METRIC REGULARITY—A SURVEY PART II. APPLICATIONS

Metric regularity theory lies in the very heart of variational analysis, a relatively new discipline whose appearance was, to a large extent, determined by the needs of modern optimization theory in which such phenomena as nondifferentiability and set-valued mappings naturally appear. The roots of the theory go back to such fundamental results of the classical analysis as the implicit function theorem, Sard theorem and some others. The paper offers a survey of the state of the art of some principal parts of the theory along with a variety of its applications in analysis and optimization.

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METRIC REGULARITY—A SURVEY PART 1. THEORY

Journal of the Australian Mathematical Society ◽

10.1017/s1446788715000701 ◽

2016 ◽

Vol 101 (2) ◽

pp. 188-243 ◽

Cited By ~ 23

Author(s):

A. D. IOFFE

Keyword(s):

Variational Analysis ◽

State Of The Art ◽

Metric Regularity ◽

Optimization Theory ◽

Regularity Theory ◽

Implicit Function ◽

Classical Analysis ◽

Set Valued Mappings ◽

Principal Parts ◽

Modern Optimization

Metric regularity theory lies at the very heart of variational analysis, a relatively new discipline whose appearance was, to a large extent, determined by the needs of modern optimization theory in which such phenomena as nondifferentiability and set-valued mappings naturally appear. The roots of the theory go back to such fundamental results of the classical analysis as the implicit function theorem, Sard theorem and some others. This paper offers a survey of the state of the art of some principal parts of the theory along with a variety of its applications in analysis and optimization.

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Elements of Optimization Theory and Variational Analysis

Newton-Type Methods for Optimization and Variational Problems - Springer Series in Operations Research and Financial Engineering ◽

10.1007/978-3-319-04247-3_1 ◽

2014 ◽

pp. 1-60 ◽

Cited By ~ 1

Author(s):

Alexey F. Izmailov ◽

Mikhail V. Solodov

Keyword(s):

Variational Analysis ◽

Optimization Theory

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Optimal metric regularity in general relativity follows from the RT-equations by elliptic regularity theory in $L^p$ -spaces

Methods and Applications of Analysis ◽

10.4310/maa.2020.v27.n3.a1 ◽

2020 ◽

Vol 27 (3) ◽

pp. 199-242

Author(s):

Moritz Reintjes ◽

Blake Temple

Keyword(s):

General Relativity ◽

Metric Regularity ◽

Regularity Theory ◽

Elliptic Regularity ◽

Elliptic Regularity Theory

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Problems of the Modern Optimization Theory

Automation and Remote Control ◽

10.1023/b:aurc.0000014711.84584.32 ◽

2004 ◽

Vol 65 (2) ◽

pp. 157

Author(s):

I. I. Eremin ◽

A. I. Kibzun

Keyword(s):

Optimization Theory ◽

Modern Optimization

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Towards variational analysis in metric spaces: metric regularity and fixed points

Mathematical Programming ◽

10.1007/s10107-009-0316-3 ◽

2009 ◽

Vol 123 (1) ◽

pp. 241-252 ◽

Cited By ~ 26

Author(s):

A. D. Ioffe

Keyword(s):

Fixed Points ◽

Variational Analysis ◽

Metric Spaces ◽

Metric Regularity

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Metric regularity, fixed points and some associated problems of variational analysis

Journal of Fixed Point Theory and Applications ◽

10.1007/s11784-014-0186-z ◽

2014 ◽

Vol 15 (1) ◽

pp. 67-99 ◽

Cited By ~ 8

Author(s):

Alexander D. Ioffe

Keyword(s):

Fixed Points ◽

Variational Analysis ◽

Metric Regularity

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Modern Optimization and Simulation Methods in Managerial and Business Economics: A Review

Administrative Sciences ◽

10.3390/admsci10030047 ◽

2020 ◽

Vol 10 (3) ◽

pp. 47

Author(s):

Laura Calvet ◽

Rocio de la Torre ◽

Anita Goyal ◽

Mage Marmol ◽

Angel Juan

Keyword(s):

State Of The Art ◽

Global Market ◽

Imperfect Markets ◽

Simulation Methods ◽

Business Economics ◽

Scarce Resources ◽

Optimization Simulation ◽

Optimization And Simulation ◽

Fierce Competition ◽

Modern Optimization

Managerial and Business Economics (ME/BE) aims at using quantitative and computational methods to make an efficient (ideally optimal) assignment of the scarce resources owned by firms and organizations. In the current global market, characterized by a fierce competition, an optimal use of the available resources is more important than ever for guaranteeing the economical sustainability of organizations and enterprises of any size. Heuristic optimization algorithms and simulation methods have been successfully employed to analyze and enhance complex systems and processes in a myriad of ME/BE-related fields. This paper reviews recent works on the use of these methodologies in competitive markets, as well as in imperfect markets considering externalities. The paper also discusses open challenges and how state-of-the art methods combining optimization, simulation, and machine learning can contribute to properly address them.

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On tangential approximations of the solution set of set-valued inclusions

Journal of Applied Analysis ◽

10.1515/jaa-2021-2049 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Amos Uderzo

Keyword(s):

Crucial Role ◽

Variational Analysis ◽

Necessary Optimality Conditions ◽

Solution Set ◽

Inclusion Problem ◽

Feasible Region ◽

Contingent Cone ◽

Constrained Problems ◽

Outer Approximations ◽

Set Valued Mappings

Abstract In the present paper, the problem of estimating the contingent cone to the solution set associated with certain set-valued inclusions is addressed by variational analysis methods and tools. As a main result, inner (resp. outer) approximations, which are expressed in terms of outer (resp. inner) prederivatives of the set-valued term appearing in the inclusion problem, are provided. For the analysis of inner approximations, the evidence arises that the metric increase property for set-valued mappings turns out to play a crucial role. Some of the results obtained in this context are then exploited for formulating necessary optimality conditions for constrained problems, whose feasible region is defined by a set-valued inclusion.

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