Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Constructions of Generalized Differentiation Read chapter Read first chapter

2018 | OriginalPaper | Chapter

2. Fundamental Principles of Variational Analysis

Author : Boris S. Mordukhovich

Published in: Variational Analysis and Applications

Publisher: Springer International Publishing

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

This chapter is devoted to the exposition and developments of the fundamental principles of variational analysis, which play a crucial role in resolving many issues of variational theory and applications by employing optimization ideas and techniques.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter Constructions of Generalized Differentiation
next chapter Well-Posedness and Coderivative Calculus
Literature
50.
T. Q. Bao (2013), On a nonconvex separation theorem and the approximate extremal principle in Asplund spaces, Acta Math. Vietnam. 38, 279–291.
71.
H. H. Bauschke, D. R. Luke, H. M. Phan and X. Wang (2013), Restricted normal cones and the method of alternating projections: theory, Set-Valued Var. Anal. 21, 431–473.
72.
H. H. Bauschke, D. R. Luke, H. M. Phan and X. Wang (2013), Restricted normal cones and the method of alternating projections: applications, Set-Valued Var. Anal. 21, 475–501.
73.
H. H. Bauschke, D. R. Luke, H. M. Phan and X. Wang (2014), Restricted normal cones and sparsity optimization with affine constraints, Found. Comput. Math. 14, 63–83.
102.
J. M. Borwein and A. Jofré (1988), Nonconvex separation property in Banach spaces, Math. Methods Oper. Res. 48, 169–179.
106.
J. M. Borwein, Y. Lucet and B. S. Mordukhovich (2000), Compactly epi-Lipschitzian convex sets and functions in normed spaces, J. Convex Anal. 7, 375–393.
107.
J. M. Borwein, B. S. Mordukhovich and Y. Shao (1999), On the equivalence of some basic principles of variational analysis, J. Math. Anal. Appl. 229, 228–257.
108.
J. M. Borwein and D. Preiss (1987), A smooth variational principle with applications to subdifferentiability and differentiability of convex functions, Trans. Amer. Math. Soc. 303, 517–527.
109.
J. M. Borwein and H. M. Strójwas (1985), Tangential approximations, Nonlinear Anal. 9, 1347–1366.
114.
J. M. Borwein and Q. J. Zhu (2005), Techniques of Variational Analysis, Springer, New York.MATH
187.
M. Cúth and M. Fabian (2016), Asplund spaces characterized by rich families and separable reduction of Fréchet subdifferentiability, J. Funct. Anal. 270, 1361–1378.
205.
R. Deville, G. Godefroy and V. Zizler (1993), Smoothness and Renorming in Banach Spaces, Wiley, New York.MATH
207.
J. Diestel (1984), Sequences and Series in Banach Spaces, Springer, New York.CrossRef
229.
D. Drusvyatskiy, A. D. Ioffe and A. S. Lewis (2015), Transversality and alternating projections for nonconvex sets, Found. Comput. Math. 15, 1637–1651.
234.
A. Y. Dubovitskii and A. A. Milyutin (1965), Extremum problems in the presence of restrictions, USSR Comput. Maths. Math. Phys. 5, 1–80.
245.
G. Eichfelder (2014), Variable Ordering Structures in Vector Optimization, Springer, Berlin.CrossRef
248.
I. Ekeland (1972), Sur les problémes variationnels, C. R. Acad. Sci. Paris 275, 1057–1059.
249.
I. Ekeland (1974), On the variational principle, J. Math. Anal. Appl. 47, 324–353.
250.
I. Ekeland (1979), Nonconvex minimization problems, Bull. Amer. Math. Soc. 1, 432–467.
254.
M. Fabian (1989), Subdifferentiability and trustworthiness in the light of a new variational principle of Borwein and Preiss, Acta Univ. Carolina, Ser. Math. Phys. 30, 51–56.
257.
M. Fabian and B. S. Mordukhovich (1998), Smooth variational principles and characterizations of Asplund spaces, Set-Valued Anal. 6, 381–406.
258.
M. Fabian and B. S. Mordukhovich (2002), Separable reduction and extremal principles in variational analysis, Nonlinear Anal. 49, 265–292.
259.
M. Fabian and B. S. Mordukhovich (2003), Sequential normal compactness versus topological normal compactness in variational analysis, Nonlinear Anal. 54, 1057–1067.
265.
S. D. Flåm (2006), Upward slopes and inf-convolutions, Math. Oper. Res. 31, 188–198.
278.
C. Gerstewitz (Tammer) (1983), Ninchtkonvexe dualität in der vectoroptimierung, Wissenschaftliche Zeitschrift der TH Leuna-Merseburg 25, 357–364 (in German).
279.
C. Gerth (Tammer) and P. Weidner (1990), Nonconvex separation theorems and some applications in vector optimization, J. Optim. Theory Appl. 67, 297–320.
300.
A. Göpfert, H. Riahi, C. Tammer and C. Zălinescu (2003), Variational Methods in Partially Ordered Spaces, Springer, New York.MATH
321.
T. X. D. Ha (2012), The Fermat rule and Lagrange multiplier rule for various effective solutions to set-valued optimization problems expressed in terms of coderivatives, in Recent Developments in Vector Optimization, edited by Q. H. Ansari and J.-C. Yao, pp. 417–466, Springer, Berlin.CrossRef
349.
J.-B. Hiriart-Urruty (1983), A short proof of the variational principle for approximate solutions of a minimization problem, Amer. Math. Monthly 90, 206–207.
364.
A. D. Ioffe (1981), Calculus of Dini subdifferentials, CEREMADE Publication 8110, Universiteé de Paris IX “Dauphine”.
365.
A. D. Ioffe (1981), Approximate subdifferentials of nonconvex functions, CEREMADE Publication 8120, Universiteé de Paris IX “Dauphine.”
367.
A. D. Ioffe (1983), On subdifferentiability spaces, Ann. New York Acad. Sci. 410, 107–119.
368.
A. D. Ioffe (1984), Approximate subdifferentials and applications, I: the finite dimensional theory, Trans. Amer. Math. Soc. 281, 389–415.
369.
A. D. Ioffe (1989), Approximate subdifferentials and applications, III: the metric theory, Mathematika 36, 1–38.
371.
A. D. Ioffe (2000), Codirectional compactness, metric regularity and subdifferential calculus, in Constructive, Experimental and Nonlinear Analysis, edited by M. Théra, pp. 123–164, American Mathematical Society, Providence, Rhode Island.MATH
375.
A. D. Ioffe (2017), Variational Analysis of Regular Mappings: Theory and Applications (2017), Springer, Cham, Switzerland.CrossRef
376.
A. D. Ioffe and J. V. Outrata (2008), On metric and calmness qualification conditions in subdifferential calculus, Set-Valued Anal. 16, 199–228.
385.
J. Jahn (2004), Vector Optimization: Theory, Applications and Extensions, Springer, Berlin.CrossRef
389.
V. Jeyakumar and D. T. Luc (2008), Nonsmooth Vector Functions and Continuous Optimization, Springer, New York.MATH
398.
A. Jourani and L. Thibault (1996), Metric regularity and subdifferential calculus in Banach spaces, Set-Valued Anal. 3, 87–100.
399.
A. Jourani and L. Thibault (1996), Extensions of subdifferential calculus rules in Banach spaces, Canad. J. Math. 48, 834–848.
407.
R. Kasimbeyli (2010), A nonlinear cone separation theorem and scalarization in nonconvex vector optimization, SIAM J. Optim. 20, 1591–1619.
409.
A. A. Khan, C. Tammer and C. Zălinescu (2015), Set-Valued Optimization. An Introduction with Applications, Springer, Berlin.MATH
428.
A. Y. Kruger (1981), Generalized differentials of nonsmooth functions, Depon. VINITI #1332-81, Moscow (in Russian).
430.
A. Y. Kruger (1985), Generalized differentials of nonsmooth functions and necessary conditions for an extremum, Siberian Math. J. 26, 370–379.
433.
A. Y. Kruger (2009), About stationarity and regularity in variational analysis, Taiwanese J. Math. 13, 1737–1785.
436.
A. Y. Kruger and M. A. López (2012), Stationarity and regularity of infinite collections of sets, J. Optim. Theory Appl. 154, 339–369.
437.
A. Y. Kruger and M. A. López (2012), Stationarity and regularity of infinite collections of sets. Applications to infinitely constrained optimization, J. Optim. Theory Appl. 155, 390–416.
438.
A. Y. Kruger, D. R. Luke and N. M. Thao (2017), Set regularities and feasibility problems, Math. Program., DOI 10.1007/s10107-016-1039-x.CrossRefMATH
440.
A. Y. Kruger and B. S. Mordukhovich (1980), Generalized normals and derivatives, and necessary optimality conditions in nondifferential programming, I&II, Depon. VINITI: I# 408-80, II# 494-80, Moscow (in Russian).
441.
A. Y. Kruger and B. S. Mordukhovich (1980), Extremal points and the Euler equation in nonsmooth optimization, Dokl. Akad. Nauk BSSR 24, 684–687 (in Russian).
458.
A. S. Lewis, D. R. Luke, and J. Malick (2009), Local linear convergence for alternating and averaged nonconvex projections, Found. Comput. Math. 9, 485–513.
459.
A. S. Lewis and J. Malick (2008), Alternating projection on manifolds, Math. Oper. Res. 33, 216–234.
468.
G. Li, K. F. Ng and X. Y. Zheng (2007), Unified approach to some geometric results in variational analysis, J. Funct. Anal. 248 (2007), 317–343.
480.
D. R. Luke (2012), Local linear convergence of approximate projections onto regularized sets, Nonlinear Anal. 75, 1531–1546.
481.
D. R. Luke, N. H. Thao and M. K. Tam (2017), Quantitative convergence analysis of iterated expansive, set-valued mappings, to appear in Math. Oper. Res., arXiv:1605.05725.
502.
B. S. Mordukhovich (1976), Maximum principle in problems of time optimal control with nonsmooth constraints, J. Appl. Math. Mech. 40, 960–969.
504.
B. S. Mordukhovich (1980), Metric approximations and necessary optimality conditions for general classes of extremal problems, Soviet Math. Dokl. 22, 526–530.
505.
B. S. Mordukhovich (1984), Nonsmooth analysis with nonconvex generalized differentials and adjoint mappings, Dokl. Akad. Nauk BSSR 28, 976–979 (in Russian).
507.
B. S. Mordukhovich (1988), Approximation Methods in Problems of Optimization and Control, Nauka, Moscow (in Russian).MATH
511.
B. S. Mordukhovich (1994), Generalized differential calculus for nonsmooth and set-valued mappings, J. Math. Anal. Appl. 183, 250–288.
515.
B. S. Mordukhovich (2000), Abstract extremal principle with applications to welfare economics, J. Math. Anal. Appl. 251, 187–216.
518.
B. S. Mordukhovich (2004), Coderivative analysis of variational systems, J. Global Optim. 28, 347–362.
522.
B. S. Mordukhovich (2006), Variational Analysis and Generalized Differentiation, I: Basic Theory, Springer, Berlin.
523.
B. S. Mordukhovich (2006), Variational Analysis and Generalized Differentiation, II: Applications, Springer, Berlin.
533.
B. S. Mordukhovich and N. M. Nam (2009), Variational analysis of generalized equations via coderivative calculus in Asplund spaces, J. Math. Anal. Appl. 350, 663–679.
538.
B. S. Mordukhovich and N. M. Nam (2017), Extremality of convex sets with some applications, Optim. Lett. 11, 1201–1215.
541.
B. S. Mordukhovich, N. M. Nam, R. B. Rector and T. Tran (2017), Variational geometric approach to generalized differential and conjugate calculi in convex analysis, Set-Valued Var. Anal. 25, 731–755.
546.
B. S. Mordukhovich, N. M. Nam and N. D. Yen (2006), Fréchet subdifferential calculus and optimality conditions in nondifferentiable programming, Optimization 55, 685–396.
567.
B. S. Mordukhovich and H. M. Phan (2011), Rated extremal principle for finite and infinite systems with applications to optimization, Optimization 60, 893–924.
568.
B. S. Mordukhovich and H. M. Phan (2012), Tangential extremal principle for finite and infinite systems, I: basic theory, Math. Program. 136, 31–63.
569.
B. S. Mordukhovich and H. M. Phan (2012), Tangential extremal principle for finite and infinite systems, II: applications to semi-infinite and multiobjective optimization, Math. Program. 136, 31–63.
578.
B. S. Mordukhovich and Y. Shao (1995), Differential characterizations of covering, metric regularity, and Lipschitzian properties of multifunctions between Banach spaces, Nonlinear Anal. 25, 1401–1424.
579.
B. S. Mordukhovich and Y. Shao (1996), Extremal characterizations of Asplund spaces, Proc. Amer. Math. Soc. 124, 197–205.
580.
B. S. Mordukhovich and Y. Shao (1996), Nonsmooth sequential analysis in Asplund spaces, Trans. Amer. Math. Soc. 348, 1235–1280.
581.
B. S. Mordukhovich and Y. Shao (1996), Nonconvex coderivative calculus for infinite-dimensional multifunctions, Set-Valued Anal. 4, 205–136.
582.
B. S. Mordukhovich and Y. Shao (1997), Stability of multifunctions in infinite dimensions: point criteria and applications, SIAM J. Control Optim. 35, 285–314.
583.
B. S. Mordukhovich and Y. Shao (1997), Fuzzy calculus for coderivatives of multifunctions, Nonlinear Anal. 29, 605–626.
586.
B. S. Mordukhovich, J. S. Treiman and Q. J. Zhu (2003), An extended extremal principle with applications to multiobjective optimization, SIAM J. Optim. 14, 359–379.
587.
B. S. Mordukhovich and B. Wang (2002), Necessary optimality and suboptimality conditions in nondifferentiable programming via variational principles, SIAM J. Control Optim. 41, 623–640.
588.
B. S. Mordukhovich and B. Wang (2002), Extensions of generalized differential calculus in Asplund spaces, J. Math. Anal. Appl. 272, 164–186.
589.
B. S. Mordukhovich and B. Wang (2003), Calculus of sequential normal compactness in variational analysis, J. Math. Anal. Appl. 282, 63–84.
610.
N. V. Ngai and M. Théra (2001), Metric regularity, subdifferential calculus and applications, Set-Valued Anal. 9, 187–216.
617.
D. Noll and A. Rondepierre (2016), On local convergence of the method of alternating projections, Found. Comput. Math. 16, 425–455.
635.
J.-P. Penot (1998), Compactness properties, openness criteria and coderivatives, Set-Valued Anal. 6, 363–380.
637.
638.
R. R. Phelps (1993), Convex Functions, Monotone Operators and Differentiability, 2nd edition, Springer, Berlin.MATH
667.
R. T. Rockafellar (1970), Convex Analysis, Princeton University Press, Princeton, New Jersey.CrossRef
669.
R. T. Rockafellar (1979), Directional Lipschitzian functions and subdifferential calculus, Proc. London Math. Soc. 39, 331–355.
670.
R. T. Rockafellar (1980), Generalized directional derivatives and subgradients of nonconvex functions, Canad. J. Math. 32, 157–180.
675.
R. T. Rockafellar (1985), Extensions of subgradient calculus with applications to optimization, Nonlinear Anal. 9, 665–698.
678.
R. T. Rockafellar and R. J-B. Wets (1998), Variational Analysis, Springer, Berlin.CrossRef
685.
731.
B. Wang and X. Yang (2016), Weak differentiability with applications to variational analysis, Set-Valued Var. Anal. 24, 299–321.
732.
B. Wang and D. Wang (2015), Generalized sequential normal compactness in Asplund spaces, Applic. Anal. 94, 99–107.
773.
X. Y. Zheng and K. F. Ng (2005), The Fermat rule for multifunctions in Banach spaces, Math. Program. 104, 69–90.
774.
X. Y. Zheng and K. F. Ng (2006), The Lagrange multiplier rule for multifunctions in Banach spaces, SIAM J. Optim. 17, 1154–1175.
777.
X. Y. Zheng and K. F. Ng (2011), A unified separation theorem for closed sets in a Banach spaces and optimality conditions for vector optimization, SIAM J. Optim. 21, 886–911.
786.
Q. J. Zhu (1998), The equivalence of several basic theorems for subdifferentials, Set-Valued Anal. 6, 171–185.
787.
Q. J. Zhu (2004), Nonconvex separation theorem for multifunctions, subdifferential calculus and applications, Set-Valued Anal. 12, 275–290.
Metadata
Title
Fundamental Principles of Variational Analysis
Author
Boris S. Mordukhovich
Copyright Year
2018
Book
Variational Analysis and Applications

Print ISBN: 978-3-319-92773-2

Electronic ISBN: 978-3-319-92775-6

Copyright Year: 2018

DOI
https://doi.org/10.1007/978-3-319-92775-6_2

Premium Partner

    Image Credits