Luận án Nonlinear metric regularity of set-valued mappings on a fixed set and applications

MINISTRY OF EDUCATION AND TRAINING
QUY NHON UNIVERSITY
DAO NGOC HAN
NONLINEAR METRIC REGULARITY OF SET-VALUED
MAPPINGS ON A FIXED SET AND APPLICATIONS
DOCTORAL THESIS IN MATHEMATICS
Binh Dinh - 2021
MINISTRY OF EDUCATION AND TRAINING
QUY NHON UNIVERSITY
DAO NGOC HAN
NONLINEAR METRIC REGULARITY OF SET-VALUED
MAPPINGS ON A FIXED SET AND APPLICATIONS
Speciality: Mathematical Analysis
Speciality code: 9 46 01 02
Reviewer 1: Assoc. Prof. Dr. Phan Nhat Tinh
Reviewer 2: Assoc. Prof. Dr. Nguyen Huy Chieu
Reviewer 3: Assoc. Prof. Dr. Pham Tien Son
Supervisors:
Assoc. Prof. Dr. Habil. Huynh Van Ngai
Dr. Nguyen Huu Tron
Binh Dinh - 2021
Declaration
This dissertation was completed at the Department of Mathematics and Statis-
tics, Quy Nhon University under the guidance of Assoc. Prof. Dr. Habil. Huynh
Van Ngai and Dr. Nguyen Huu Tron. I hereby declare that the results presented in
here are new and original. Most of them were published in peer-reviewed journals,
others have not been published elsewhere. For using results from joint papers I
have gotten permissions from my co-authors.
Binh Dinh, July 16, 2021
Advisors PhD student
Assoc. Prof. Dr. Habil. Huynh Van Ngai Dao Ngoc Han
i
Acknowledgments
The dissertation was carried out during the years I have been a PhD student
at the Department of Mathematics and Statistics, Quy Nhon University. On the
occasion of completing the thesis, I would like to express the deep gratitude to
Assoc. Prof. Dr. Habil. Huynh Van Ngai not only for his teaching and scientific
leadership, but also for the helping me access to the academic environment through
the workshops, mini courses that assist me in broadening my thinking to get the
entire view on the related issues in my research.
I wish to express my sincere gratitude to my second supervisor, Dr. Nguyen
Huu Tron, for accompanying me in study. Thanks to his enthusiastic guidance, I
approached the problems quickly, and this valuable support helps me to be more
mature in research.
A very special thank goes to the teachers at the Department of Mathematics
and Statistics who taught me wholeheartedly during the time of study, as well as
all the members of the Assoc. Prof. Huynh Van Ngai’s research group for their
valuable comments and suggestions on my research results. I would like to thank
the Department of Postgraduate Training, Quy Nhon University for creating the
best conditions for me to complete this work within the schedules.
I also want to thank my friends, PhD students and colleagues at Quy Nhon
University for their sharing and helping in the learning process. Especially, I am
grateful to Mrs. Pham Thi Kim Phung for her constant encouragement giving me
the motivation to overcome difficulties and pursue the PhD program.
I wish to acknowledge my mother, my parents in law for supporting me in
every decision. And, my enormous gratitude goes to my husband and sons for their
love and patience during the time I was working intensively to complete my PhD
program. Finally, my sincere thank goes to my father for guiding me to math and
this thesis is dedicated to him.
ii
Contents
Table of Notations 1
Introduction 3
1 Preliminaries 11
1.1 Some related classical results . . . . . . . . . . . . . . . . . . . . . . 11
1.2 Basic tools from variational analysis and nonsmooth analysis . . . . 13
1.2.1 Ekeland’s variational principles . . . . . . . . . . . . . . . . 13
1.2.2 Subdifferentials and some calculus rules . . . . . . . . . . . . 15
1.2.3 Coderivatives of set-valued mappings . . . . . . . . . . . . . 18
1.2.4 Duality mappings . . . . . . . . . . . . . . . . . . . . . . . . 20
1.2.5 Strong slope and some error bound results . . . . . . . . . . 22
1.3 Metric regularity and equivalent properties . . . . . . . . . . . . . . 27
1.3.1 Local metric regularity . . . . . . . . . . . . . . . . . . . . . 27
1.3.2 Nonlocal metric regularity . . . . . . . . . . . . . . . . . . . 29
1.3.3 Nonlinear metric regularity . . . . . . . . . . . . . . . . . . . 30
1.4 Metric regularity criteria in metric spaces . . . . . . . . . . . . . . . 33
iii
1.5 The infinitesimal criteria for metric
regularity in metric spaces . . . . . . . . . . . . . . . . . . . . . . . 36
2 Metric regularity on a fixed set: definitions and characterizations 38
2.1 Definitions and equivalence of the nonlinear metric regularity concepts 39
2.2 Characterizations of nonlinear metric
regularity via slope . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.3 Characterizations of nonlinear metric
regularity via coderivative . . . . . . . . . . . . . . . . . . . . . . . 54
3 Perturbation stability of Milyutin-type regularity and applications 64
3.1 Perturbation stability of Milyutin-type
regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.2 Application to fixed double-point problems . . . . . . . . . . . . . . 78
4 Star metric regularity 84
4.1 Definitions and characterizations of nonlinear star metric regularity 84
4.2 Stability of Milyutin-type regularity under perturbation of star reg-
ularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5 Stability of generalized equations governed by composite multi-
functions 97
5.1 Notation and some related concepts . . . . . . . . . . . . . . . . . . 98
5.2 Regularity of parametrized epigraphical and composition set-valued
mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.3 Stability of implicit set-valued mappings . . . . . . . . . . . . . . . 118
iv
5.3.1 Stability of implicit set-valued mappings associated to the
epigraphical set-valued mapping . . . . . . . . . . . . . . . . 118
5.3.2 Stability of implicit set-valued mappings associated to a com-
posite mapping . . . . . . . . . . . . . . . . . . . . . . . . . 122
Conclusions 128
List of Author’s Related Publications 130
References 131
v
Table of Notations
N : the set of natural numbers
R : the set of real numbers
R+ : the set of non-negative real numbers
∅ : the empty set
Rn : the n-dimensional Euclidean vector space
〈x, y〉 : the scalar product in an Euclidean space
||x|| : norm of a vector x
B(x, r) : the open ball centered x with radius r
B(x, r) : the closed ball centered x with radius r
BX : the open unit ball of X
BX : the closed unit ball of X
B(A, r) : the open ball around a set A with radius r > 0
e(A,B) : the excess of a set A over other one B
dom f : the domain of f
epi f : the epigraph of f
X∗ : the dual space of a Banach space X
X∗∗ : the dual space of X∗
A∗ : Y ∗ → X∗ : the adjoint operator of a bounded linear operator
A : X → Y
d(x,Ω) : the distance from x to a set Ω
N̂(x¯; Ω) : the Fre´chet normal cone of Ω at x¯
N(x¯; Ω) : the Mordukhovich normal cone of Ω at x¯
x
Ω→ x¯ : x→ x¯ and x ∈ Ω
F : X ⇒ Y : a set-valued map between X and Y
GraphF : the graph of F
D̂∗F (x¯, y¯)(·) : the Fre´chet coderivative of F at (x¯, y¯)
D∗F (x¯, y¯)(·) : the Mordukhovich coderivative of F at (x¯, y¯)
∇f(x¯) : the Fre´chet derivative of f : X → Y at x¯
EH : the epigraphical-type set-valued mapping associated to
the set-valued mapping H
SEH : the solution mapping associated to EH
1
SH : the solution mapping associated to the set-valued
mapping H
ϕFy (x) : the lower semicontinuous envelop function of the
distance function d(y, F (x))
ϕ∗Fy (x) : the lower semicontinuous envelop function of the
distance function d(y, F (x) ∩ V )
ϕpT (x, y) : the lower semicontinuous envelop function of the
distance function d(y, T (x, p))
surF : the modulus of openness of F
surγ F : the modulus of γ-openness of F
regF : the modulus of metric regularity of F
regγ F : the modulus of γ-metric regularity of F
reg(γ,κ) F : the modulus of (γ, κ)-Milyutin regularity of F
reg∗γ F : the modulus of γ-Milyutin regularity
∗ of F
reg∗(γ,κ) F : the modulus of (γ, κ)-Milyutin regularity
∗ of F
lipF : the Lipschitz modulus of F
lipγ F : the γ-Lipschitz modulus of F
2
Introduction
In mathematics, solving many problems leads to the formation of equations
and solving them. The basis question dealing with the equations is that whether
a solution exists or not. If exists, how to identify a such solution? And, how
does the solution set change when the input data are perturbed? One of the
powerful frameworks to consider the existence of solutions of equations is metric
regularity. For equations of the form f(x) = y, where f : X → Y is a single-valued
mapping between metric spaces, the condition ensuring the existence of solutions
of equations is the surjectivity of f . In the case of f being a single-valued mapping
between Banach spaces and strictly differentiable at x¯, the problem of regularity of
f is reduced to that of its linear approximation ∇f(x¯) and the regularity criterion
is the surjectivity of ∇f(x¯). This result is obtained from the Lyusternik–Graves
theorem, which is considered as one of the main results of nonlinear analysis.
Actually, a large amount of practical problems interested in outrun equations.
For instance, systems of inequalities and equalities, variational inequalities or
systems of optimality conditions are covered by the solvability of an inclusion
y ∈ F (x),
where F : X ⇒ Y is a set-valued mapping between metric spaces. These inclusions
are named as generalized equations or variational systems, which were initiated
by Robinson in 1970s, see [74, 75] for details. They cover many problems and
phenomena in mathematics and other science areas, such as equations, variational
inequalities, complementary problems, dynamical systems, optimal control, and
necessary/sufficient conditions for optimization and control problems, fixed point
theory, coincidence point theory and so on. Nowadays, generalized equations have
attracted the attention of many experts (see, for instance, [7, 27, 51, 56, 57, 74, 75]
and the references given therein). And thus, variational analysis has appeared in
response to the strong development.
A central issue of variational analysis is to investigate the existence and
behavior of the solution set F−1(y) of generalized equations when y and/ or F itself
are perturbed, where the mapping F may lack of smoothness: non-differentiable
3
functions or set-valued mappings, etc. Then, it is almost impossible to approximate
F by simple objects, like linear operators (see, Ioffe [51]), and so the condition of
surjectivity of the derivative mapping at the point in this case is not useful. However,
this can be replaced by giving an estimation of the distance from a certain point
x near a given solution x¯ to the solution set F−1(y) (unknown quantity) of the
generalized equation through distance d(y, F (x)) from a point y near y¯ ∈ F (x¯) to
the image of F at x. In applications, the distance d(y, F (x)) is able to calculate
or estimate, meanwhile finding the exact solution set might be considerably more
complicated. Then, F : X ⇒ Y satisfying the above estimation is said to be local
metric regularity around (x ... iy S. E. (2015), “Stability theorems
for estimating the distance to a set of coincidence points”, SIAM J. Optim.,
25 (2), 807–828.
[6] Asplund E. (1967), “Positivity of duality mappings”, Bull. Amer. Math. Soc.,
73, 200–203.
[7] Aubin J. P., Cellina A. (1984), Differential Inclusions: Set-Valued Maps and
Viability Theory, Springer-Verlag, Berlin.
[8] Aubin J. P., Ekeland I. (1984), Applied Nonlinear Analysis, John Wiley &
Sons, New York.
131
[9] Aze´ D. (2003), “A survey on eror bounds for lower semicontinuous functions”,
ESAIM Proc., 13, 1–17.
[10] Aze´ D., Corvellec J. N. (2004), “Characterizations of error bounds for lower
semicontinuous functions on metric spaces”, ESAIM Control Optim. Calc.
Var., 10 (3), 409–425.
[11] Aze´ D. (2006), “A unified theory for metric regularity of multifunctions”, J.
Convex Anal., 13 (2), 225–252.
[12] Aze´ D., Benahmed S. (2008), “On implicit set-valued mapping theorems”,
Set-Valued Var. Anal., 16, 129–155.
[13] Banach S. (1932), The´orie des ope´rations line´aries, Monografje Matematyczne,
Warszawa.
[14] Beurling A., Livingston A. E. (1962), “A theorem on duality mappings in
Banach spaces”, Ark. Mat., 4, 405–411.
[15] Borwein J. M., Zhuang D. M. (1988), “Verifiable necessary and sufficient
conditions for openness and regularity for set-valued and single-valued mapps”,
J. Math. Anal. Appl., 134, 441–459.
[16] Borwein J. M., Zhu Q. J. (2005), Techniques of variational analysis, Springer,
New York.
[17] Cibulka R., Fabian M., Kruger A. Y. (2019), “On semiregularity of mappings”,
J. Math. Anal. Appl., 473 (2), 811–836.
[18] Cibulka R., Durea M., Pantiruc M., Strugariu R. (2020), “On the stability of
the directional regularity”, Set-Valued Var. Anal., 28, 209–237.
[19] Chidume C. (2009), Geometric Properties of Banach Spaces and Nonlinear
Interations, Lecture Notes in Mathematics 1965, Springer-Verlag, London.
[20] Chieu N. H., Yao J., C., Yen N. D. (2010), “Relationships between Robinson
metric regularity and Lipschitz-like behavior of implicit set-valued mappings”,
Nonlinear Anal., 72, 3594–3601.
132
[21] Cioranescu I. (1990), Geometry of Banach spaces, Duality Mappings and
Nonlinear Problems, Kluwer Academic Publishers, Dordecht.
[22] De Giorgi E., Marino A., Tosques M. (1980), “Problemi di evoluzione in spazi
metrici e curve di massima pendenza (Evolution problems in metric spaces
and curves of maximal slope)”, Atti. Accad. Naz. Lincei rend. Cl. Sci. fis.
Mat. Natur., 68, 180–187.
[23] Dmitruk A. V., Milyutin A. A., Osmolovskii, N. P. (1980), “Lyusternik’s
theorem and the theory of extrema”, Russian Math. Surveys, 35 (6), 11–51.
[24] Dmitruk A. V., Kruger A. Y. (2008), “Metric regularity and systems of
generalized equations”, J. Math. Anal. Appl., 342 (2), 864–873.
[25] Dontchev A. L., Lewis A.S., Rockafellar R. T. (2003), “The radius of metric
regularity”, Trans. Amer. Math. Soc. Ser. B, 355 (2), 49–517.
[26] Dontchev A. L., Lewis A. S. (2005), “Perturbations and metric regularity”,
Set-Valued Var. Anal., 13, 417–438.
[27] Dontchev A. L., Rockafellar R. T. (2009), Implicit Functions and Solution
Mappings. A View from Variational Analysis, Springer, New York.
[28] Dontchev A. L., Frankowska H. (2011), “Lusternik-Graves theorem and fixed
points”, Proc. Amer. Math. Soc., 139, 521–534.
[29] Durea M., Strugariu R. (2012), “Openness stability and implicit multifunction
theorems: Applications to variational systems”, Nonlinear Anal., 75,
1246–1259.
[30] Durea M., Strugariu R. (2012), “Chain rules for linear openness in general
Banach spaces”, SIAM J. Optim., 22 (3), 899–913.
[31] Durea M., Strugariu R. (2014), “Chain rules for linear openness in metric
spaces. Applications to parametric variational systems”, Math. Program., 143A
(1), 147–176.
133
[32] Durea M., Strugariu R. (2014), “Chain rules for linear openness in metric
spaces. Applications to parametric variational systems”, Math. Program., 143
(1-2), 147–176.
[33] Durea M., Strugariu R. (2016), “Metric subregularity of composition set-valued
mappings with applications to fixed point theory”, Set-Valued Var. Anal., 24
(2), 231–251.
[34] Durea M., Tron N. H., Strugariu R. (2013), “Metric regularity of epigraphical
multivalued mappings and applications to vector optimization”, Math.
Program., 139 (1-2), 139–159.
[35] Durea M., Ngai H. V, Tron N. H, Strugariu R. (2014), “Metric regularity of
composition set-valued mappings: Metric setting and coderivative conditions”,
J. Math. Anal. Appl., 412, 41–62.
[36] Ekeland I. (1974), “On the variational principle”, J. Math. Anal. Appl., 47,
324–353.
[37] Fabian M. (1990), “Subdifferentials, local ε-supports and Asplund spaces”, J.
Lond. Math. Soc., 41, 175–192.
[38] Frankowska H., Quincampoix M. (2012), “Ho¨lder metric regularity of
set-valued maps”, Math. Program., 132 (1-2), 333–354.
[39] Gfrerer H. (2013), “On directional metric regularity, subregularity and
optimality conditions for nonsmooth mathematical programs”, Set-Valued Var
Anal., 21, 151–176.
[40] Gfrerer H. (2013), “On directional metric subregularity and second-order
optimality conditions for a class of nonsmooth mathematical programs”, SIAM
J. Optim., 23, 632–665.
[41] Gfrerer H. (2014), “On metric pseudo-(sub)regularity of multifunctions and
optimality conditions for degenerated mathematical programs”, Set-Valued
Var. Anal., 22, 79–115.
134
[42] Ioffe A. D. (2000), “Metric regularity and subdifferential calculus”, Russian
Math. Surveys, 55 (3), 501–558.
[43] Ioffe A. D. (2001), “On perturbation stability of metric regularity”, Set-Valued
Var. Anal., 9, 101–109.
[44] Ioffe A. D. (2010), “On regularity concepts in variational analysis”. J. Fixed
Point Theory Appl., 8 (2), 339–363.
[45] Ioffe A. D. (2010), “Towards variational analysis in metric spaces: metric
regularity and fixed points”, Math. Program., 123, 241–252.
[46] Ioffe A. D. (2011), “Regularity on fixed sets”, SIAM J. Optim., 21 (4),
1345–1370.
[47] Ioffe A. D. (2013), “Nonlinear regularity models”, Math. Program., 139 (1),
223–242.
[48] Ioffe A. D. (2016), “Metric regularity-a survey. Part I. Theory”, J. Aust. Math.
Soc., 101, 188–243.
[49] Ioffe A. D. (2016), “Metric regularity-a survey. Part II. Applications”, J. Aust.
Math. Soc., 101, 376–417.
[50] Ioffe A. D. (2017), “Implicit functions: a metric theory”, Set-Valued Var. Anal.,
25 (4), 679–699.
[51] Ioffe A. D. (2017), Variational Analysis of Regular Mappings: Theory and
Applications, Springer Monographs in Mathematics, Switzerland.
[52] Kruger A. Y. (1988), “A covering theorem for set-valued mappings”,
Optimization, 19 (6), 763–780.
[53] Kruger A. Y. (2003), “On Fre´chet subdifferentials”, J. Math. Sci. (N. Y.), 116
(3), 3325–3358.
[54] Kruger A. Y. (2016), “Nonlinear metric subregularity”, J. Optim. Theory
Appl., 171, 820–855.
135
[55] Li G., Mordukhovich B. S. (2012), “Ho¨lder metric subregularity with
applications to proximal point method”, SIAM J. Optim., 22 (4), 1655–1684.
[56] Mordukhovich B. S. (2006), Variational Analysis and Generalized
Differentiation. I. Basic theory, Springer-Verlag, Berlin.
[57] Mordukhovich B. S. (2006), Variational Analysis and Generalized
Differentiation. II. Applications, Springer-Verlag, Berlin.
[58] Mordukhovich B. S. (2018), Variational Analysis and Applications, Springer
Monographs in Mathematics, New York.
[59] Mordukhovich B. S., Shao Y. (1995), “Differential characterizations of
covering, metric regularity and Lipschitzian multifunction”, Nonlinear Anal.,
25, 1401–1428.
[60] Ngai H. V., The´ra M. (2004), “Error bounds and implicit set-valued mappings
in smooth Banach spaces and applications to optimization”, Set-Valued Var.
Anal., 12, 195–223.
[61] Ngai H. V., The´ra M. (2008), “Error bounds in metric spaces and application
to the perturbation stability of metric regularity”, SIAM J. Optim., 19 (1),
1–20.
[62] Ngai H. V, The´ra M. (2009), “Error bounds for systems of lower semicontinuous
functions in Asplund spaces”, Math. Program., 116 (1-2), 397–427.
[63] Ngai H.V., Tron N. H., The´ra M. (2013), “Implicit multifunction theorem in
complete metric spaces”, Math. Program., 139 (1-2), 301–326.
[64] Ngai H.V., Tron N. H., The´ra M. (2014), “Metric regularity of the sum of
multifunctions and applications”, J. Optim. Theory Appl., 160 (2), 355–390.
[65] Ngai H. V, The´ra M. (2015), “Directional metric regularity of multifunctions”,
Math. Oper. Res., 40 (4), 969–991.
136
[66] Ngai H. V., Tron N. H., The´ra M. (2016), “Directional Ho¨lder metric
regularity”, J. Optim. Theory Appl., 171, 785–819.
[67] Ngai H. V., Tron N. H., Tinh P. N. (2017), “Directional Holder metric
subregularity and applications to tangent cones”, J. Convex Anal., 24 (2),
417–457.
[68] Ngai H. V., Tron N. H., Vu N. V., The´ra M. (2020), “Directional metric pseudo
subregularity of set-valued mappings: a general model”, Set-Valued Var. Anal.,
28, 61–87.
[69] Ngai H. V., Tron N. H., Han D. N. (2021), “Metric perturbation of Milyutin
regularity on a fixed set and application to fixed point theorems”, preprint.
[70] Ngai H. V., Tron N. H., Han D. N. (2021), “Star metric regularity on a fixed
set”, preprint.
[71] Nghia T. T. A. (2014), “A note on implicit multfunction theorem”, Optim.
Lett., 8 (1), 329–341.
[72] Penot J. P. (1989), “Metric regularity, openness and Lipschitzian behavior of
multifunctions”, Nonlinear Anal., 13 (6), 629–643.
[73] Penot J.P. (2013), Calculus Without Derivatives, Springer (Graduate Texts in
Mathematics).
[74] Robinson S. M. (1975), “Theory for systems of inequalities. Part I: Linear
systems”, SIAM J. Numer. Anal., 12, 754–769.
[75] Robinson S. M. (1976), “Stability theory for systems of inequalities. Part II:
Differentiable nonlinear systems”, SIAM J. Numer. Anal., 13, 497–513.
[76] Rockafellar R. Tyrrell, Wets Roger J-B (2005), Variational Analysis, Springer,
New York.
[77] Schauder J. (1930), “Uber die Umkerhrung linearer, stetiger
Funktionaloperationen”, Studia Math., 2, 1–6.
137
[78] Schirotzek W. (2007), Nonsmooth Analysis, Springer, Berlin.
[79] Tron N. H., Han D. N., Ngai H. V. (2020), “Nonlinear metric regularity on
fixed sets”, In revision, submitted to Optimization.
[80] Tron N. H., Han D. N. (2020), “Stability of generalized equations governed by
composite multifunctions”, Pac. J. Optim., 16 (4), 641–662.
[81] Tron N. H., Han D. N. (2020), “On the Milyutin regularity of set-valued
mappings”, QNU. J. Sci., 14 (3), 37–45.
[82] Ursescu C. (1996), “Inherited openness”, Rev. Roumaine Math. Pures Appl.,
41, 401–416.
[83] Yen N. D., Yao J. C. (2009), “Point-based sufficient conditions for metric
regularity of implicit multifunctions”, Nonlinear Anal., 70 (7), 2806–2815.
[84] Zheng X. Y., Ng K. F. (2009), “Metric regularity of composite multifunctions
in Banach spaces”, Taiwanese J. Math., 13 (6A), 1723–1735.
[85] Zheng X. Y., Zhu A. J. (2016), “Generalized metric subregularity and
regularity with respect to an admissible functions”, SIAM J. Optim., 26 (1),
535–563.
138