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MINISTRY OF EDUCATION AND TRAINING
VINH UNIVERSITY

LE VAN HIEN

SOME RESULTS ON METRIC SUBREGULARITY
IN VARIATIONAL ANALYSIS
AND APPLICATIONS

Speciality: Mathematical Analysis
Code: 9 46 01 02

SUMMARY OF MATHEMATICS DOCTORAL THESIS

NGHE AN - 2019


Work is completed at Vinh University

Supervisors:
1. Dr. Nguyen Huy Chieu

2. Assoc. Prof. Dr. Dinh Huy Hoang

Reviewer 1:

Reviewer 2:

Reviewer 3:

Thesis will be defended at school-level thesis evaluating council at Vinh University
at ... date ... month ... year ...

Thesis can be found at:
1. Nguyen Thuc Hao Library and Information Center - Vinh University
2. Vietnam National Library


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PREFACE

1. Rationale
In order to implement more tools to investigate optimization and related problems, R. T. Rockafellar and J.-J. Moreau proposed and studied the subdifferential for
convex functions in the early 1960s. In the mid-1970s, F. H. Clarke and B. S. Mordukhovich independently introduced the concepts of the subdifferential for possibly
non-convex functions. Derivatives and coderivatives of set-valued mappings appeared
in the early 1980s. Besides, many other generalized differential concepts were also
presented and examined in the literature. In 1998, R. T. Rockafellar and R. J.-B. Wets
published a monograph book namely “Variational Analysis” based on summarizing,
systematizing and complementing basic results in this research direction, marking the
birth of Variational analysis.
Up to now, the first-order variational analysis has been quite perfect, while the
second-order variational analysis has been intensively examined and rapidly developed. Recently, this field has attracted the attention of many mathematicians.
The generalization differentiation plays a vital role in variational analysis and its
application. To any generalized differential structures, there are always two fundamental problems naturally raised: firstly, which feature of the function, mapping or
set is reflected by the structure; secondly, how we can calculate or estimate that
structure in terms of the initial data. In fact, in order to thoroughly address each of
these problems, we all need some information about certain regularity of the involved
functions, mappings or sets. That is why regularity properties are important research
objects in variational analysis.
The metric subregularity is one of the remarkable regularity properties in the firstorder variational analysis. Recently, there have been various studies on this property
in the second-order variational analysis. However, its role in second-order variational
analysis is still an interesting and not fully understood issue that requires further
investigation.
With such reasons, we have selected and studied the topic “Some results on
metric subregularity in variational analysis and applications”.


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2. Research Objectives
The purpose of the thesis is to establish new research results based on the investigation of the two aforementioned fundamental problems, contributing to clarify the
role of the metric subregularity in variational analysis and applications.
3. Research Subjects
The research subjects of this thesis are regularity properties in variational analysis,
subgradient graphical derivative, tilt stability and isolated calmness.
4. Research Scopes
For the first problem, the thesis focuses on studying the possibility of the subgradient graphical derivative in recognizing tilt stability for unconstrained optimization
problems in which the objective function is prox-regular. At the same time, the
thesis is also interested in nonlinear programs under metric subregularity constraint
qualification with the objective and constraint functions being twice continuously
differentiable functions.
For the second one, the thesis focuses on calculating the subgradient graphical
derivative for a normal cone mapping under the metric subregularity condition and
using this result to investigate the isolated calmness property of the solution mapping
for a broad class of generalized equations.
5. Research Methodology
In this thesis, we use the variational approach and some techniques from functional
analysis, convex analysis, set-valued analysis, variational analysis, optimization.
6. Scientific and Practical Meaning
The thesis contributes to enrich the calculation rules in variational analysis; proposes a new approach to study the tilt stability as well as improves some results of
tilt stability for nonlinear programming problems; thereby clarifies the role of metric
subregularity in variational analysis and application. Moreover, the thesis may be a
good reference for those who are interested in variational analysis, optimization and
their applications.
7. Research Organization
7.1. Research Overview
The regularity properties play an important role in variational analysis and its
application. On the one hand, these properties are used to establish optimality conditions and study stabilities for optimization and related problems. On the other
hand, they are used to develop calculus rules in variational analysis. In addition,
they are also utilized to investigate the convergence of algorithms in numerical optimization.


5

In variational analysis, mathematicians have proposed and studied many different
regularity concepts for sets, extended-real-valued functions and set-valued mappings.
One of the vital regularity properties in the study of optimal conditions and calculation rules of generalized differentiation is the metric subregularity. In 1979, A. D. Ioffe
used this property to define the concept of regular points and set first-order necessary optimality conditions for a class of optimization problems. The term “metric
subregularity” was suggested by A. L. Dontchev and R. T. Rockafellar in 2004. The
metric subregularity of the set-valued mapping is equivalent to calmness of the inverse. In 2008, A. D. Ioffe and J. V. Outrata established a system of calculation rules
for the first-order generalized differentiation in the form of duality using the metric
subregularity. Recently, researchers have also established many calculation rules for
the second-order generalized differentiation structures under the metric subregularity.
Graphical derivative of a set-valued mapping at a point in its graph is the setvalued mapping whose graph is the tangent cone to the graph of the given set-valued
mapping at the point in question. This concept was introduced by J. -P. Aubin
in 1981, who called it the contingent derivative. The term “graphical derivative” was
used in the book “Variational Analysis” by R. T. Rockafellar and R. J. -B. Wets. The
graphical derivative is a powerful tool in variational analysis. One can use it to investigate the stability and sensitivity of constraint and variational systems, and more
general, generalized equations. The graphical derivative can also be used to characterize some nice properties of set-valued mappings, such as the metric regularity, the
Aubin property, the isolated calmness and the strong metric subregularity. In spite of
being the key in tackling some important issues in variational analysis, calculation of
the graphical derivative of a set-valued mapping is generally a challenging task. The
problem has been studied by many researchers for a long time, and many interesting
results in the direction have been established.
Consider the set Γ given by the formula Γ := x ∈ Rn | q(x) ∈ Θ , where
q : Rn → Rm , q(x) = (q1 (x), q2 (x), ..., qm (x)) is a twice continuously differentiable
mapping and Θ ⊂ Rm is a nonempty closed set. Set Mq (x) := q(x) − Θ with
x ∈ Rn . If Θ = Rm
− then Γ is the feasible set of the nonlinear programming problem and, in this case, the Mangasarian–Fromovitz constraint qualification (MFCQ)
holds at x¯ ∈ Γ iff the mapping Mq is metrically regular around (¯
x, 0). Moreover, if
n
adding the assumption qi : R → R, i = 1, 2, ..., m, are convex functions, the Slater
condition holds iff Mq is metrically regular. If Θ is a closed convex cone, then Γ is the
feasible set of the cone programming and the Robinson constraint qualification (RCQ)
is equivalent to the metric regularity of Mq . The Slater condition, MFCQ and RCQ
are the crucial qualification conditions in optimization theory and its application.
These conditions are originally the metric regularity of the set-valued mapping Mq .
Therefore, it is possible to collectively refer these conditions as the metric regularity


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constraint qualification. In 2015, for Γ to be the feasible set of a nonlinear programming problem, H. Gfrerer and B. S. Mordukhovich defined the metric subregularity
constraint qualification (MSCQ) as the metric subregularity of Mq . The concept has
been extended for Θ to be an arbitrary closed set.
This thesis, we concern the computation the graphical derivative DNΓ of the
normal cone mapping NΓ : Rn ⇒ Rn , x → NΓ (x), with Θ being a nonempty polyhedral
convex set. The first result in this direction was established by A. L. Dontchev and R.
T. Rockafellar in 1996, where these authors accurately described the graph of DNΓ ,
with the assumption that Γ is a polyhedral convex set, in terms of the input data
of the problem. The result was then utilized to calculate the second-order limiting
subdifferential of the indicator function of Γ.
In 2013, combining some calculus rules available in variational analysis, R. Henrion
et al. revealed a nice formula for computing the graphical derivative DNΓ under the
metric regularity of the set-valued mapping Mq (x) := q(x) − Θ around the reference
point. In 2014, H. Gfrerer and J. V. Outrata proved that this formula holds if Θ := Rm

and the metric regularity is replaced by the metric subregularity at the reference
point plus a uniform metric regularity around this point. Among other things, their
important contribution is that they proposed a scheme allowing us to directly prove
the formula for calculating the graphical derivative of the normal cone mapping,
which paves the way for satisfactorily solving the problem of compuating the graphical
derivation of the normal cone mapping. In 2015, following this scheme for the case
1
Θ := {0Rm1 } × Rm−m
under the metric subregularity constraint qualification, H.

Gfrerer and B. S. Mordukhovich showed that the same result remain to be hold if the
uniform metric regularity condition is replaced by the weaker condition, which is the
the bounded extreme point property (BEPP).
Generally, the result of calculating the graphical derivation by A. L. Dontchev
and R. T. Rockafellar is independent to the results set later. However, basically they
all have the assumption under the metric subregularity qualification and a certain
additional property. This leads to the following natural question: Can we unify the
results of the calculating the graphical derivative of the normal cone mapping by removing the additional property? In other words , whether the formulas for calculating
the graphical derivative of the normal cone mapping mentioned above are still hold
if Mq only assumed to be metric subregular?
In Chapter 2, with the assumption that Mq is metric subregular at the reference
point and Θ is a polyhedral convex set, removing the additional property, we successfully proved that the mentioned formula for calculating the graphical derivative
of the normal cone mapping is still hold and thus responds affirmatively to the above
question. To establish this formula, we used the proof scheme of H. Gfrerer and


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J. V. Outrata combining with an idea of A. D. Ioffe and J. V. Outrata. Thank to
this formula, we obtained formulas for computing the graphical derivative of solution mappings and characterized the isolated calmness of the solution mappings for a
generalized equation class. Our results incorporate with many related results in this
research direction.
Tilt stability is a property of local minimizers guaranteeing the minimizing point
shifts in a Lipschitzian manner under linear perturbations on the objective function
of an optimization problem. This notion was introduced by R. A. Poliquin and R.
T. Rockafellar for problems of unconstrained optimization with extended-real-valued
objective function. Tilt stability is basically equivalent to a uniform second-order
growth condition as well as strong metric regularity of the subdifferential.
The first characterization of tilt stability using second-order generalization differentiation was due to R. A. Poliquin and R. T. Rockafellar in 1998. They proved that
for an unconstrained optimization problem, under mild assumptions of prox-regularity
and subdifferential continuity, a stationary point is a tilt-stable local minimizer if and
only if the second-order limiting subdifferential is positive-definite at the point in
question. Furthermore, using this result together with a formula of A. L. Dontchev
and R. T. Rockafellar for the second-order limiting subdifferential of the indicator
function of a polyhedral convex set, they obtained a second-order characterization of
tilt stability for nonlinear programming problems with linear constraints.
In 2012, by establishing new second-order subdifferential calculi, B. S. Mordukhovich
and R. T. Rockafellar derived second-order characterizations of tilt-stable minimizers for some classes of constrained optimization problems. Among other important
things, they showed that for C 2 -smooth nonlinear programming problems, under the
linear independence constraint qualification (LICQ), a stationary point is a tilt-stable
local minimizer if and only if the strong second-order sufficient condition (SSOSC)
holds. In the same year, under the validity of both the MFCQ and CRCQ, B. S.
Mordukhovich and J. V. Outrata proved that SSOSC is a sufficient condition for a
stationary point to be a tilt-stable local minimizer in nonlinear programming. In
2015, B. S. Mordukhovich and T. T. A. Nghia showed that SSOSC is indeed not a
necessary condition for tilt stability and then introduced the uniform second-order sufficient condition (USOSC) to characterize tilt stability when both MFCQ and CRCQ
occur. Recently, H. Gfrerer and B. S. Mordukhovich obtained some point-based
second-order sufficient conditions for tilt-stable local minimizers under the validity of
both the MSCQ and BEPP. Furthermore, when supplementing either nondegeneracy
in critical directions or the 2-regularity, the point-based second-order characterization
of tilt stability were established.
Instead of using the second-order subdifferential, we mainly use the subgradient


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graphical derivative of an extended-real-valued function to characterize tilt stability.
This tilt stability approach has never been applied by other researchers. We note
that one of the biggest advantages of this approach is the workable computation
of the graphical derivative in various important cases under very mild assumptions
in initial data. Furthermore, several results on tilt stability were established based
on the calculation of the subgradient graphical derivative as a mediate step. These
observations lead us to the following natural questions:
Is it possible to use the subgradient graphical derivative to characterize tilt stability of local minimizers for unconstrained optimization problems in which the objective
function is prox-regular and subdifferentially continuous? If yes, is such a characterization useful in helping us to improve the knowledge of tilt stability for nonlinear
programming problems? Is it possible to remove prox-regular condition?
Chapter 3 of the thesis will answer these questions in a sufficient way, as follows:
We have established tilt stability characteristics of local minimizer for the unconstrained optimal problem via the subgradient graphical derivative. Applying this
result to the nonlinear programming problem under MSCQ, we obtained the necessary and sufficient conditions for tilt-stable local minimizer.
7.2. Research Organization
The contents of this dissertation are divided into three chapters.
Chapter 1 is devoted to present the preparatory knowledge as a basis for introducing the main results of the thesis in the remaining chapters.
Chapter 2 focuses on studying the formula for computing the graphical derivative
of normal cone mapping in case Θ is a polyhedral convex set with Mq which is metric
subregular and its applications. Section 2.1, we present the formula for computing
the graphical derivative of the normal cone mappings. Then, in section 2.2, we show
how to use this formula to compute the graphical derivative of solution mappings as
well as derive the new results on the isolated calmness for generalized equations and
stationary point mappings.
Chapter 3 presents the results on tilt stability of local minimizer of for optimization problem. In section 3.1, we establish a new second-order characterization of
tilt-stable local minimizers for unconstrained optimization problems in which the objective function is prox-regular and subdifferentially continuous. Based on the results
obtained in section 3.1, section 2.1 and some other authors’ results, section 3.2 establishes the necessary and sufficient conditions in order that a stationary point of a
nonlinear programming problem under MSCQ is a tilt-stable local minimizer.


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CHAPTER 1
PRELIMINARIES

In this thesis, all spaces are assumed to be Euclidean spaces with scalar product
·, · and Euclidean norm · .

1.1

Basic notions

This section recalls some notions and their properties from variational analysis,
noted from which are used in the sequel.
1.1.1 Definition. Mapping F for each x ∈ Rn to one and only one set F (x) ⊂ Rm
is called set-valued mapping from Rn to Rm and denoted by F : Rn ⇒ Rm .
If for every x ∈ Rn set F (x) has only one element, then we say F is a single
mapping from Rn to Rm . We usually use the standard notation F : Rn → Rm .
The domain, range and graph to F : Rn ⇒ Rm is defined by
domF := x ∈ Rn | F (x) = ∅ ,
rgeF := y ∈ Rm | ∃x ∈ Rn such that y ∈ F (x) ,
gphF := (x, y) ∈ Rn × Rm | y ∈ F (x) ,
respectively. The inverse mapping F −1 : Rm ⇒ Rn to F is defined by
F −1 (y) = x ∈ Rn | y ∈ F (x) , for all y ∈ Rm .
1.1.2 Definition. Let Ω be a nonempty subset of Rn .
(i) The (Bouligand-Severi)tangent/contingent cone to Ω at x¯ ∈ Ω is given by
TΩ (¯
x) := v ∈ Rn | there exist tk ↓ 0, vk → v with x¯ + tk vk ∈ Ω for all k ∈ N .
(ii) The (Fr´echet) regular normal cone to Ω at x¯ ∈ Ω is defined by
NΩ (¯
x) := v ∈ Rn | lim sup


x→¯
x

v, x − x¯
≤0 ,
x − x¯


10


where x → x¯ means that x → x¯ with x ∈ Ω.
(iii) The (Mordukhovich) limiting/basic normal cone to Ω at x¯ ∈ Ω is defined by
NΩ (¯
x) = v ∈ Rn | there exist xk → x¯ and vk ∈ NΩ (xk ) with vk → v .
If x¯ ∈ Ω, put NΩ (¯
x) = NΩ (¯
x) := ∅ by convention.
1.1.4 Definition. Consider the set-valued mapping F : Rn ⇒ Rm with domF = ∅.
(i) Given a point x¯ ∈ domF, the graphical derivative of F at x¯ for y¯ ∈ F (¯
x) is the
n
m
set-valued mapping DF (¯
x|¯
y ) : R ⇒ R defined by
DF (¯
x|¯
y )(v) := w ∈ Rm | (v, w) ∈ TgphF (¯
x, y¯)

for all v ∈ Rn ,

that is, gphDF (¯
x|¯
y ) := TgphF (¯
x, y¯).
(ii) The regular coderivative of F at a given point (¯
x, y¯) ∈ gphF is the set-valued
mapping D∗ F (¯
x, y¯) : Rm ⇒ Rn defined by
D∗ F (¯
x, y¯)(y ∗ ) := x∗ ∈ Rn | (x∗ , −y ∗ ) ∈ NgphF (¯
x, y¯)

for all y ∗ ∈ Rm .

In the case F (¯
x) = {¯
y }, one writes DF (¯
x) and D∗ F (¯
x) for DF (¯
x|¯
y ) and D∗ F (¯
x, y¯),
respectively.
We note that if F : Rn → Rm is a single-valued mapping that is differentiable at
x¯, then DF (¯
x) = ∇F (¯
x) and D∗ F (¯
x) = ∇F (¯
x)∗ .
1.1.6 Definition. Let ϕ : Rn → R := R ∪ {±∞} and x¯ ∈ Rn with y¯ := ϕ(¯
x) finite.
(i) The regular subdifferential of ϕ at x¯ is defined by
∂ϕ(¯
x) := x∗ ∈ Rn | (x∗ , −1) ∈ Nepiϕ (¯
x, y¯) ,
where epiϕ := (x, α) ∈ Rn × R | α ≥ ϕ(x) is the epigraph of ϕ.
(ii) The limiting subdifferential of ϕ at x¯ is defined by
∂ϕ(¯
x) := x∗ ∈ Rn | (x∗ , −1) ∈ Nepiϕ (¯
x, y¯) .
If |ϕ(¯
x)| = ∞, then put ∂ϕ(¯
x) = ∂ϕ(¯
x) := ∅ by convention.
Note that ∂ϕ(¯
x) ⊂ ∂ϕ(¯
x) and if ϕ is a convex function, then both ∂ϕ(¯
x) and ∂ϕ(¯
x)
coincide with the subdifferential in the sense of convex analysis:
∂ϕ(¯
x) = ∂ϕ(¯
x) = x∗ ∈ Rn | x∗ , x − x¯ ≤ ϕ(x) − ϕ(¯
x) for all x ∈ Rn .
1.1.8 Definition. Let f : Rn → R be an extended-real-valued function.
(i) The domain of f is defined by domf := x ∈ Rn | f (x) < ∞ .
(ii) The function f is said to be proper if domf = ∅ and f (x) > −∞, ∀x ∈ Rn .


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(iii) We say f is lower semicontinuous (l.s.c.) at x if lim inf f (u) ≥ f (x).
u→x

(iv) The function f is called prox-regular at x¯ ∈ dom f for v¯ ∈ ∂f (¯
x) if there are
reals r, ε > 0 such that for all x, u ∈ Bε (¯
x) with |f (u) − f (¯
x)| < ε we have
f (x) ≥ f (u) + v, x − u −

r
x − u 2,
2

(1.1)

for all v ∈ ∂f (x) ∩ Bε (¯
v ).
(v) The function f is called subdifferentially continuous at x¯ for v¯ ∈ ∂f (¯
x) for all
sequences xi → x¯ and vi → v¯ with vi ∈ ∂f (xi ), we have f (xi ) → f (¯
x).

1.2

Regularty properties and qualification conditions

Firstly, we recall one important property of set-valued mapping known by the
name of the metric regularity as follow.
1.2.1 Definition. The set-valued mapping F : Rn ⇒ Rm is said to be metrically
regular around (¯
x, y¯) ∈ gph F with modulus κ > 0 if there exist neighborhoods U of
x¯ and V of y¯ such that
dF −1 (y) (x) ≤ κdF (x) (y),

for all (x, y) ∈ U × V.

(1.2)

The regular property of much interest in this thesis is the metric subregularity,
which is given by A. D. Ioffe and defined as follow.
1.2.5 Definition. The set-valued mapping F : Rn ⇒ Rm is said to be metrically
subregular at (¯
x, y¯) ∈ gphF with modulus κ > 0 if there exist r > 0, such that
dF −1 (¯y) (x) ≤ κdF (x) (¯
y ), for all x ∈ Br (¯
x).

(1.3)

The infimum of all such κ is the modulus of metric subregularity and is denoted by
subreg F (¯
x|¯
y ).
Using the metric subregularity, H. Gfrerer and B. S. Mordukhovich introduced
metric subregularity constraint qualification in the nonlinear programming setting
and based on this basis, we introduced for the general case as follow.
1.2.8 Definition. Consider the constraint set
Γ := {x ∈ Rn | q(x) ∈ Θ},
where q : Rn → Rm is a continuously differentiable mapping and Θ is a nonempty
closed set in Rm . One says that the metric subregularity constraint qualification
(MSCQ) holds at x¯ ∈ Γ if Mq (x) := q(x) − Θ is metrically subregular at (¯
x, 0).


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Next, we recall some well-known constraint qualifications in nonlinear programming.
1.2.9 Definition. Considering Γ is the feasible set of the nonlinear programming
Γ := x ∈ Rn | q(x) ∈ Rm
− ,
where q(x) := q1 (x), ..., qm (x)) with qi : Rn → R is a continuously differentiable
mapping, for all i = 1, 2..., m.
(i) The Mangasarian–Fromovitz constraint qualification (MFCQ) is said to hold
at point x¯ ∈ Γ if there exists a vector d ∈ Rn such that
∇qi (¯
x), d < 0 for all i ∈ I(¯
x),
where I(¯
x) := i ∈ {1, . . . , m} | qi (¯
x) = 0 is the active index set at x¯ ∈ Γ.
(ii) The constant rank constraint qualification (CRCQ) is said to hold at x¯ ∈ Γ if
there is a neighborhood U of x¯ such that the gradient system {∇qi (x)| i ∈ J} has the
same rank in U for any index J ⊂ I(¯
x).
(iii) The linear independence constraint qualification (LICQ) is said to hold at
x¯ ∈ Γ if the gradient system {∇qi (¯
x), i ∈ I(¯
x)} are linearly independent.
(iv) The constraint set Γ is said to have the bounded extreme point property
(BEPP) at x¯ ∈ Γ if there exist real numbers κ > 0 and r > 0 such that
E(x, x∗ ) ⊂ κ x∗ B for all x ∈ Γ ∩ Br (¯
x) and x∗ ∈ Rn ,
where E(x, x∗ ) denotes the set of extremal points of Λ(x, x∗ ), with Λ(x, x∗ ) denotes
the set of multipliers
T

Λ(x, x∗ ) := λ ∈ Rm
/ I(x) .
+ | ∇q(x) λ = x , λi = 0 for i ∈


13

CHAPTER 2
GRAPHICAL DERIVATIVE OF NORMAL CONE MAPPING UNDER
THE METRIC SUBREGULARITY CONDITION

This chapter presents the formula for computation the graphical derivative of
normal cone mapping under the metric subregularity constraint qualification and its
applications.

2.1

Computation of graphical derivative for a class of normal
cone mappings

In this section, we suppose that
Γ := {x | q(x) ∈ Θ},
where q : Rn → Rm is a twice continuously differentiable mapping and Θ is a
nonempty polyhedral convex set in Rm
−.

For each x¯ ∈ Γ and x¯ ∈ NΓ (¯
x), put
Λ := {λ ∈ NΘ (¯
y ) | ∇q(¯
x)T λ = x¯∗ },
with y¯ := q(¯
x). We denote by
Iq (¯
x) := {i = 1, 2, . . . , | bi , y¯ = αi }
the active index set of Γ at x¯ and
K := TΓ (¯
x) ∩ {¯
x∗ }⊥
the critical cone of Γ at x¯.
To proceed, we need the following result, which provides a useful formula for
computing the normal cone to the critical cone in terms of the initial data.
2.1.1 Lemma. Suppose that MSCQ is valid at x¯ and y¯ := q(¯
x). Then, for each
v ∈ K and λ ∈ Λ, one has
NK (v) = ∇q(¯
x)T µ | µT ∇q(¯
x)v = 0, µ ∈ TNΘ (¯y) (λ) ,

(2.1)


14

where NΘ (¯
y ) = pos{bi | i ∈ Iq (¯
x) and TNΘ (¯y) (λ) = pos{bi | i ∈ Iq (¯
x) − R+ λ. Consequently, for v ∈ K, one has





T

(2.2)
NK (v) =
ti bi ∇q(¯
x) − t0 x¯ | t0 , ti ∈ R+ , i ∈ Iq (¯
x) ∩ v .


i∈Iq (¯
x)

We now arrive at the main result of this section, which provides a formula for
the graphical derivative of the normal cone mapping NΓ in the case where Θ is a
nonempty polyhedral convex set under a very weak condition (MSCQ).
2.1.10 Theorem. Let MSCQ be satisfied at x¯ ∈ Γ and x¯∗ ∈ NΓ (¯
x). Then, one has
TgphNΓ (¯
x, x¯∗ ) = (v, v ∗ ) ∈ Rn × Rn | ∃λ ∈ Λ(v) :
v ∗ ∈ ∇2 λT q (¯
x)v + NK (v) .

(2.3)

Therefore, the graphical derivative of the normal cone mapping NΓ (x) is given by
DNΓ (¯
x|¯
x∗ )(v) = ∇2 λT q (¯
x)v | λ ∈ Λ(v) + NK (v).

(2.4)

Here Λ(v) is the optimal solution set of the linear programming LP(v), and the cone
NK (v) can be computed by (2.2).
For the case where Γ is a feasible set of a nonlinear programming, it may happen
that the assumption of Theorem 2.1.10 is fulfilled, while the bounded extreme point
property is invalid.
2.1.12 Example. Let q : R2 ⇒ R2 be given by q(x) := (−x1 , x1 − x21 x22 ),
Θ := {(0, 0)}, Γ := x ∈ R2 | q(x) ∈ Θ = {0} × R and x¯ := (0, 0).
Then, the assumption of Theorem 2.1.10 is fulfilled, while the bounded extreme point
property is invalid at x¯.
The next result gives us a formula for computing the regular coderivative of the
normal cone mapping, which is a direct consequence of Theorem 2.1.10.
2.1.13. Corollary. Under the assumption of Theorem 2.1.10, one has
D∗ NΓ (¯
x, x¯∗ )(u∗ ) = u |

u, v − u∗ , ∇2 λT q (¯
x)v ≤ 0,
for all v ∈ K, λ ∈ Λ(v), −u∗ ∈ TK (v) .

2.2

Application to generalized equation

We first consider the parametric generalized equation of the form:
0 ∈ F (x, y) + NΓ (x),

(2.5)


15

where F : Rn × Rs → Rn is a continuous differentiable mapping, x is a variable, y is
a parameter, and Γ := {x ∈ Rn | q(x) ∈ Θ}, Θ is a nonempty polyhedral set in Rm ,
and q : Rn → Rm is a twice continuously differentiable mapping. Denote by S the
solution mapping to (2.5) given by
S(y) := x ∈ Rn | 0 ∈ F (x, y) + NΓ (x) .
2.2.2 Theorem. Let (¯
y , x¯) ∈ gphS and let Mq be metrically subregular at (¯
x, 0).
Then, one has
DS(¯
y |¯
x)(z) ⊂ v | − ∇y F (¯
x, y¯)z ∈ ∇x F (¯
x, y¯)v +

∇2 λT q (¯
x)v :
λ ∈ Λ(v) + NK (v) ,

(2.6)

for all z ∈ Rs . Inclusion (2.6) holds as equality if assume further that ∇y F (¯
x, y¯) is
∗ ⊥

surjective. Here K := TΓ (¯
x) ∩ {¯
x } with x¯ := −F (¯
x, y¯), and Λ(v) is the optimal
solution set of LP(v).
If q is an affine mapping, then {∇2 λT q (¯
x)v | λ ∈ Λ(v)} = {0} and Mq is automatically metrically subregular. Hence, in this case, formula (2.6) can be much more
simplified.
2.2.3 Corollary. Consider the generalized equation (2.5) with q : Rn → Rm being an
affine mapping. For any (¯
y , x¯) ∈ gphS and x¯∗ := −F (¯
x, y¯), one has
DS(¯
y |¯
x)(z) ⊂ v | − ∇y F (¯
x, y¯)z ∈ ∇x F (¯
x, y¯)v + NK (v) , for all z ∈ Rs .

(2.7)

Inclusion (2.7) holds as equality if in addition ∇y F (¯
x, y¯) is surjective.
1
2.2.6 Corollary. Consider (2.5) with Θ := {0Rm1 }×Rm−m
v`a (¯
y , x¯) ∈ gphS. Assume

that CRCQ is fulfilled at x¯. Then, one has

x)v + NK (v) ,
DS(¯
y |¯
x)(z) ⊂ v | − ∇y F (¯
x, y¯)z ∈ ∇x F (¯
x, y¯)v + ∇2 λT q (¯

(2.8)

for all z ∈ Rs and λ ∈ Λ.
Inclusion (2.8) holds as equality if in addition ∇y F (¯
x, y¯) is surjective.
Next, we consider the so-called isolated calmness of S. This property introduced
by A. L. Dontchev, which is an important property in variational analysis.
2.2.7 Definition. The set-valued mapping F : Rs ⇒ Rn is said to be isolated calm
at (¯
y , x¯) ∈ gphF if there exist κ, r > 0 such that
F (y) ∩ Br (¯
x) ⊂ {¯
x} + κ y − y¯ BRn ,

for all y ∈ Br (¯
y ).


16

The following theorem gives a characterization of the isolated calmness of the
solution mapping.
2.2.9 Theorem. Let (¯
y , x¯) ∈ gphS and let Mq be metrically subregular at (¯
x, 0). If
the implication
0 ∈ ∇x L(¯
x, y¯, λ)v + NK (v)
⇒ v = 0.
(2.9)
λ ∈ Λ(v), v ∈ Rn
is valid, then S is isolated calm at (¯
y , x¯). The reverse statement also holds if ∇y F (¯
x, y¯)
n
s
m
n
is surjective. Here L : R × R × R → R is given by
L(x, y, λ) := F (x, y) + ∇q(x)T λ.
2.2.10 Corollary. Consider the generalized equation (2.5) with Γ := Θ, n = m and
q := In the identity mapping in Rn . Let (¯
y , x¯) ∈ gphS and x¯∗ := −F (¯
x, y¯). Then, if
(∇x F (¯
x, y¯) + NK )−1 (0) = {0}

(2.10)

then S is isolated calm at (¯
y , x¯).
Moreover, if, in addition, rank∇y F (¯
x, y¯) = n then property (2.10) is necessary and
sufficient for S to have the isolated calmness at (¯
y , x¯).
Next, we now consider the parametric generalized equation
w ∈ F (x, y) + NΓ (x),

(2.11)

where F : Rn × Rs → Rn is a continuous differentiable mapping, x is the variable,
and p := (y, w) represents the parameter, and Γ := {x ∈ Rn | q(x) ∈ Θ} with
Θ ⊂ Rm being a polyhedron and q : Rn → Rm being a twice continuously differentiable
mapping. Let S : Rs × Rn ⇒ Rn be the solution mapping of (2.11), that is,
S(p) := x ∈ Rn | w ∈ F (x, y) + NΓ (x)

for all p := (y, w) ∈ Rs × Rn .

(2.12)

The following result gives us a characterization of the isolated calmness of the
mapping S(p).
2.2.11 Theorem. Let (¯
p, x¯) ∈ gphS and let Mq be metrically subregular at (¯
x, 0).
Then, the following assertions are equivalent.
(i) The implication
0 ∈ ∇x L(¯
x, p¯, λ)v + NK (v)
λ ∈ Λ(v), v ∈ Rn

⇒v=0

is valid.
(ii) The solution mapping S(p) is isolated calm at (¯
p, x¯).
Here L : Rn × Rs × Rn × Rm → Rn is defined by
L(x, p, λ) := F (x, y) − w + ∇q(x)T λ with p := (y, w).


17

Finally, we consider the parametric optimization problem
min g(x, y) − w, x | x ∈ Γ ,

(2.13)

where g : Rn × Rs → R is twice continuously differentiable, the feasible set Γ := {x ∈
Rn | q(x) ∈ Θ}, Θ is a nonempty polyhedral convex set in Rm , q : Rn → Rm is twice
continuously differentiable, x is a variable, and y ∈ Rs and w ∈ Rn are parameters.
Noting that the set-valued mapping XKKT : Rs × Rn ⇒ Rn defined by
XKKT (p) := x ∈ Rn | 0 ∈ ∇x g(x, y) − w + NΓ (x) , p := (y, w) ∈ Rs × Rn ,
is called the stationary point mapping of (2.13).
Obviously, the stationary point mapping XKKT (p) is a special case of the setvalued mapping S(p) given by (2.12). So, by Theorem 2.2.11, we get the corresponding
characterization of isolated calmness of the stationary point mapping of (2.13).
2.2.12 Corollary.Let (¯
p, x¯) ∈ gphXKKT and let Mq be metrically subregular at (¯
x, 0).
Then, the following assertions are equivalent.
(i) The implication
0 ∈ ∇x L(¯
x, p¯, λ)v + NK (v)
λ ∈ Λ(v), v ∈ Rn

⇒v=0

is valid.
(ii) The mapping XKKT (p) is isolated calm at (¯
p, x¯).
Here L : Rn × Rs × Rn × Rm → Rn is defined by
L(x, p, λ) := ∇x g(x, y) − w + ∇q(x)T λ with p := (y, w).


18

CHAPTER 3
TILT STABILITY VIA SUBGRADIENT GRAPHICAL DERIVATIVE
FOR A CLASS OF OPTIMIZATION PROBLEMS WITH THE
PROX-REGULARITY ASSUMPTION

In this chapter, we provide a new second-order characterization via the subgradient graphical derivative of tilt-stable local minimizers for unconstrained optimization
problems in which the objective function is prox-regular and subdifferentially continuous. In the next step, applying the feature set above to nonlinear programming
under MSCQ, we obtained a second-order tilt stable characteristic via the relaxed
uniform second-order sufficient condition and we continuously obtained the pointbased second-order sufficient condition so that the stationary point of the problem is
a tilt stable local minimizer. Finally, when applying to the quadratic program with
a quadratic inequality constraint, we obtained a simpler feature of the tilt stability.

3.1

Second-order characterizations of tilt stability for a class
of unconstrained optimization problems

First we recall the definition of tilt stability, this concept due to R. A. Poliquin
and R. T. Rockafellar is defined in 1998.
3.1.1 Definition. Given f : Rn → R, a point x¯ ∈ dom f is a tilt-stable local minimizer
of f with modulus κ > 0 if there is a number γ > 0 such that the mapping
Mγ : v → argmin f (x) − v, x x ∈ Bγ (¯
x)
is single valued and Lipschitz continuous with constant κ on some neighborhood of
0 ∈ Rn with Mγ (0) = x¯.
In this case, we denote
tilt (f, x¯) := inf κ| x¯ is a tilt-stable minimizer of f with modulus κ > 0 .
The following theorem provides the characterizations for tilt stability via the
subgradient graphical derivative, which will be the main tool in investigating tilt
stability for nonlinear programming problems in section 3.2.


19

3.1.3 Theorem. Let f : Rn → R be an l.s.c. proper function with x¯ ∈ dom f and
0 ∈ ∂f (¯
x). Assume that f is both prox-regular and subdifferentially continuous at x¯
for v¯ = 0. Then the following assertions are equivalent.
(i) The point x¯ is a tilt-stable local minimizer of f with modulus κ > 0.
(ii) There is a constant η > 0 such that for all w ∈ Rn we have
1
w 2 whenever z ∈ D∂f (u, v)(w), (u, v) ∈ gph ∂f ∩ Bη (¯
x, 0).
κ
Furthermore, we have
z, w ≥

tilt (f, x¯) = inf sup
η>0

w 2
z ∈ D∂f (u|v)(w), (u, v) ∈ gph ∂f ∩ Bη (¯
x, 0)
z, w

(3.1)

(3.2)

with the convention that 0/0 = 0.
The following two examples show that the prox-regularity assumption is essential
for both (i) ⇒ (ii) and (ii) ⇒ (i) in Theorem 3.1.3.
3.1.4 Example . Let f : R → R be the function defined by

1
1

1
1
1
if
≤ |x| ≤ ,


,
 min 1 + |x| −
n+1
n

n
n(n
+
1)
n
n

N
,
f (x) :=


if x = 0,

 0
1
if |x| > 1.
Then x¯ = 0 is a tilt-stable local minimizer and f is subdifferentially continuous but
not prox-regular at x¯ = 0 for v¯ = 0, while assertion (ii) of Theorem 3.1.3 is invalid.
3.1.5 Example. Let f : R2 → R be the function defined by
f (x) := x21 + x22 + δΩ (x1 , x2 ),
when Ω := {(x1 , x2 ) ∈ R2 | x1 x2 = 0} and x = (x1 , x2 ). Then x¯ = 0 is not a tilt-stable
local minimizer and f is subdifferentially continuous but not prox-regular at x¯ = 0
for v¯ = 0 ∈ ∂f (0), while assertion (ii) of Theorem 3.1.3 holds.

3.2

Tilt stability in nonlinear programming under the metric
subregular condition

Consider the nonlinear programming problem
min g(x) | qi (x) ≤ 0, i = 1, 2, ..., m ,

(3.3)

where g : Rn → R and qi : Rn → R are twice continuously differentiable functions.
Let q(x) := q1 (x), q2 (x), ..., qm (x) for x ∈ Rn and let Γ := {x ∈ Rn | q(x) ∈ Rm
− }.
Based on Definition 3.1.1, people define the tilt stable local minimizer of problem
(3.3) as follows.


20

3.2.1 Definition. We say the point x¯ ∈ Γ is a tilt stable local minimizer of problem
(3.3) with modulus κ > 0 if there exists γ > 0 such that the solution mapping
˜ γ (v) := argmin g(x) − v, x | q(x) ∈ Rm
M
x)
− , x ∈ Bγ (¯
is single valued and Lipschitz continuous with constant κ on some neighborhood of
˜ γ (0) = x¯.
0 ∈ Rn with M
Thus, x¯ is a tilt stable local minimizer of problem (3.3) if and only if it is a tilt
stable local minimizer of the function f := g + δΓ . Denote
tilt(g, q, x¯) := tilt(f, x¯).
For x ∈ Γ, x∗ ∈ NΓ (x), denote I(x) := i ∈ {1, ..., m} | qi (x) = 0 ,
T

Λ(x, x∗ ) := λ ∈ Rm
/ I(x) ,
+ | ∇q(x) λ = x , λi = 0 for i ∈

K(x, x∗ ) := TΓ (x) ∩ {x∗ }⊥ ;

I + (λ) := {i = 1, . . . , m | λi > 0} for λ ∈ Rm
+.

Next we introduce a new second-order sufficient condition, which is motivated by
the so-called uniform second-order sufficient condition (USOSC) introduced by B. S.
Mordukhovich and T. T. A. Nghia in 2015.
3.2.2 Definition . We say that the relaxed uniform second-order sufficient condition
(RUSOSC) holds at x¯ ∈ Γ with modulus > 0 if there exists η > 0 such that
∇2xx L(x, λ)w, w ≥

w 2,

(3.4)

whenever (x, v) ∈ gphΨ ∩ Bη (¯
x, 0), here Ψ : Rn ⇒ Rn , Ψ(x) := ∇g(x) + NΓ (x) and
λ ∈ Λ x, v − ∇g(x); w with w ∈ Rn satisfying
∇qi (x), w = 0 for i ∈ I + (λ), ∇qi (x), w ≥ 0 for i ∈ I(x)\I + (λ).

(3.5)

We now arrive at the first result of this section, which gives us a fuzzy characterization of tilt stable local minimizers in terms of RUSOSC and its modification for
nonlinear programming problems.
3.2.5 Theorem. Given a stationary point x¯ ∈ Γ and real numbers κ, γ > 0, suppose
that MSCQ is fulfilled at x¯ and γ > subreg Mq (¯
x|0). Then the following assertions
are equivalent.
(i) The point x¯ is a tilt-stable local minimizer of problem (3.3) with modulus κ.
(ii) The RUSOSC is satisfied at x¯ with modulus

:= κ−1 .

(iii) There exists η > 0 such that
∇2xx L(x, λ)w, w ≥

1
w 2,
κ


21

whenever (x, v) ∈ gphΨ ∩ Bη (¯
x, 0) and λ ∈ Λ x, v − ∇g(x); w ∩ γ v − ∇g(x) BRm
for w ∈ Rn satisfying
∇qi (x), w = 0 for i ∈ I + (λ) and ∇qi (x), w ≥ 0 for i ∈ I(x)\I + (λ),
where Ψ : Rn ⇒ Rn , Ψ(x) := ∇g(x) + NΓ (x).
3.2.6 Corollary. Let x¯ be a stationary point of (3.3) at which CRCQ holds. Then,
the following assertions are equivalent.
(i) The point x¯ is a tilt stable local minimizer of (3.3) with modulus κ > 0.
(ii) There exists η > 0 such that
∇2xx L(x, λ)w, w ≥

1
w
κ

2

whenever (x, v) ∈ gphΨ∩ Bη (¯
x, 0), λ ∈ Λ x, v−∇g(x) , ∇qi (x), w = 0 for i ∈ I + (λ)
and ∇qi (x), w ≥ 0 for i ∈ I(x)\I + (λ).
We next establish a point-based sufficient condition for tilt stability under MSCQ.
3.2.9 Theorem. Given a stationary point x¯ ∈ Γ and real numbers κ, γ > 0, suppose
that MSCQ is fulfilled at x¯ and γ > subreg Qq (¯
x|0) and that the following second-order
condition holds:
x, λ)w, w > κ1 w 2
∇2xx L(¯
whenever w = 0 with ∇qi (¯
x), w = 0, i ∈ I + (λ), and λ ∈ ∆(¯
x),
Λ x¯, −∇g(¯
x); v

where ∆(¯
x) :=

(3.6)

γ ∇g(¯
x) BRm .

0=v∈K x
¯,−∇g(¯
x)

Then x¯ is a tilt-stable local minimizer of (3.3) with modulus κ. Furthermore, we have
the estimation:
tilt(g, q, x¯) ≤ sup

w 2
| λ ∈ ∆(¯
x), ∇qi (¯
x), w = 0, i ∈ I + (λ)
2
∇xx L(¯
x, λ)w, w

<∞
(3.7)

with the convention that 0/0 := 0 in (3.7).
The following theorem provides another second-order sufficient condition for tiltstable local minimizers by surpassing the appearance κ trong (3.6).
3.2.11 Theorem. Given a stationary point x¯ ∈ Γ and a real number γ > 0, suppose
that MSCQ is fulfilled at x¯ and γ > subreg Qq (¯
x|0) and that the following second-order
condition holds:
w, ∇2xx L(¯
x, λ)w > 0 whenever w = 0 with ∇qi (¯
x), w = 0, i ∈ I + (λ),
and
λ ∈ ∆(¯
x) :=
Λ x¯, −∇g(¯
x); v
γ ∇g(¯
x) BRm .
(3.8)
0=v∈K x
¯,−∇g(¯
x)

Then, x¯ is a tilt-stable local minimizer for (3.3).


22

3.2.12 Definition. We say that the strong second-order sufficient condition
(SSOSC) holds at x¯ ∈ Γ if for all λ ∈ Λ x¯, −∇g(¯
x) we have
w, ∇2xx L(¯
x, λ)w > 0

(3.9)

whenever w = 0 with ∇qi (¯
x), w = 0, i ∈ I + (λ).
In 2015, under MFCQ and CRCQ, B. S. Mordukhovich and J. V. Outrata proved
that the tilt-stability is satisfied under SSOSC. In the following corollary we also
obtain this property but under condition MSCQ.
3.2.13 Corollary. Let x¯ be a stationary point of (3.3) at which MSCQ is valid.
Then, x¯ is a tilt-stable local minimizer of (3.3) provided SSOSC is satisfied at x¯.
3.2.15 Definition. The twice differentiable mapping g : Rm → Rs is said to be
2-regular at a given point x¯ ∈ Rm in direction v ∈ Rm if for any p ∈ Rs , the system
∇g(¯
x)u + [∇2 g(¯
x)v, w] = p,

∇g(¯
x)w = 0

admits a solution (u, w) ∈ Rm × Rm , where [∇2 g(¯
x)v, w] denotes the s-vector column
with the entrices ∇2 gi (¯
x)v, w , i = 1, ..., s.
x), define
For each x¯ ∈ Γ and v ∈ TΓlin (¯
I(¯
x, v) := i ∈ I(¯
x)
Ξ(¯
x, v) := z ∈ Rn
C(¯
x, v) := C

∇qi (¯
x), v = 0 ,

∇qi (¯
x), z + v, ∇2 qi (¯
x)v ≤ 0 for i ∈ I(¯
x) ,
∇qi (¯
x), z + v, ∇2 qi (¯
x)v = 0

C = i ∈ I(¯
x, v) |

with z ∈ Ξ(¯
x, v) .
3.2.16 Definition. Given a point x¯ ∈ Γ and v ∈ K(¯
x, −∇g(¯
x)). The point x¯ is said
to be nondegenerate in the direction v if the set Λ x¯, −∇g(¯
x); v is a singleton.
The following result provides a second-order necessary condition for tilt-stability,
which shows that, under either directional nondegeneracy or 2-regularity, the pointbased second-order sufficient condition established in Theorem 3.2.9 is “not too far”
from the necessary one.
3.2.20 Theorem. Given positive real numbers κ and γ, let x¯ be a tilt-stable local
minimizer of (3.3) with modulus κ, and let MSCQ hold at x¯ and γ > subreg Qq (¯
x|0).
Suppose that for every v ∈ K x¯, −∇g(¯
x) \{0} one of the following conditions is
satisfied:
(a) x¯ is nondegenerate in the direction v;
(b) for each λ ∈ Λ x¯, −∇g(¯
x); v ∩ γ ∇g(¯
x) BRm there is a maximal element
C ∈ C(¯
x, v) with I + (λ) ⊂ C such that the mapping (qi )i∈C is 2-regular at x¯ in the
direction v.


23

Then, one has
w, ∇2xx L(¯
x, λ)w ≥

1
w
κ

2

whenever ∇qi (¯
x), w = 0, i ∈ I + (λ), λ ∈ ∆(¯
x), (3.10)
Λ x¯, −∇g(¯
x); v ∩γ ∇g(¯
x) BRm . Furthermore, we have

where ∆(¯
x) :=
0=v∈K x
¯,−∇g(¯
x)

tilt(g, q, x¯) = sup

w 2
w, ∇2xx L(¯
x, λ)w

λ ∈ ∆(¯
x), ∇qi (¯
x), w = 0, i ∈ I + (λ)
(3.11)

with the convention that 0/0 := 0 in (3.11).
By combining Theorem 3.2.9, Theorem 3.2.11 and Theorem 3.2.20, we arrive at
the following result, which provides second-order characterizations of tilt-stable local
minimizers for (3.3).
3.2.21 Corollary. Let x¯ be a stationary point of (3.3) at which MSCQ is valid,
and let γ > subreg Qq (¯
x|0). Suppose that for every 0 = v ∈ K x¯, −∇g(¯
x) one of
the conditions (a) and (b) given in Theorem 3.2.20 is satisfied. Then, the following
assertions hold:
(i) Given κ > 0, the point x¯ is a tilt-stable local minimizer of (3.3) with any
modulus κ > κ if and only if the second-order condition (3.10) is valid;
(ii) The point x¯ is a tilt-stable minimizer of (3.3) if and only if the positivedefiniteness condition (3.8) is valid.
Finally, we consider the quadratic program of the form:
min g(x) | q(x) ≤ 0 ,

x∈Rn

(3.12)

where g(x) := 21 xT Ax + aT x, q(x) = q0 (x) := 12 xT B0 x + bT0 x + β0 , with A, B0 ∈ S n ,
a, b0 ∈ Rn and β0 ∈ R.
By using some known results on tilt stability in nonlinear programming, we establish quite simple characterizations of tilt-stable local minimizer for (3.12) under
the metric subregularity constraint qualification.
3.2.23 Theorem. Let x¯ be a stationary point of (3.12) with q(¯
x) = 0. Then the
following assertions hold:
(i) When ∇q(¯
x) = 0 and ∇g(¯
x) = 0, x¯ is a tilt stable local minimizer if and only if
A is positively definite.
(ii) When ∇q(¯
x) = 0 and ∇g(¯
x) = 0, x¯ is a tilt-stable local minimizer if and only if
w,

B0 x¯ + b0 A + A¯
x + a B0 w > 0,

for all w ∈ Rn \{0} with B0 x¯ + b0 , w = 0.
(iii) When ∇q(¯
x) = 0 and MSCQ is valid at x¯, x¯ is a tilt-stable local minimizer if
and only if A is positively definite while −B0 is positively semidefinite.


24

GENERAL CONCLUSIONS AND RECOMMENDATIONS

1. General conclusions
This thesis is intended to study the metric subregularity and its applications. The
main results of the thesis include:
- Establishing a formula for exactly computing the graphical derivative of the
normal cone mapping under the metric subregular constraint qualification. At the
same time, we exhibit formulas for computing the graphical derivative of solution
mappings and present characterizations of the isolated calmness for a broad class of
generalized equations. Our results incorporate many important results in this research
direction.
- Setting up the characterization of the tilt-stable local minimizers for a class
of unconstrained optimization problems with the objective function is prox-regular
and subdifferentially continuous via the uniform positive-definiteness of the subgradient graphical derivative of objective function. Instead of using the second-order
subdifferential, here we used the subgradient graphical derivative to examine tilt stability. This is a new, unprecedented approach used by previous authors. Moreover,
we proved that the prox-regularity of the objective function is essential not only for
the necessary implication but also for the sufficient one.
- Obtaining some second-order necessary and sufficient conditions for tilt stability
in nonlinear programming under the metric subregularity constraint qualification to
be a tilt-stable local minimizer. In particular, we show that each stationary point of
a nonlinear programming problem satisfying MSCQ is a tilt-stable local minimizer
if strong second-order sufficient condition is satisfied. In addition, the quadratic
program with one quadratic inequality constraint satisfies the metric subregular constraint qualification, by exploiting the specificity of the problem, we have come up
with a simple and more explicit characterization of tilt-stable local minimizers.


25

2. Recommendations
We find that the topic of this thesis is still able to continuously develop in the
following directions:
- Using the approach to tilt stability via graphical derivative, examining the tiltstability for the nonpolyhedral conic programs. Recently, Benko et al obtained some
results for the second-order cone programs with this approach. For other cone programs classes, this issue needs further research.
- Investigating whether it is possible to study the full stability according to LevyPoliquin-Rockafellar by using subgradient graphical derivative. Currently, no actual
results have been set in this reseacrh direction except from some characterization of
full stability via the second-order subdifferential.


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