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

3

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

6

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

7

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

8

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

Ω

x→¯

x

v, x − x¯

≤0 ,

x − x¯

10

Ω

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.

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

3

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”.

4

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

6

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

7

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

8

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.

9

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

Ω

x→¯

x

v, x − x¯

≤0 ,

x − x¯

10

Ω

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 .

11

(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).

12

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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