesting and useful to all specialists (as well as graduate and postgraduate students) in classical and functional analysis and optimization theory. This part gives a fairly complete picture of many aspects of metric regularity theory.

### MSC:

 49-02 Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control 58C15 Implicit function theorems; global Newton methods on manifolds 49N60 Regularity of solutions in optimal control 49J53 Set-valued and variational analysis 90C26 Nonconvex programming, global optimization 47H04 Set-valued operators 47J07 Abstract inverse mapping and implicit function theorems involving nonlinear operators 54C60 Set-valued maps in general topology 58K05 Critical points of functions and mappings on manifolds
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