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VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY

INSTITUTE OF MATHEMATICS

DUONG THI KIM HUYEN

STABILITY OF SOME CONSTRAINT SYSTEMS

AND OPTIMIZATION PROBLEMS

Speciality: Applied Mathematics

Speciality code: 9 46 01 12

SUMMARY

DOCTORAL DISSERTATION IN MATHEMATICS

HANOI - 2019

The dissertation was written on the basis of the author’s research works carried at Institute

of Mathematics, Vietnam Academy of Science and Technology.

Supervisor: Prof. Dr.Sc. Nguyen Dong Yen

First referee: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..........................................

Second referee: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.......................................

Third referee: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.........................................

To be defended at the Jury of Institute of Mathematics, Vietnam Academy of Science and

Technology:

...........................................................................

...........................................................................

on . . . . . . . . . . . . . . . . . . . . . , at . . . . . . . . . . . . o’clock . . . . . . . . . . . . . . . . . . . . . . . . . . .

The dissertation is publicly available at:

• The National Library of Vietnam

• The Library of Institute of Mathematics

Introduction

Many real problems lead to formulating equations and solving them. These equations may

contain parameters like initial data or control variables. The solution set of a parametric

equation can be considered as a multifunction (that is, a point-to-set function) of the parameters involved. The latter can be called an implicit multifunction. A natural question is that

“What properties can the implicit multifunction possess?”.

Under suitable differentiability assumptions, classical implicit function theorems have addressed thoroughly the above question from finite-dimensional settings to infinite-dimensional

settings.

Nowadays, the models of interest (for instance, constrained optimization problems) outrun

equations. Thus, Variational Analysis has appeared to meet the need of this increasingly

strong development.

J.-P. Aubin, J.M. Borwein, A.L. Dontchev, B.S. Mordukhovich, H.V. Ngai, S.M. Robinson,

R.T. Rockafellar, M. Th´era, Q.J. Zhu, and other authors, have studied implicit multifunctions

and qualitative aspects of optimization and equilibrium problems by different approaches. In

particular, with the two-volume book “Variational Analysis and Generalized Differentiation”

(2006) and a series of research papers, Mordukhovich has given basic tools (coderivatives,

subdiffentials, normal cones, and calculus rules), fundamental results, and advanced techniques

for qualitative studies of optimization and equilibrium problems. Especially, the fourth chapter

of the book is entirely devoted to such important properties of the solution set of parametric

problems as the Lipschitz stability and metric regularity. These properties indicate good

behaviors of the multifunction in question. The two models considered in that chapter of

Mordukhovich’s book bear the names parametric constraint system and parametric variational

system. More discussions and references on implicit multifunction theorems can be found in

the books by Borwein and Zhu (2015), Dontchev and Rockafellar (2009), and Klatte and

Kummer (2002).

Stability properties like lower semicontinuity, upper semicontinuity, Hausdorff semicontinuity/continuity, H¨older continuity of solution maps and of approximate solution maps can

be studied for very general optimization problems and equilibrium problems (for example,

vector optimization problems, vector variational inequalities, vector equilibrium problems).

The locally convex Hausdorff topological vector spaces setting can be also adopted. Here, it

is not necessary to use the tools from variational analysis and generalized differentiation. We

refer to the works by P.Q. Khanh, L.Q. Anh, and their coauthors for some typical results in

this direction.

Within this dissertation we use coderivative to study three properties of solution maps

in finite-dimensional settings, which include Aubin property (Lipschitz-like property), metric

regularity, and the Robinson stability of solution maps of constraint and variational systems.

Results on these stability properties are applied to studying the solution stability of linear

complementarity problem, affine variational inequalities, and a typical parametric optimization problem. The dissertation has four chapters and a list of references.

1

Chapter 1 collects some basic concepts from Set-Valued Analysis and Variational Analysis and gives a first glance at some properties of multifunctions and key results on implicit

multifunctions.

In Chapter 2, we investigate the Lipschitz-like property and the Robinson stability of the

solution map of a parametric linear constraint system by means of normal coderivative, the

Mordukhovich criterion, and a related theorem due to Levy and Mordukhovich (2004). Among

other things, the obtained results yield uniform local error bounds and traditional local error

bounds for the linear complementarity problem and the general affine variational inequality

problem, as well as verifiable sufficient conditions for the Lipschitz-like property of the solution

map of the linear complementarity problem and a class of affine variational inequalities, where

all components of the problem data are subject to perturbations.

Chapter 3 shows analogues of the results of the previous chapter for the case where the

linear constraint system undergoes linear perturbations.

Finally, in Chapter 4, we analyze the sensitivity of the stationary point set map of a C 2 smooth parametric optimization problem with one C 2 -smooth functional constraint under

total perturbations by applying some results of Levy and Mordukhovich (2004), and Yen

and Yao (2009). We not only show necessary and sufficient conditions for the Lipschitzlike property of the stationary point set map, but also sufficient conditions for its Robinson

stability. These results lead us to new insights into the preceding deep investigations of Levy

and Mordukhovich (2004) and of Qui (2014, 2016) and allow us to revisit and extend several

stability theorems in indefinite quadratic programming.

Chapter 1

Preliminaries

1.1

Basic Concepts from Variational Analysis

In this chapter, several concepts and tools from Variational Analysis are recalled. As a

preparation for the investigations in Chapters 2–4, we present lower and upper estimates for

coderivatives of implicit multifunctions given by Levy and Mordukhovich (2004), Lee and Yen

(2011), as well as the sufficient conditions of Yen and Yao (2009) for the Robinson stability

property of implicit multifunctions.

The Fr´echet normal cone (also called the prenormal cone, or the regular normal cone) to a

set Ω ⊂ Rs at v¯ ∈ Ω is given by

NΩ (¯

v) =

v ∈ Rs : lim sup

Ω

v−

→v¯

Ω

v , v − v¯

≤0 ,

v − v¯

where v −

→ v¯ means v → v¯ with v ∈ Ω. By convention, NΩ (¯

v ) := ∅ when v¯ ∈

/ Ω.

2

Provided that Ω is locally closed around v¯ ∈ Ω, one calls

NΩ (¯

v ) = Lim sup NΩ (v)

v→¯

v

:= v ∈ Rs : ∃ sequences vk → v¯, vk → v ,

with vk ∈ NΩ (vk ) for all k = 1, 2, . . .

the Mordukhovich (or limiting/basic) normal cone to Ω at v¯. If v¯ ∈

/ Ω, then one puts NΩ (¯

v ) = ∅.

A multifunction Φ : Rn ⇒ Rm is said to be locally closed around a point z¯ = (¯

x, y¯) from

n

m

gph Φ := {(x, y) ∈ R × R : y ∈ Φ(x)} if gph Φ is locally closed around z¯. Here, the

product space Rn+m = Rn × Rm is equipped with the topology generated by the sum norm

(x, y) = x + y .

For any z¯ = (¯

x, y¯) ∈ gph Φ, the Fr´echet coderivative of Φ at z¯ is the multifunction D∗ Φ(¯

z)

m

n

from R to R with the values

D∗ Φ(¯

z )(y ) := x ∈ Rn : (x , −y ) ∈ Ngph Φ (¯

z)

(y ∈ Rm ).

Similarly, the Mordukhovich coderivative (limiting coderivative) of Φ at z¯ is the multifunction

D∗ Φ(¯

z ) : Rm ⇒ Rn with the values

D∗ Φ(¯

z )(y ) := x ∈ Rn : (x , −y ) ∈ Ngph Φ (¯

z)

(y ∈ Rm ).

One says that Φ is graphically regular at z¯ if D∗ Φ(¯

z )(y ) = D∗ Φ(¯

z )(y ) for any y ∈ Rm .

Suppose that X, Y , and Z are finite-dimensional Euclidean spaces. Consider a function

¯ , where R

¯ := R ∪ {+∞} ∪ {−∞}, and suppose that |ψ(¯

ψ:X→R

x)| < ∞. The set

∂ψ(¯

x) := {x ∈ X ∗ : (x , −1) ∈ Nepi ψ (¯

x, ψ(¯

x))}

is the Mordukhovich subdifferential of ψ at x¯. If |ψ(¯

x)| = ∞, then we put ∂ψ(¯

x) = ∅. The set

∂ ∞ ψ(¯

x) := {x∗ ∈ X ∗ : (x∗ , 0) ∈ Nepi ψ (¯

x, ψ(¯

x))}

is the singular subdifferential of ψ at x¯. For a set Ω ⊂ X and a point x¯ ∈ Ω, we have

N (¯

x, Ω) = ∂δΩ (¯

x) = ∂ ∞ δΩ (¯

x),

where δΩ (¯

x) is the indicator function of Ω. If ψ depends on two variables x and y, and

|ψ(¯

x, y¯)| < ∞, then ∂x ψ(¯

x, y¯) denotes the Mordukhovich subdifferential of ψ(., y¯) at x¯. For

any v¯ ∈ ∂ψ(¯

x),

∂ 2 ψ(¯

x|¯

v )(u) := D∗ (∂ψ)(¯

x|¯

v )(u) (u ∈ X ∗∗ = X)

is the limiting second-order subdifferential (or the generalized Hessian) of ψ at x¯ in direction

v¯.

1.2

Properties of Multifunctions and Implicit Multifunctions

A multifunction G : Y ⇒ X is said to be Lipschitz-like around a point (¯

y , x¯) ∈ gph G if

there exist a constant > 0 and neighborhoods U of x¯, V of y¯ such that

G(y ) ∩ U ⊂ G(y) +

¯X

y −y B

∀y, y ∈ V,

¯X denotes the closed unit ball in X. The infimum of all such moduli

where B

exact Lipschitzian bound of G around (¯

y , x¯).

3

is called the

Theorem 1.1 (Mordukhovich Criterion 1) If G is locally closed around (¯

y , x¯), then G is

Lipschitz-like around (¯

y , x¯) if and only if

D∗ G(¯

y |¯

x)(0) = {0}.

We say that a multifunction F : X ⇒ Y is metrically regular around (¯

x, y¯) ∈ gph F with

modulus r > 0 if there exist neighborhoods U of x¯, V of y¯, and a number γ > 0 such that

d(x, F −1 (y)) ≤ r d(y, F (x))

(1.1)

for any (x, y) ∈ U × V with d(y, F (x)) < γ.

The condition d(y, F (x)) < γ can be omitted when F is inner semicontinuous at (¯

x, y¯) in

the graph of F .

Theorem 1.2 (Mordukhovich Criterion 2) If F is locally closed around (¯

x, y¯) ∈ gph F ,

then F is metrically regular around (¯

x, y¯) if and only if

0 ∈ D∗ F (¯

x|¯

y )(v ) =⇒ v = 0.

Given a multifunction F : X × Y ⇒ Z and a pair (¯

x, y¯) ∈ X × Y satisfying 0 ∈ F (¯

x, y¯).

We say that the implicit multifunction G : Y ⇒ X given by

G(y) = {x ∈ X : 0 ∈ F (x, y)}

(1.2)

has the Robinson stability at ω0 = (¯

x, y¯, 0) if there exist constants r > 0, γ > 0, and neighborhoods U of x¯, V of y¯ such that

d(x, G(y)) ≤ rd(0, F (x, y))

(1.3)

for any (x, y) ∈ U × V with d(0, F (x, y)) < γ. The infimum of all such moduli r is called the

exact Robinson regularity bound of the implicit multifunction G at ω0 = (¯

x, y¯, 0).

Note that, in (1.3), the condition d(0, F (x, y)) < γ can be omitted if F is inner semicontinuous at (¯

x, y¯, 0).

1.3

An Overview on Implicit Function Theorems for

Multifunctions

Consider an implicit multifunction of the form

S(w) = {x ∈ Rn : 0 ∈ G(x, w) + M (x, w)},

(1.4)

with G : Rn+d → Rm being a continuously Fr´echet differentiable function and M a multifunction with closed graph from Rn+d to Rm . Let (w,

¯ x¯) ∈ gph S and τ¯ = (w,

¯ x¯, −G(¯

x, w)).

¯

Theorem 1.3 (see Levy and Mordukhovich (2004)) If the constraint qualification

0 ∈ ∇G(¯

x, w)

¯ ∗ v1 + D∗ M (¯

τ )(v1 ) =⇒ v1 = 0

is satisfied, then the upper estimate

D∗ S(w|¯

¯ x)(x ) ⊂ Γ(x ),

4

(C1)

where

w ∈ Rd : (−x , w ) ∈ ∇G(¯

x, w)

¯ ∗ v1 + D∗ M (¯

τ )(v1 ) ,

Γ(x ) :=

v1 ∈Rn

is valid for any x ∈ Rn . If, in addition, either M is graphically regular at τ¯, or M = M (x)

and ∇w G(¯

x, w)

¯ has full rank, then

D∗ S(w|¯

¯ x)(x ) = Γ(x ).

Theorem 1.4 (see Lee and Yen (2011)) The lower estimates

Γ(x ) ⊂ D∗ S(w|¯

¯ x)(x ) ⊂ D∗ S(w|¯

¯ x)(x ),

(1.5)

where

w ∈ Rd : (−x , w ) ∈ ∇G(¯

x, w)

¯ ∗ v1 + D∗ M (¯

τ )(v1 ) ,

Γ(x ) :=

(1.6)

v1 ∈Rn

hold for any x ∈ Rn .

Yen and Yao (2009) gave a couple of conditions guaranteeing the Robinson stability of

implicit multifunctions. In Chapters 2 and 3, it is shown that, for the linear constraint

systems, these conditions are also necessary.

Theorem 1.5 (see Yen and Yao (2009)) Let S be the implicit multifunction defined by

S(w) = {x ∈ Rn : 0 ∈ M (x, w)}.

(1.7)

If gph M is locally closed around the point ω0 := (¯

x, w,

¯ 0) and

(a) ker D∗ M (¯

τ ) = {0},

(b) w ∈ Rd : ∃v1 ∈ Rn with (0, w ) ∈ D∗ M (ω0 )(v1 ) = {0},

then S has the Robinson stability around ω0 .

Thanks to the sum rule for the Mordukhovich coderivative (see Mordukhovich (2006)), for

any v ∈ Rm , we have

D∗ M (ω0 )(v ) = ∇G(¯

x, w)

¯ ∗ v + D∗ M (¯

τ )(v ).

5

Chapter 2

Linear Constraint Systems under Total

Perturbations

The present chapter is devoted to stability analysis of linear constraint systems, linear

complementarity systems, and affine variational inequalities under total perturbations. It is

written on the basis of the paper of Huyen and Yen (2016), where a new concept of linear

constraint system was proposed. In that paper, the first time, the concept “uniform local

error bounds” for linear complementarity problems and affine variational inequality has been

defined. Recently, the paper has been cited by C. Li and K.F. Ng (2018).

2.1

An Introduction to Parametric Linear Constraint

Systems

In this chapter, we study the Lipschitz-like property and the Robinson metric regularity of

the solution map of a parametric linear constraint system in the form

Ax + b ∈ K,

(2.1)

with A ∈ Rm×n being an m × n matrix, b ∈ Rm a vector, and K ⊂ Rm a closed set. When

K is a cone, (2.1) can be formally rewritten as

Ax + b ≥K 0,

(2.2)

where v ≥K u means that v − u ∈ K. In addition, if K is convex then the partial order

m

m

“≥K ” is transitive. For K = Rm

+ , where R+ denotes the nonnegative orthant in R , (2.2) is

a standard linear inequality system.

Unlike the traditional considerations, here K needs not to be convex.

The multifunction S : Rm×n × Rm ⇒ Rn with

S(A, b) := {x ∈ Rn | Ax + b ∈ K}

is said to be the solution map of (2.1). We interpret the pair (A, b) as a parameter. With K

being fixed, in the sequel, we will allow both the linear part (that is vector b) and the nonlinear

part (matrix A) of the data set {A, b} of (2.1) to change. It is easy to see that the solution

map (A, b) → S(A, b) is a special case of the implicit multifunction y → G(y) defined by (1.2).

The aim of this chapter will be achieved by using the Mordukhovich Criterion 1 and a

formula for computing exactly the limiting coderivative of implicit multifunctions obtained by

6

Levy and Mordukhovich (Theorem 1.3), as well as a result from Yen and Yao on the Robinson

stability of implicit multifunctions (Theorem 1.5).

The abstract stability results of (2.1) can be effectively applied to

(a) traditional inequality systems,

(b) linear complementarity problems,

(c) affine variational inequalities

to yield necessary and sufficient conditions for the Lipschitz-like property and the Robinson

stability of the related solution maps as well as uniform local error bounds and traditional

local error bounds.

2.2

The Solution Maps of Parametric Linear Constraint

Systems

Let K ⊂ Rm be a fixed closed set. For any pair (A, b) ∈ Rm×n × Rm , we consider the

parametric linear constraint system (2.1). Put W = Rm×n × Rm . For every w = (A, b) ∈ W ,

we set G(x, w) = −Ax − b, M (x, w) = K, and M (x, w) = G(x, w) + M (x, w). Then, the

solution map of (2.1) is given by

S(w) = {x ∈ Rn | 0 ∈ M (x, w)}.

¯ ¯b) and suppose that x¯ = x¯1 , . . . x¯n

From now on, let us fix an element w¯ = (A,

to S(w).

¯ Here and in the sequel, the superscript T denotes matrix transposition.

T

belongs

Theorem 2.1 The mapping S is Lipschitz-like around (w,

¯ x¯) if and only if

ker A¯T ∩ N (¯

v ; K) = {0},

(2.3)

¯x + ¯b and ker A¯T := {v ∈ Rm | A¯T v = 0} is the kernel of A¯T .

where v¯ = A¯

Until now, no criterion for the Robinson stability has been found. However, with two

following lemmas, we can show that the Lipschitz-like property and the Robinson stability of

for our parametric linear constraint systems are equivalent.

Lemma 2.1 If S has the Robinson stability at ω0 := (w,

¯ x¯, 0), then it is Lipschitz-like around

(w,

¯ x¯).

Lemma 2.2 If condition (2.3) is satisfied, then S has the Robinson stability at ω0 = (w,

¯ x¯, 0).

The above results lead us to the following main theorem of this chapter.

Theorem 2.2 The Lipschitz-like property of S around (w,

¯ x¯) and its Robinson stability at

ω0 = (w,

¯ x¯, 0) are equivalent. Moreover, these properties appear if and only if condition (2.3)

is satisfied.

7

2.3

Stability Properties of Generalized Linear Inequality Systems

With K specially being a closed convex cone, applying the results of the previous section

to the generalized linear inequality system (2.2), we can describe necessary and sufficient

conditions for the Lipschitz-like property and the Robinson stability of the solution map S as

follows.

Theorem 2.3 If K is a closed convex cone, then the following properties are equivalent:

(a) S is Lipschitz-like around (w,

¯ x¯);

(b) S has the Robinson stability at ω0 = (w,

¯ x¯, 0);

(c) ker A¯T ∩ N (¯

v ; K) = {0};

¯x + b,

(d) (ker A¯T ) ∩ K ∗ ∩ (¯

v )⊥ = {0}, where K ∗ = {v ∈ Rm | v , v ≤ 0, ∀v ∈ K}, v¯ = A¯

⊥

m

and (¯

v ) := {v ∈ R | v , v¯ = 0};

(e) 0 ∈ int(rge A¯ + K − v¯), where rge A¯ := A¯ (Rn ) is the range of A¯ and int Ω denotes the

interior of Ω;

(f) rge A¯ + cone(K − v¯) = Rm , where cone C is the cone generated by C.

Thus, each one of the following conditions is sufficient for (a) and (b) to hold:

(ker A¯T ) ∩ K ∗ = {0};

(ker A¯T ) ∩ (¯

v )⊥ = {0};

K ∗ ∩ (¯

v )⊥ = {0}.

2.4

The Solution Maps of Linear Complementarity Problems

In this section, we apply results on the stability of solution map of parametric linear

constraint system (2.1) to investigate parametric linear complementarity problems.

Given a vector q in Rn , and a matrix M in Rn×n , the linear complementarity problem

(LCP) aims at finding a vector x in Rn such that

M x + q ≥ 0, x ≥ 0

xT (M x + q) = 0.

The solution set of this problem is denoted by Sol(M, q).

Let

A=

M

∈ R(2n)×n ,

E

b=

q

0

∈ R2n ,

where E ∈ Rn×n is the unit matrix, and

K=

u

v

∈ Rn × Rn : u ≥ 0, v ≥ 0, v T u = 0 .

8

It is clear that x ∈ Sol(M, q) if and only if Ax + b ∈ K. Put W = R(2n)×n × R2n and consider

the multifunction S : W ⇒ Rn defined by

S(w) = {x ∈ Rn : Ax + b ∈ K} ∀w = (A, b) ∈ W.

¯ , q¯) ∈ Rn×n × Rn . Let x¯ ∈ Sol(M

¯ , q¯). Then x¯ ∈ S(w),

¯ ¯b). Put

Fix a pair (M

¯ where w¯ := (A,

¯ x¯ + q¯)i > 0}, I2 = {i ∈ I : x¯i > 0, (M

¯ x¯ + q¯)i = 0},

I = {1, 2, . . . , n}, I1 = {i ∈ I : x¯i = 0, (M

¯ x¯ + q¯)i = 0},

I3 = {i ∈ I : x¯i = 0, (M

¯x + ¯b =

y¯ = A¯

¯ x¯ + q¯

M

,

x¯

and note that y¯ ∈ K.

We present the above cone K in the form

K=

u

v

∈ Rn × Rn :

ui

vi

Ki =

ui

vi

∈ R2 : ui ≥ 0, vi ≥ 0, ui vi = 0 .

∈ Ki , i ∈ I

,

(2.4)

where

(2.5)

Definition 2.1 We say that the solution map Sol(.) of LCP satisfies the uniform local error

¯ , q¯), x¯) if there exist constants r > 0, δ > 0, and a neighborhood V of x¯ such

bound at ((M

that

n

(M x + q)i

d(x; Sol(M, q)) ≤ r

d

; Ki ,

(2.6)

xi

i=1

¯ + q − q¯ < δ.

for any x ∈ V and (M, q) satisfying M − M

From (2.6) we can infer that

n

¯ , q¯)) ≤ r

d(x; Sol(M

d

i=1

¯ x + q¯)i

(M

xi

; Ki

,

(2.7)

for any x ∈ V .

Let us consider a regularity condition: If u = (u1 , . . . , un )T ∈ Rn and if

For each i ∈ I1 , ui = 0;

For each i ∈ I , (M

¯ T u )i = 0;

2

¯ T u )i = 0, or u ≤ 0

For each i ∈ I3 , either ui = 0, or (M

i

¯T

(2.8)

and (M u )i ≥ 0,

then u = 0.

The major result of this section reads as follows.

¯ , q¯). If the regularity condition (2.8) is satisfied, then

Theorem 2.4 Suppose that x¯ ∈ Sol(M

the problem LCP has the uniform local error bound (2.6) and the traditional local error bound

¯ , q¯), x¯) and its solution map Sol(.) is Lipschitz-like around ((M

¯ , q¯), x¯).

(2.7) at ((M

9

2.5

The Solution Maps of Affine Variational Inequalities

Let M ∈ Rn×n , q ∈ Rn , and let ∆ ⊂ Rn be a polyhedral convex set defined by

∆ = {x ∈ Rn : Cx ≥ d},

where C ∈ Rm×n , d ∈ Rm . The problem finding x¯ ∈ ∆ such that

M x¯ + q, y − x¯ ≥ 0 ∀y ∈ ∆,

is called an affine variational inequality (AVI). Denote the solution set of AVI by Sol(M, q, C, d).

¯ According to Gowda and Pang (1994) (see also Theorem 5.3

¯ , q¯, C,

¯ d).

Fix a vector field (M

¯ if and only if there exists a Lagrange multiplier

¯ , q¯, C,

¯ d)

of Lee and Yen (2005)), x¯ ∈ Sol(M

λ ∈ Rm satisfying

¯ x¯ − C¯ T λ + q¯ = 0

M

λ ≥ 0, C¯ x¯ ≥ d¯

(C¯ x¯ − d)

¯ T λ = 0.

Put

¯ −C¯ T

M

q¯

(n+m+m)×(n+m) ¯

¯

A=

∈R

, b=

0

E

0 ∈ Rn+m+m ,

C¯

0

−d¯

K = (s, u, v) ∈ Rn × Rm × Rm : s = 0, u ≥ 0, v ≥ 0, v T u = 0 ,

and

K0 = (u, v) ∈ Rm × Rm : u ≥ 0, v ≥ 0, v T u = 0 .

¯ if and only if there exists

¯ , q¯, C,

¯ d)

Here E is the unit matrix of order m. Thus, x¯ ∈ Sol(M

m

λ ∈ R such that

x¯

A¯

+ ¯b ∈ K.

λ

We consider the multifunction S : W ⇒ Rn × Rm defined by

S(w) =

x

λ

(x, λ) ∈ Rn × Rm : A

+b∈K

∀w = (A, b) ∈ W,

¯ ∈ S(w).

¯ ¯b) and suppose that (¯

where W := R(n+m+m)×(n+m) × Rn+m+m . Put w

¯ = (A,

x, λ)

¯

¯

¯

¯

¯

¯

¯

Then, x¯ ∈ Sol(M , q¯, C, d). Set I = {1, 2, . . . , m}, I1 = {i ∈ I | (C x¯ − d)i = 0, λi > 0},

¯ i > 0, λ

¯ i = 0}, I3 = {i ∈ I | (C¯ x¯ − d)

¯ i = 0, λ

¯ i = 0},

I2 = {i ∈ I | (C¯ x¯ − d)

¯ + q¯

¯ x¯ − C¯ T λ

M

¯

¯x + ¯b =

,

z¯ = A¯

λ

¯

¯

C x¯ − d

¯ + q¯ = 0, and

¯ x¯ − C¯ T λ

and note that M

¯

λ

∈ K0 . Similarly as it was done for (LCP),

C¯ x¯ − d¯

we decompose the cone K0 by writing

K0 =

u

v

∈ Rm × Rm :

10

ui

vi

∈ Ki , i ∈ I

,

where

Ki =

ui

vi

∈ R2 : ui ≥ 0, vi ≥ 0, ui vi = 0 .

Definition 2.2 We say that the problem AVI has the uniform local error bound at the point

¯ x¯) if there exist λ

¯ ∈ Rm , positive constants r and δ, and a neighborhood V of x¯

¯ , q¯, C,

¯ d),

((M

such that

m

d(x; Sol(M, q, C, d)) ≤ r

T¯

Mx − C λ + q +

d

i=1

¯i

λ

(Cx − d)i

; Ki

,

(2.9)

¯ + q − q¯ + C − C¯ + d − d¯ < δ.

for any x ∈ V and (M, q, C, d) with M − M

¯ into (2.9) yields

¯ , q¯, C,

¯ d)

Substituting (M, q, C, d) = (M

m

¯ ≤r

¯ , q¯, C,

¯ d))

d(x; Sol(M

¯ + q¯ +

¯ x − C¯ T λ

M

d

i=1

¯i

λ

¯i

¯ − d)

(Cx

; Ki

,

(2.10)

for any x ∈ V . This is a local error bound for the unperturbed AVI in the traditional form.

¯ T z + C¯ T z = 0

Consider the regularity condition: If vector (z1 , z3 ) ∈ Rn × Rm satisfies M

1

3

and the system

¯ )i = 0 if i ∈ I1 ;

(Cz

1

(z ) = 0 if i ∈ I ;

2

3 i

either (Cz

¯ )i = 0, or (z )i = 0, or (Cz

¯ )i ≤ 0 and (z )i ≤ 0 if i ∈ I3 ,

1

3

1

3

(2.11)

then (z1 , z3 ) = (0, 0).

The next result is applied to a class of AVIs where rank C = m.

¯ and λ

¯ is a Lagrange multiplier corresponding

¯ , q¯, C,

¯ d)

Theorem 2.5 Suppose that x¯ ∈ Sol(M

to x¯. If the regularity condition (2.11) is satisfied, then there are the local error bounds for

AVI in the forms (2.9) and (2.10). Moreover, if rank C¯ = m then the solution map Sol (.) of

¯ x¯).

¯ , q¯, C,

¯ d),

AVI is Lipschitz-like around ((M

Chapter 3

Linear Constraint Systems under

Linear Perturbations

Linear perturbations are just special cases of total perturbations. However, they have their

own significance.

11

As in Chapter 2, consider the parametric linear constraint system (2.1). When b is subject

to change, the solution map S : Rm ⇒ Rn of (2.1) is defined by

S(b) = {x ∈ Rn : b ∈ −Ax + K}.

Put Ψ(x) = −Ax + K and note that Ψ : Rn ⇒ Rm is the inverse multifunction of S.

In what follows, we fix a vector ¯b and a solution x¯ = x¯1 , . . . x¯n

belongs to the graph of Ψ. Let v¯ := A¯

x + ¯b.

T

∈ S(¯b). Then, (¯

x, ¯b)

Theorem 3.1 The following statements are equivalent:

(a) Ψ is metrically regular around (¯

x, ¯b) ∈ gph Ψ with modulus ;

(b) S is Lipschitz-like around (¯b, x¯) ∈ gph S with modulus ;

(c) ker AT ∩ N (¯

v ; K) = {0}.

Moreover, when K is a closed convex cone, these statements are equivalent to each one of the

following:

(d) (ker AT ) ∩ K ∗ ∩ (¯

v )⊥ = {0}, where K ∗ = {v ∈ Rm : v , v ≤ 0, ∀v ∈ K}, v¯ = A¯

x + b,

⊥

and (¯

v ) := {v ∈ Rm : v , v¯ = 0};

(e) 0 ∈ int(rge A + K − v¯), where rge A := A (Rn ) is the range of A and int Ω denotes the

interior of Ω;

(f) rge A + cone(K − v¯) = Rm , where cone C is the cone generated by C.

3.1

Stability properties of Linear Constraint Systems

under Linear Perturbations

Following the approach adopted in Chapter 2, we transform LCP into the linear constraint

system in the form (2.1) by setting

A=

M

∈ R(2n)×n ,

E

b=

q

0

∈ R2n ,

and

K=

u

v

∈ Rn × Rn : u ≥ 0, v ≥ 0, v T u = 0 .

It is clear that x ∈ SolM (q) if and only if Ax + b ∈ K.

Unlike Chapter 2, here we consider the case where only vector q of the problem LCP is

subject to change. Hence, in the corresponding linear constraint system (2.1), only vector b is

perturbed. Thus various results on the solution stability of LCP under perturbations of q can

be obtained by considering the multifunction S : R2n ⇒ Rn , S(b) = {x ∈ Rn : Ax + b ∈ K},

where A and K have been defined above.

q¯

and suppose that x¯ ∈ S(¯b) or x¯ ∈ SolM (¯

q ). For convenience, we put

0

I = {1, 2, . . . , n}, I1 = {i ∈ I : x¯i = 0, (M x¯ + q¯)i > 0}, I2 = {i ∈ I : x¯i > 0, (M x¯ + q¯)i = 0},

Fix a vector ¯b =

12

I3 = {i ∈ I : x¯i = 0, (M x¯ + q¯)i = 0} and

y¯ = A¯

x + ¯b =

M x¯ + q¯

.

x¯

Recall that y¯ ∈ K. Denoting

Ki =

ui

vi

∈ R × R : ui ≥ 0, vi ≥ 0, ui vi = 0 ,

we have

K=

u

v

∈ Rn × Rn :

ui

vi

∈ Ki , i ∈ I

.

(3.1)

Definition 3.1 (see Chapter 2) We say that the problem LCP satisfies the uniform local error

bound at (¯

q , x¯) if there exist constants > 0, δ > 0 and a neighborhood V of x¯ such that

n

d(x; SolM (q)) ≤

d

(M x + q)i

xi

; Ki

,

(3.2)

d

(M x + q¯)i

xi

; Ki

,

(3.3)

i=1

for any x ∈ V and q satisfying q − q¯ < δ.

From (3.2) we can infer that

n

d(x; SolM (¯

q )) ≤

i=1

for any x ∈ V .

The following verifiable regularity condition plays a central role in our study of LCP: If

u = (u1 , . . . , un )T ∈ Rn and if

For each i ∈ I1 , ui = 0;

T

For each i ∈ I2 , (M u )i = 0;

For each i ∈ I3 , either ui = 0, or (M T u )i = 0,

T

(3.4)

or (ui ≤ 0 and (M u )i ≥ 0),

then u = 0.

The linear perturbations permit us to go to a further result as follows.

Theorem 3.2 Suppose that x¯ ∈ SolM (¯

q ). The following statements are equivalent:

(a) The problem LCP satisfies the uniform local error bound (3.2) at (¯

q , x¯);

(b) The solution map SolM (.) is Lipschitz-like around (¯

q , x¯) ∈ gph S;

(c) The regularity condition (3.4) holds.

Looking back to the regularity condition (3.4), we can see that If I = I1 , then (3.4) is

automatically satisfied; If I = I2 , then (3.4) holds if and only if M is nonsingular. The

following result helps us to open condition (3.4).

Proposition 3.1 If I3 = I, then (3.4) is satisfied if and only if M is P -matrix.

13

We now obtain a sufficient condition for the fulfillment of (3.4). This result can be presented

similarly for the case of total perturbations considered in Chapter 2.

Theorem 3.3 If M = diag[MI1 I1 , MI2 I2 , MI3 I3 ] with MI2 I2 being a nonsingular matrix and

MI3 I3 ∈ P, then (3.4) is satisfied.

3.2

Solution Stability of Affine Variational Inequalities

under Linear Perturbations

As it has been noted in Chapter 2, x ∈ SolM,C (q, d) if and only if there exists a Lagrange

multiplier λ ∈ Rm such that

x

A

+ b ∈ K,

λ

where

M −C T

q

A= 0

E ∈ R(n+m+m)×(n+m) , b = 0 ∈ Rn+m+m ,

C

0

−d

and

K = (s, u, v) ∈ Rn × Rm × Rm | s = 0, u ≥ 0, v ≥ 0, v T u = 0 .

Here, as before, E is the unit matrix of order m. We are going to study the multifunction

S : Rm ⇒ Rn × Rm defined by

S(b) =

(x, λ) ∈ Rn × Rm : A

x

λ

+b∈K

.

q¯

¯

¯

¯

In what follows, we fix a pair (¯

q , d) and suppose that x¯ ∈ SolM,C (¯

q , d). Put b =

0

−d¯

¯ is a Lagrange multiplier corresponding to x¯. Then, (¯

¯ ∈ S(¯b). Set

and suppose that λ

x, λ)

I = {1, . . . , m} and consider the index sets

¯ i = 0, λ

¯ i > 0},

I1 := {i ∈ I : (C x¯ − d)

¯

¯ i = 0},

I2 := {i ∈ I : (C x¯ − d)i > 0, λ

¯ i = 0, λ

¯ i = 0}.

I3 := {i ∈ I : (C x¯ − d)

Let

(3.5)

¯ + q¯

M x¯ − C T λ

¯

.

z¯ = A¯

x + ¯b =

λ

C x¯ − d¯

¯ + q¯ = 0 and

Note that M x¯ − C T λ

¯

λ

∈ K0 , where K0 is defined by

C x¯ − d¯

K0 = (u, v) ∈ Rm × Rm : u ≥ 0, v ≥ 0, v T u = 0 .

The set K0 can be decomposed by writing

K0 =

u

v

∈ Rm × Rm :

14

ui

vi

∈ Ki , i ∈ I

,

where

Ki =

ui

vi

∈ R2 : ui ≥ 0, vi ≥ 0, ui vi = 0 .

Definition 3.2 (see Chapter 2) We say that the problem AVI has the uniform local error

¯ x¯) if there exist λ

¯ ∈ Rm , positive constants and δ, and a neighborhood V

bound at ((¯

q , d),

of x¯ such that

m

T¯

d(x; SolM,C (q, d)) ≤

Mx − C λ + q +

d

¯i

λ

(Cx − d)i

; Ki

,

(3.6)

d

¯i

λ

¯i

(Cx − d)

; Ki

,

(3.7)

i=1

for any x ∈ V and (q, d) with q − q¯ + d − d¯ < δ.

¯ into (3.6) yields

Substituting (q, d) = (¯

q , d)

m

¯ ≤

d(x; SolM,C (¯

q , d))

T¯

M x − C λ + q¯ +

i=1

for any x ∈ V . This is a local error bound for the unperturbed AVI in the traditional form.

We now consider the regularity condition: If vector (u , η ) ∈ Rn × Rm satisfies equality

M T u + C T η = 0 and the system

(Cu )i = 0 if i ∈ I1 ;

η = 0 if i ∈ I ;

2

i

either (Cu ) = 0, or η = 0, or ((Cu ) ≤ 0 and η ≤ 0) if i ∈ I ,

3

i

i

i

i

(3.8)

then (u , η ) = (0, 0).

¯ and λ

¯ is a Lagrange multiplier corresponding

Theorem 3.4 Suppose that x¯ ∈ SolM,C (¯

q , d)

to x¯. If the regularity condition (3.8) is satisfied, then the local error bounds for AVI in the

forms (3.6) and (3.7) are valid. Moreover, if rank C = m, then the solution map

(q, d) → SolM,C (q, d)

¯ x¯).

is Lipschitz-like around ((¯

q , d),

The next result appears by the linear perturbations.

Proposition 3.2 If rank C = m, then the solution map (q, d) → SolM,C (q, d) is Lipschitz-like

¯ x¯) if and only if (3.8) is satisfied.

around ((¯

q , d),

The next proposition gives a set of general conditions for the validity of (3.8).

Proposition 3.3 Let the index sets I1 , I2 , and I3 be defined by (3.5). If

M = diag[MI1 I1 , MI2 I2 , MI3 I3 ]

and C = diag[CI1 I1 , CI2 I2 , CI3 I3 ], where MI2 I2 and CI1 I1 are nonsingular, MI3 I3 ∈ P, and

CI3 I3 is a diagonal matrix with positive diagonal elements, then (3.8) is fulfilled.

15

Chapter 4

Sensitivity Analysis of a Stationary

Point Set Map under Total

Perturbations

4.1

Problem Formulation

Let f0 and F be twice continuously differentiable real-valued functions (C 2 -functions for

brevity) defined on the product Rn × Rd of two Euclidean spaces. For every w ∈ Rd , we

consider the parametric optimization problem

(Pw )

Minimize f0 (x, w) subject to x ∈ Rn and F (x, w) ≤ 0.

The constraint set of (Pw ) is C(w) := {x ∈ Rn : F (x, w) ≤ 0}. The stationary point set of

(Pw ) is defined by

S(w) = {x ∈ Rn : 0 ∈ ∇x f0 (x, w) + NC(w) (x)}.

(4.1)

When w varies on Rd , one has a multifunction S : Rd ⇒ Rn with S(w) being calculated

by (4.1). Setting f (x, w) = g(F (x, w)) = (g ◦ F )(x, w), where g(y) = δR− (y), i.e., g(y) = 0 for

y ≤ 0 and g(y) = +∞ for y > 0, we can rewrite (4.1) as

S(w) = {x ∈ Rn : 0 ∈ ∇x f0 (x, w) + ∂x f (x, w)}.

(4.2)

Fix a vector w = w¯ ∈ Rd and suppose that x¯ ∈ S(w).

¯ Note that (Pw¯ ) has a single smooth

inequality constraint. The Mangasarian-Fromovitz Constraint Qualification is fulfilled at the

point x¯ ∈ C(w)

¯ if and only if

If F (¯

x, w)

¯ = 0, then ∇x F (¯

x, w)

¯ = 0.

(MFCQ)

In what follows, we assume that (MFCQ) is valid.

Put D = {(x, w) ∈ Rn × Rd : F (x, w) ≤ 0}. If F (¯

x, w)

¯ < 0, then (¯

x, w)

¯ is an interior

point of D. If F (¯

x, w)

¯ = 0, then (¯

x, w)

¯ is a boundary point of D.

4.2

4.2.1

Lipschitzian Stability of the Stationary Point Set

Map

Interior Points

Suppose that F (¯

x, w)

¯ < 0. Consider a couple of conditions:

16

ker ∇2xx f0 (¯

x, w)

¯ ∩ ker ∇2wx f0 (¯

x, w)

¯ = {0}

(4.3)

ker ∇2xx f0 (¯

x, w)

¯ ⊂ ker ∇2wx f0 (¯

x, w).

¯

(4.4)

and

Theorem 4.1 Suppose that F (¯

x, w)

¯ < 0. The following assertions are valid:

(a) If S is Lipschitz-like around (w,

¯ x¯), then condition (4.4) holds;

(b) If conditions (4.3) and (4.4) are simultaneously fulfilled, i.e.,

ker ∇2xx f0 (¯

x, w)

¯ = {0},

(4.5)

then S is Lipschitz-like around (w,

¯ x¯);

(c) If condition (4.3) is satisfied, then S is Lipschitz-like around (w,

¯ x¯) if and only if condition

(4.4) holds.

4.2.2

Boundary Points

Suppose that F (¯

x, w)

¯ = 0.

Case 1: The Nondegenerate Case (λ > 0)

Consider a couple of conditions:

ker A1 ∩ ker A2 ∩ (ker ∇x F (¯

x, w)

¯ × R) = {(0, 0)}

(4.6)

ker A1 ∩ (ker ∇x F (¯

x, w)

¯ × R) ⊂ ker A2 ,

(4.7)

and

where

A1 :=

∇2xx f0 (¯

x, w)

¯ + λ∇2xx F (¯

x, w)

¯

∇x F (¯

x, w)

¯

∈ Rn×(n+1)

x, w)

¯ + λ∇2wx F (¯

x, w)

¯

∇2wx f0 (¯

∇w F (¯

x, w)

¯

∈ Rd×(n+1) .

and

A2 :=

Theorem 4.2 Suppose that F (¯

x, w)

¯ = 0 and the Lagrange multiplier λ corresponding to the

stationary point x¯ ∈ S(w)

¯ is positive. Then the following assertions are true:

(a) S is Lipschitz-like around (w,

¯ x¯) if (4.6) and (4.7) hold simultaneously, i.e.,

ker A1 ∩ (ker ∇x F (¯

x, w)

¯ × R) = {0}.

(b) If (4.6) holds, then S is Lipschitz-like around (w,

¯ x¯) if and only if (4.7) is satisfied.

17

(4.8)

Case 2: The Degenerate Case (λ = 0)

Consider the system of conditions:

ker A1 ∩ ker A2 = {0},

ker A1 ∩ (ker ∇x F (¯

x, w)

¯ × R) ⊂ ker A2 ,

ker A1 ∩ ∆1 ⊂ ker A2 ,

2

(4.9)

ker ∇xx f0 (¯

x, w)

¯ ∩ ∆2 ⊂ ker ∇2wx f0 (¯

x, w)

¯

and

ker A1 ∩ ∆3 ⊂ ker A2 .

(4.10)

Here

A1 :=

∇2xx f0 (¯

x, w)

¯

∇x F (¯

x, w)

¯

∈ Rn×(n+1) ,

A2 :=

x, w)

¯

∇2wx f0 (¯

∇w F (¯

x, w)

¯

∈ Rd×(n+1) ,

x, w)v

¯ 1 > 0, γ ≥ 0},

∆1 := {(v1 , γ) ∈ Rn × R : ∇x F (¯

∆2 := {v1 ∈ Rn : ∇x F (¯

x, w)v

¯ 1 < 0},

and

∆3 := {(v1 , γ) ∈ Rn × R : ∇x F (¯

x, w)v

¯ 1 ≥ 0, γ ≥ 0}.

Theorem 4.3 Suppose that F (¯

x, w)

¯ = 0 and the Lagrange multiplier λ corresponding to the

stationary point x¯ ∈ S(w)

¯ is zero. The following assertions are true:

(a) If S is Lipschitz-like around (w,

¯ x¯), then condition (4.10) holds;

(b) If condition (4.9) is fulfilled, then S is Lipschitz-like around (w,

¯ x¯).

4.3

The Robinson Stability of the Stationary Point Set

Map

We have shown that the sufficient conditions for S being Lipschitz-like around (w,

¯ x¯) in

each case also guarantee for S having the Robinson stability at ω0 .

Theorem 4.4 The stationary point set map S of (Pw ) has the Robinson stability at the point

ω0 = (¯

x, w,

¯ 0) if one of the following is valid:

(a) F (¯

x, w)

¯ < 0 and the condition (4.5) holds, i.e,

ker ∇2xx f0 (¯

x, w)

¯ = {0};

(b) F (¯

x, w)

¯ = 0, the Lagrange multiplier λ corresponding to the stationary point x¯ ∈ S(w)

¯ is

positive, and condition (4.8) is satisfied, i.e.,

ker A1 ∩ (ker ∇x F (¯

x, w)

¯ × R) = {0};

18

(c) F (¯

x, w)

¯ = 0, the Lagrange multiplier λ corresponding to the stationary point x¯ ∈ S(w)

¯

equals to zero, and condition (4.9) satisfied, i.e,

ker A1 ∩ ker A2 = {0},

ker A ∩ (ker ∇ F (¯

x, w)

¯ × R) ⊂ ker A2 ,

x

1

ker A1 ∩ ∆1 ⊂ ker A2 ,

2

2

x, w).

¯

x, w)

¯ ∩ ∆2 ⊂ ker ∇wx f0 (¯

ker ∇xx f0 (¯

4.4

Applications to Quadratic Programming

In this section, the above general results are applied to a class of nonconvex quadratic programming problems. Namely, we will consider the problems of minimizing a linear-quadratic

function under one linear-quadratic functional constraint. Special cases of such problems have

been considered by Lee, Tam, and Yen (2012), Lee and Yen (2014), Qui and Yen (2014), etc.

Nonconvex quadratic programming under linear constraints was studied by many authors;

see, e.g., the dissertation of N.N. Tam (2000), and the book by Lee, Tam, and Yen (2005) and

the references therein.

Denote by Sn the space of n × n symmetric matrices. Let D, A ∈ Sn , c and b be vectors

in Rn , and α a real number. Put w = (w1 , w2 ) with w1 := (D, c) and w2 := (A, b, α). Denote

the problem (Pw ) with f0 (x, w) = 21 xT Dx + cT x and F (x, w) = 21 xT Ax + bT x + α by (QPw ).

For convenience, we put W1 = Sn × Rn , W2 = Sn × Rn × R, and W = W1 × W2 . Fix a

¯ c¯), w¯2 = (A,

¯ ¯b, α

vector w¯ = (w¯1 , w¯2 ) ∈ W with w¯1 = (D,

¯ ), and suppose that a stationary

point x¯ ∈ S(w)

¯ is given.

Consider the following conditions:

det

¯ + λA¯ A¯

¯x + ¯b

D

¯x + ¯b)T

(A¯

0

= 0,

(4.11)

det

¯

¯x + ¯b

D

A¯

¯x + ¯b)T

(A¯

0

= 0,

(4.12)

¯ 1 + γ(A¯

¯x + ¯b) = 0, γ ≥ 0] =⇒ (A¯

¯x + ¯b)T v1 ≤ 0,

[Dv

(4.13)

¯ 1 = 0 =⇒ (A¯

¯x + ¯b)T v1 = 0,

Dv

(4.14)

and

¯ + γ(A¯

¯x + ¯b) = 0

Dv

1

¯x + ¯b)T v ≥ 0, γ ≥ 0

(A¯

1

=⇒

v1 = 0

γ = 0.

(4.15)

Theorem 4.5 The following assertions are true:

¯ = 0. Moreover,

(a) If F (¯

x, w)

¯ < 0, then S is Lipschitz-like around (w,

¯ x¯) if and only if det D

under this condition, S has the Robinson stability at ω0 ;

(b) If F (¯

x, w)

¯ = 0 and the Lagrange multiplier λ corresponding to x¯ ∈ S(w)

¯ is positive, then

S is Lipschitz-like around (w,

¯ x¯) if and only if (4.11) is fulfilled. This condition is sufficient

for S having the Robinson stability at ω0 ;

19

(c) If F (¯

x, w)

¯ = 0 and the Lagrange multiplier λ corresponding to x¯ ∈ S(w)

¯ is zero, then

(4.15) is necessary for S being Lipschitz-like around (w,

¯ x¯). Meanwhile, the fulfillment of

(4.12)–(4.14) is sufficient for the Lipschitz-like property of S around (w,

¯ x¯), as well as for the

Robinson stability of S at ω0 .

4.5

Results Obtained by Another Approach

Following the detailed hints of one referee of this paper, we will compare our results with

those which can be obtained by using the theory of strongly regular generalized equations of

Robinson (1980).

Suppose that x¯ ∈ S(w)

¯ and the condition (MFCQ) is satisfied. It is not difficult to show

that, thanks to (MFCQ), there exist a neighborhood W0 of w¯ and a neighborhood U0 of

x¯ such that for every (x, w) ∈ U0 × W0 one has NC(w) (x) = {λ∇x F (x, w) | λ ≥ 0} when

F (x, w) = 0 and NC(w) (x) = {0} when F (x, w) < 0. Hence, for every (x, w) ∈ U0 × W0 , the

condition

0 ∈ ∇x f0 (x, w) + NC(w) (x)

is equivalent to the existence of a Lagrange multiplier α ∈ R such that

0∈

∇x L(x, α, w)

−F (x, w)

+ NRn ×R+ (x, α),

where L(x, α, w) := f0 (x, w) + αF (x, w). Setting g(x, α, w) =

∇x L(x, α, w)

, we consider

−F (x, w)

the parametric generalized equation

0 ∈ g(x, α, w) + NRn ×R+ (x, α) (w ∈ Rd )

(4.16)

and denote its solution set by S(w). Then,

S(w) = {(x, α) ∈ Rn × Rd : 0 ∈ g(x, α, w) + NRn ×R+ (x, α)}

and S(.) is the implicit multifunction defined by (4.16). From what has been said we have

S(w) ∩ U0 = {x ∈ U0 : ∃ α s.t. (x, α) ∈ S(w)} (∀w ∈ W0 ).

(4.17)

As in preceding sections, we will denote by λ the unique multiplier corresponding to x¯

belonging to S(w).

¯ Consider the following three cases.

Case 1: F (¯

x, w)

¯ < 0. This case has been analyzed in Subsection 4.2.1.

Thanks to a criterion for the strong regularity of the generalized equation (4.16), we obtain

results for the cases of boundary points.

Case 2: F (¯

x, w)

¯ = 0 and λ > 0.

Proposition 4.1 Suppose that F (¯

x, w)

¯ = 0 and the Lagrange multiplier λ corresponding to

the stationary point x¯ ∈ S(w)

¯ is positive. If condition (4.8) is satisfied, then S has a Lipschitz

continuous single-valued localization around w¯ for x¯.

Clearly, Proposition 4.1 encompasses claim (a) in Theorem 4.2, which gives a sufficient

condition for the Lipschitz-like property of S around (w,

¯ x¯).

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Case 3: F (¯

x, w)

¯ = 0 and λ = 0.

Consider the four conditions:

ker A1 ∩ ker A2 = {0},

(4a)

x, w)

¯ × R) ⊂ ker A2 = {0},

ker A1 ∩ (ker∇x F (¯

(4b)

ker A1 ∩ ∆1 ⊂ ker A2 ,

(4c)

x, w).

¯

x, w)

¯ ∩ ∆2 ⊂ ker ∇2wx f0 (¯

ker ∇2xx f0 (¯

(4d)

and

and

Proposition 4.2 Suppose that F (¯

x, w)

¯ = 0 and the Lagrange multiplier λ corresponding to

the stationary point x¯ ∈ S(w)

¯ is zero. If (4a)–(4c) are satisfied, then S has a Lipschitz

continuous single-valued localization around w¯ for x¯.

The result stated in Proposition 4.2 is better than assertion (b) of Theorem 4.3, which says

that if (4.9) is fulfilled, i.e., (4a)–(4d) are valid, then S is Lipschitz-like around (w,

¯ x¯).

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General Conclusions

The main results of this dissertation include:

1) Criterion for the Lipschitz-like property and the Robinson stability of the solution map

of a parametric linear constraint system under total perturbations and applications to the

linear complementarity problems and affine variational inequality problems;

2) Analogues of the above results for the case when the linear constraint system undergoes

linear perturbations;

3) The Lipschitz-like property of the stationary point set map of a smooth parametric

optimization problem with one smooth functional constraint under total perturbations;

4) The Robinson stability of the above stationary point set map and applications to

quadratic programming.

Recently, we have developed a part of the results in Chapter 4, which were obtained for

an optimization problem with just one inequality constraint, for the case where finitely many

equality and inequality constraints are involved.

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List of Author’s Related Papers

1. D.T.K. Huyen and N.D. Yen, Coderivatives and the solution map of a linear constraint system, SIAM Journal on Optimization 26 (2016), pp. 986–1007. (SCI)

2. D.T.K. Huyen and J.-C. Yao, Solution stability of a linearly perturbed constraint

system and applications, Set-Valued and Variational Analysis, 27 (2019), pp. 169–189.

(SCIE)

3. D.T.K. Huyen, J.-C. Yao, and N.D. Yen, Sensitivity analysis of an optimization

problem under total perturbations. Part 1: Lipschitzian stability, Journal of Optimization

Theory and Applications, 180 (2019), pp. 91–116. (SCI)

4. D.T.K. Huyen, J.-C. Yao, and N.D. Yen, Sensitivity analysis of an optimization

problem under total perturbations. Part 2: Robinson stability, Journal of Optimization

Theory and Applications, 180 (2019), pp. 117–139. (SCI)

The results of this dissertation have been presented at

- The weekly seminar of the Department of Numerical Analysis and Scientific Computing,

Institute of Mathematics, Vietnam Academy of Science and Technology;

- Workshop “International Workshop on Nonlinear and Variational Analysis” (August

7–9, 2015, Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung,

Taiwan);

- “Taiwan-Vietnam 2015 Winter Mini-Workshop on Optimization” (November 17, 2015,

National Cheng Kung University, Tainan, Taiwan);

- The 15th Workshop on “Optimization and Scientific Computing” (April 21–23, 2016,

Ba Vi, Hanoi);

- Seminar of Prof. Xiao-qi Yang’s research group (June 2016, Department of Applied

Mathematics, Hong Kong Polytechnic University, Hong Kong);

- “Vietnam-Korea Workshop on Selected Topics in Mathematics” (February 20–24, 2017,

Danang, Vietnam);

- “Taiwan-Vietnam Workshop on Mathematics” (May 9–11, 2018, Department of Applied

Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan).

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