Stability of some constraint systems and optimization problems

con-Within this dissertation we use coderivative to study three properties ofsolution maps in finite-dimensional settings, which include Aubin propertyLipschitz-like property, metric reg

VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY

INSTITUTE OF MATHEMATICS

DUONG THI KIM HUYEN

STABILITY OF SOME CONSTRAINT SYSTEMS

AND OPTIMIZATION PROBLEMS

DISSERTATION

SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY IN MATHEMATICS

HANOI - 2019

VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY

INSTITUTE OF MATHEMATICS

DUONG THI KIM HUYEN

STABILITY OF SOME CONSTRAINT SYSTEMS

AND OPTIMIZATION PROBLEMS

Speciality: Applied MathematicsSpeciality code: 9 46 01 12

DISSERTATION FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY IN MATHEMATICS

Supervisor: Prof Dr.Sc NGUYEN DONG YEN

HANOI - 2019

This dissertation was written on the basis of my research works carried out atthe Institute of Mathematics, Vietnam Academy of Science and Technology,under the guidance of Prof Nguyen Dong Yen All results presented in thisdissertation have never been published by others

Hanoi, October 2, 2019

The author

Duong Thi Kim Huyen

I still remember very well the first time I have met Prof Nguyen DongYen at Institute of Mathematics On that day I attended a seminar of Prof.Hoang Tuy about Global Optimization At the end of the seminar, I came

to talk with Prof Nguyen Dong Yen I said to him that I wanted to learnabout Optimization Theory, and I asked him to let me be his student Hedid say yes A few days later, he sent me an email and he informed methat my master thesis would be about “openness of set-valued maps andimplicit multifunction theorems” Three years later, I defensed successfully

my master thesis under his guidance at Institute of Mathematics, VietnamAcademy of Science and Technology I would say I am deeply indebted tohim not only for his supervision, encouragement and support in my research,but also for his precious advices in life

The Institute of Mathematics is a wonderful place for studying and ing I would like to thank all the staff members of the Institute who havehelped me to complete my master thesis and this work within the schedules

work-I also would like to express my special appreciation to Prof Hoang XuanPhu, Assoc Prof Ta Duy Phuong, Assoc Prof Phan Thanh An, andother members of the weekly seminar at Department of Numerical Analysisand Scientific Computing, Institute of Mathematics, as well as all the mem-bers of Prof Nguyen Dong Yen’s research group for their valuable commentsand suggestions on my research results I greatly appreciate Dr Pham DuyKhanh and Dr Nguyen Thanh Qui, who have helped me in typing my masterthesis when I was pregnant with my first baby, and encouraged me to pursue

a PhD program

I would like to thank Prof Le Dung Muu, Prof Nguyen Xuan Tan,Assoc Prof Truong Xuan Duc Ha, Assoc Prof Nguyen Nang Tam,Assoc Prof Nguyen Thi Thu Thuy, and Dr Le Hai Yen, for their careful

readings of the first version of this dissertation and valuable comments.Financial supports from the Vietnam National Foundation for Science andTechnology Development (NAFOSTED) are gratefully acknowledged.

I am sincerely grateful to Prof Jen-Chih Yao from Department of AppliedMathematics, National Sun Yat-sen University, Taiwan, and Prof Ching-Feng Wen from Research Center for Nonlinear Analysis and Optimization,Kaohsiung Medical University, Taiwan, for granting several short-termedscholarships for my doctorate studies I would like to thank Prof Xiao-qiYang for his supervision during my stay at Department of Applied Math-ematics, Hong Kong Polytechnic University, Hong Kong, by the ResearchStudent Attachment Program

I would like to show my appreciation to Prof Boris Mordukhovich fromDepartment of Mathematics, Wayne State University, USA, and Ass Prof.Tran Thai An Nghia from Department of Mathematics and Statistics, Oak-land University, USA, for valuable comments and encouragement on my re-search works

My enormous gratitude goes to my husband and my son for their love, couragement, and especially for their patience during the time I was workingintensively to complete my PhD studies Finally, I would like to express mylove and thanks to my parents, my parents in law, my ant in law, and all mysisters and brothers for their strong encouragement and support

Chapter 2 Linear Constraint Systems under Total

2.1 An Introduction to Parametric Linear Constraint Systems 82.2 The Solution Maps of Parametric Linear Constraint Systems 112.3 Stability Properties of Generalized Linear Inequality Systems 192.4 The Solution Maps of Linear Complementarity Problems 212.5 The Solution Maps of Affine Variational Inequalities 272.6 Conclusions 33

Chapter 3 Linear Constraint Systems under Linear

3.1 Stability properties of Linear Constraint Systems under Linear

Perturbations 363.2 Solution Stability of Linear Complementarity Problems under

Linear Perturbations 383.3 Solution Stability of Affine Variational Inequalities under Lin-

ear Perturbations 48

3.4 Conclusions 57

Chapter 4 Sensitivity Analysis of a Stationary Point Set Map under Total Perturbations 59 4.1 Problem Formulation 60

4.2 Auxiliary Results 61

4.3 Lipschitzian Stability of the Stationary Point Set Map 64

4.3.1 Interior Points 64

4.3.2 Boundary Points 70

4.4 The Robinson Stability of the Stationary Point Set Map 80

4.5 Applications to Quadratic Programming 84

4.6 Results Obtained by Another Approach 92

4.7 Proof of Lemma 4.3 96

4.8 Proof of Lemma 4.4 97

4.9 Conclusions 100

Table of Notations

¯

hx, yi the scalar product in an Euclidean space

B(x, ρ) the open ball centered x with radius ρ

¯

B(x, ρ) the closed ball centered x with radius ρ

N (¯x) the family of the neighborhoods of ¯x

Rm×n the vector space of m × n real matrices

ker A the kernel of matrix A (i.e., the null space

of the operator corresponding to matrix A)

A∗ : Y∗ → X∗ the adjoint operator of a bounded

linear operator A : X → Yd(x, Ω) the distance from x to a set Ω

b

N (¯x; Ω) or NbΩ(¯ the Fr´echet normal cone of Ω at ¯x

N (¯x; Ω) or NΩ(¯ the Mordukhovich normal cone of Ω at ¯x

Limsup the Painlev´e-Kuratowski upper limit

∇f (¯x) the Fr´echet derivative of f : X → Y at ¯x

∇2f (¯x) the Hessian matrix of f : X → R at ¯x

∇xψ(¯x, ¯y) the partial derivative of ψ : X × Y → Z

in x at (¯x, ¯y)

∂f (x) the Mordukhovich subdifferential of f at x

∂∞f (x) the singular subdifferential of f at x

∂2f (¯x, ¯y) the second-order subdifferential of f at ¯x

in direction ¯y ∈ ∂f (¯x)

∂xψ(¯x, ¯y) the partial subdifferential of ψ : X × Y → R

in x at (¯x, ¯y)

F : X ⇒ Y a set-valued map between X and Y

b

D∗F (¯x, ¯y)(·) the Fr´echet coderivative of F at (¯x, ¯y)

D∗F (¯x, ¯y)(·) the Mordukhovich coderivative of F at (¯x, ¯y)

diag[Mαα, Mββ, Mγγ] a block diagonal matrix

Qualification

Many real problems lead to formulating equations and solving them Theseequations may contain parameters like initial data or control variables Thesolution 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 becalled an implicit multifunction A natural question is that “What propertiescan the implicit multifunction possess?”

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

the-Nowadays, the models of interest (for instance, constrained optimizationproblems) outrun equations Thus, Variational Analysis (see, e.g., [50, 80])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 optimiza-tion and equilibrium problems by different approaches In particular, withthe two-volume book “Variational Analysis and Generalized Differentiation”(see [50, 51]) and a series of research papers, Mordukhovich has given basictools (coderivatives, subdiffentials, normal cones, and calculus rules), funda-mental results, and advanced techniques for qualitative studies of optimiza-tion and equilibrium problems Especially, the fourth chapter of the book isentirely devoted to such important properties of the solution set of paramet-ric problems as the Lipschitz stability and metric regularity These propertiesindicate good behaviors of the multifunction in question The two modelsconsidered in that chapter of Mordukhovich’s book bear the names paramet-ric constraint system and parametric variational system More discussionsand references on implicit multifunction theorems can be found in the books

by Borwein and Zhu [10], Dontchev and Rockafellar [19], and Klatte andKummer [35].

Let us briefly review some contents of the book “Implicit Functions andSolution Mappings” [19] of Dontchev and Rockafellar The first chapter ofthis book is devoted to functions defined implicitly by equations and theauthors begin with classical inverse function theorem and classical implicitfunction theorem The book presents a very deep view from VariationalAnalysis on solution maps The authors have investigated many properties

of solution maps such as calmness, Lipschitz continuity, outer Lipschitz tinuity, Aubin property, metric regularity, linear openness, strong regularityand their applications to Numerical Analysis The main tools that have beenused in the book are graphical differentiation and coderivetive

con-Within this dissertation we use coderivative to study three properties ofsolution maps in finite-dimensional settings, which include Aubin property(Lipschitz-like property), metric regularity, and the Robinson stability of so-lution maps of constraint and variational systems Results on these stabilityproperties are applied to studying the solution stability of linear comple-mentarity problem, affine variational inequalities, and a typical parametricoptimization problem

Introduced by Aubin [5, p 98] under the name pseudo-Lipschitz erty, the Lipschitz-like property of multifunctions is a fundamental concept

prop-in stability and sensitivity analysis of optimization and equilibrium lems The Lipschitz-like property guarantees the local convergence of somevariants of Newton’s method for generalized equations [12, 17, 19] In partic-ular, from [19, Theorem 6C.1, p 328] it follows that, if a mild approximationcondition is satisfied and the solution map under right-hand-side perturba-tions is Lipschitz-like around a point in question, then there exists an iter-ative sequence Q-linearly converging to the solution Moreover, as shown

prob-by Dontchev [17, Theorem 1], the Newton method applied to a generalizedequation in a Banach space is locally convergent uniformly in the canonicalparameter if and only if the solution map of this equation is Lipschitz-likearound the reference point In addition, if the derivative of the base map

is locally Lipschitz, then the Lipschitz-likeness implies the existence of a quadratically convergent Newton sequence (see [17, Theorem 2])

Q-Metric regularity (in the classical sense) is another fundamental property

of set-valued mappings We refer to the survey of A.D Ioffe [32, 33] on thisproperty and its applications Borwein and Zhuang [11] and Penot [63] haveshown that the Lipschitz-like property of a set-valued mapping F : X ⇒ Ybetween Banach spaces around a point (¯x, ¯y) in the graph

gph F := {(x, y) ∈ X × Y : y ∈ F (x)}

of F is equivalent to the metric regularity of the inverse map F−1 : Y ⇒ Xaround (¯y, ¯x) It is also known (see Mordukhovich [49]) that the propertiesjust mentioned are equivalent to the openness with linear rate of F around(¯x, ¯y)

Let G : X ⇒ Y be an implicit multifunction defined by

G(x) := {y ∈ Y : 0 ∈ F (x, y)} (x ∈ X), (1)where F : X × Y ⇒ Z is a multifunction, X, Y , and Z are Banach spaces.Then the concept of Robinson stability of G at (¯x, ¯y, 0) ∈ gph F can be de-fined This property of an implicit multifunction, which has been called themetric regularity in the sense of Robinson by several authors, was introduced

by Robinson [75] It is a type of uniform local error bounds and it has ous applications in optimization theory and theory of equilibrium problems.Stability properties like lower semicontinuity, upper semicontinuity, Haus-dorff semicontinuity/continuity, H¨older continuity of solution maps and ofapproximate solution maps can be studied for very general optimization prob-lems and equilibrium problems (for example, vector optimization problems,vector variational inequalities, vector equilibrium problems) The locally con-vex Hausdorff topological vector spaces setting can be also adopted Here,

numer-it is not necessary to use the tools from variational analysis and generalizeddifferentiation We refer to the works by P.Q Khanh, L.Q Anh, and theircoauthors [1–4] for some typical results in this direction

The dissertation has four chapters and a list of references

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

Vari-In Chapter 2, we investigate the Lipschitz-like property and the son stability of the solution map of a parametric linear constraint system

Robin-by means of normal coderivative, the Mordukhovich criterion, and a relatedtheorem due to Levy and Mordukhovich [41] Among other things, the ob-tained results yield uniform local error bounds and traditional local errorbounds for the linear complementarity problem and the general affine vari-ational inequality problem, as well as verifiable sufficient conditions for theLipschitz-like property of the solution map of the linear complementarityproblem 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 thecase where the linear constraint system undergoes linear perturbations.Finally, in Chapter 4, we analyze the sensitivity of the stationary point setmap of a C2-smooth parametric optimization problem with one C2-smoothfunctional constraint under total perturbations by applying some results ofLevy and Mordukhovich [41], and Yen and Yao [88] We not only show neces-sary and sufficient conditions for the Lipschitz-like property of the stationarypoint set map, but also sufficient conditions for its Robinson stability Theseresults lead us to new insights into the preceding deep investigations of Levyand Mordukhovich [41] and of Qui [71, 72] and allow us to revisit and extendseveral stability theorems in indefinite quadratic programming

The dissertation is written on the basis of four published articles: paper [31]

in SIAM Journal on Optimization, paper [28] in Journal of Set-Valued andVariational Analysis, and papers [29, 30] in Journal of Optimization Theoryand Applications

The results of this dissertation have been presented at

- The weekly seminar of the Department of Numerical Analysis and tific Computing, Institute of Mathematics, Vietnam Academy of Science andTechnology;

Scien Workshop “International Workshop on Nonlinear and Variational AnalyScien sis” (August 7–9, 2015, Center for Fundamental Science, Kaohsiung MedicalUniversity, Kaohsiung, Taiwan);

Analy “TaiwanAnaly Vietnam 2015 Winter MiniAnaly Workshop on Optimization” (NovemAnaly ber 17, 2015, National Cheng Kung University, Tainan, Taiwan);

(Novem The 15th Workshop on “Optimization and Scientific Computing” (April21–23, 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” ary 20–24, 2017, Danang, Vietnam);

(Febru “Taiwan(Febru Vietnam Workshop on Mathematics” (May 9–11, 2018, Depart(Febru ment of Applied Mathematics, National Sun Yat-sen University, Kaohsiung,Taiwan)

Depart-Chapter 1

Preliminaries

In this chapter, several concepts and tools from Variational Analysis arerecalled As a preparation for the investigations in Chapters 2–4, we presentlower and upper estimates for coderivatives of implicit multifunctions given

by Levy and Mordukhovich [41], Lee and Yen [39], as well as the sufficientconditions of Yen and Yao [88] for the Robinson stability property of implicitmultifunctions

The concepts and tools discussed in this chapter can be found in the graphs of Mordukhovich [50, 52] and the classical work of Rockafellar andWets [80]

mono-1.1 Basic Concepts from Variational Analysis

Introduced by Mordukhovich [48] in 1980, the limiting coderivative is abasic concept of generalized differentiation and it has a very important role

in Variational Analysis and applications One can compare the role of thelimiting coderivative, which helps to develop the dual-space approach to opti-mization and equiliblium problems, with that of derivative in classical Math-ematical Analysis We are going to describe the finite-dimensional version ofthe concept The reader is referred to [52, Chapter 1] for a comprehensivetreatment of limiting coderivative and related notions

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

v0 ∈ Rs : ∃ sequences vk → ¯v, vk0 → v0,

with vk0 ∈ NbΩ(vk) for all k = 1, 2, the Mordukhovich (or limiting/basic) normal cone to Ω at ¯v If ¯v /∈ Ω, thenone puts NΩ(¯v) = ∅

A multifunction Φ : Rn

⇒ Rm is said to be locally closed around a point

¯

z = (¯x, ¯y) from gph Φ := {(x, y) ∈ Rn ×Rm : y ∈ Φ(x)} if gph Φ is locallyclosed around ¯z Here, the product space Rn+m = Rn ×Rm is equipped withthe topology generated by the sum norm k(x, y)k = kxk + kyk

For any ¯z = (¯x, ¯y) ∈ gph Φ, the Fr´echet coderivative of Φ at ¯z is themultifunction Db∗Φ(¯z) :Rm

⇒ Rn with the values

∂ψ(¯x) := {x0 ∈ X∗ : (x0, −1) ∈ Nepi ψ(¯x, ψ(¯x))}

is the Mordukhovich subdifferential of ψ at ¯x If |ψ(¯x)| = ∞, then we put

For set-valued mappings, being Lipschitz-like around a point in the graph

is a very nice behavior Maps with this property are considered locally stable

in a strong sense For sum rules, chain rules, etc., Lipschitz-likeness plays

a role of constraint qualification This property was originally defined byJ.-P Aubin who called it the pseudo-Lipschitz property [5, p 98] It is alsoknown under other names: the Aubin continuity property [18, p 1089], andthe sub-Lipschitzian property [79] A characterization of the Lipschitz-likeproperty via the local Lipschitz property of a distance function was given byRockafellar [79]

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

Defini-Theorem 1.1 (Mordukhovich Criterion 1) (see [49], [80, Defini-Theorem 9.40],and [50, Theorem 4.10]) 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}

As in [49, Definition 4.1], we say that a multifunction F : X ⇒ Y ismetrically regular around (¯x, ¯y) ∈ gph F with modulus r > 0 if there existneighborhoods 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 tinuous at (¯x, ¯y) ∈ gph F (This concept can be found on page 42 of themonograph [50].) Indeed, the latter means that for every neighborhood V0

semicon-of ¯y, there exits a neighborhood U0 of ¯x such that F (x) ∩ V0 6= ∅ for all

x ∈ U0 Hence, for every neighborhood V0 of ¯y, there exists γ0 > 0 suchthat d(y, F (x)) < γ0 for all x ∈ U0 and y ∈ V0 So, if (1.1) holds true withconstants r, γ and neighborhoods U and V , then for a number γ00 ∈ (0, γ0],

we can find neighborhoods U00 of ¯x and V00 of ¯y with the property (1.1) placing U by U ∩ U00, and V by V ∩ V00, we have the inequality in (1.1) Thus,

Re-if F is inner semicontinuous at (¯x, ¯y), then F is metrically regular around at(¯x, ¯y) with modulus r > 0 if and only if there exist neighborhoods V of ¯y, U

of ¯x such that

d(x, F−1(y)) ≤ r d(y, F (x))for any (x, y) ∈ U × V

Theorem 1.2 (Mordukhovich Criterion 2) (see [49] and also [19, orem 4H.1, p 246]) If F is locally closed around (¯x, ¯y) ∈ gph F , then F ismetrically regular around (¯x, ¯y) if and only if

The-0 ∈ D∗F (¯x|¯y)(v0) =⇒ v0 = 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)

By suggesting two examples, Jeyakumar and Yen [34, p 1119] have provedthat the Robinson stability of G at (¯x, ¯y, 0) ∈ gph F is not equivalent to theLipschitz-like property of G around (¯x, ¯y) We refer to [14] for a discussion

on the relationships between the Robinson stability and the Lipschitz-likebehavior of implicit multifunctions

Recently, Gfrerer and Mordukhovich [21] have given first-order and order sufficient conditions for this stability property of a parametric con-straint system and put it in the relationships with other properties, such asthe classical metric regularity and the Lipschitz-like property

second-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) Indeed, the latter means that for every

µ > 0 there exist neighborhoods Uµ of ¯x, Vµ of ¯y such that

F (x, y) ∩ B(0, µ) 6= ∅ ∀(x, y) ∈ Uµ × Vµ (1.4)

So, if (1.3) is satisfied with positive constants r, γ and neighborhoods U and

V , then for a value µ ∈ (0, γ] we can find neighborhoods Uµ of ¯x, Vµ of ¯y withthe property (1.4) Replacing U by U ∩ Uµ, and V by V ∩ Vµ, we see that theinequality in (1.3) is fulfilled because, by virtue of (1.4), d(0, F (x, y)) < µ ≤ γfor every (x, y) ∈ Uµ × Vµ Thus, if F is inner semicontinuous at (¯x, ¯y, 0),then G has the Robinson stability at ω0 = (¯x, ¯y, 0) if and only if there exist

r > 0 and neighborhoods U of ¯x, V of ¯y such that

M : Rn+d

⇒ Rm a multifunction with closed graph Let ( ¯w, ¯x) ∈ gph S and

¯

τ = ( ¯w, ¯x, −G(¯x, ¯w))

Theorem 1.3 (see [41, Theorem 2.1]) If the constraint qualification

0 ∈ ∇G(¯x, ¯w)∗v10 + D∗M (¯τ )(v10) =⇒ v01 = 0 (C1)

is satisfied, then the upper estimate

D∗S( ¯w|¯x)(x0) ⊂ Γ(x0),where

is valid for any x0 ∈ Rn If, in addition, either M is graphically regular at ¯τ ,

or M = M (x) and ∇wG(¯x, ¯w) has full rank, then

D∗S( ¯w|¯x)(x0) = Γ(x0)

Theorem 1.4 (see [39, Theorem 3.4]) The lower estimates

b

Γ(x0) ⊂ Db∗S( ¯w|¯x)(x0) ⊂ D∗S( ¯w|¯x)(x0), (1.6)where

Put M (x, w) = G(x, w) + M (x, w) From (1.5) we havef

S(w) = {x ∈ Rn : 0 ∈ M (x, w)}.f (1.8)

By the Fr´echet coderivative sum rule with equalities [50, Theorem 1.62],

b

D∗M (ωf 0)(v10) = ∇G(¯x, ¯w)∗v10 +Db∗M (¯τ )(v10)for any v10 ∈ Rn, where ω0 := (¯x, ¯w, 0) ∈ gphM Therefore, we can writef

Propo-for a Banach space setting and the closedness of gph M is an extra tion (See [39, Remark 3.2] for comments on lower estimate for the values ofthe Fr´echet coderivative of implicit multifunctions.)

assump-Yen and Yao [88] gave a couple of conditions guaranteeing the Robinsonstability of implicit multifunctions In Chapters 2 and 3, we will show that,for the linear constraint systems, these conditions are also necessary

Theorem 1.5 (see [88, Theorem 3.1]) Let S be the implicit multifunctiondefined by (1.8) If gphM is locally closed around the point ωf 0 := (¯x, ¯w, 0)and

(a) ker D∗M (¯f τ ) = {0},

(b) nw0 ∈ Rd : ∃v10 ∈ Rn with (0, w0) ∈ D∗M (ωf 0)(v10)o = {0},

then S has the Robinson stability around ω0

Now we are going to find out how the above implicit multifunction theoremscan be used to obtain our desired results

2.1 An Introduction to Parametric Linear Constraint

Systems

In this chapter, we study the Lipschitz-like property and the Robinsonstability of the solution map of a parametric linear constraint system in theform

Unlike the traditional considerations (see, e.g., [9, 75, 87]), here K needsnot to be convex This small change, seemingly, brings us a lot of benefits inusing theoretical results.

The multifunction S : Rm×n×Rm

⇒ Rn withS(A, b) := {x ∈ Rn : Ax + b ∈ K}

is said to be the solution map of (2.1) We interpret the pair (A, b) as aparameter With K being fixed, in the sequel, we will allow both the linearpart (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) 7→ S(A, b)

is a special case of the implicit multifunction y 7→ G(y) defined by (1.2).The aim of this chapter will be achieved by using the Mordukhovich Cri-terion 1 and a formula for computing exactly the limiting coderivative ofimplicit multifunctions obtained by Levy and Mordukhovich (Theorem 1.3),

as well as a result from Yen and Yao on the Robinson stability of implicitmultifunctions (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 andthe Robinson stability of the related solution maps as well as uniform localerror bounds and traditional local error bounds

According to the classification in [50, Chapter 4], (a) is a class of straint systems, while (b) and (c) are two classes of variational systems.Note that various sufficient conditions for the Lipschitz-likeness of the so-lution map of parametric constraint systems and variational systems weregiven by B S Modukhovich and other authors; see [50, Chapter 4], [53], andthe references therein It is well known that qualitative studies of variationalsystems are more difficult than those of constraint systems Interestingly,despite to the fact that (2.1) itself is a constraint system, the model can beemployed to investigate special variational systems like (b) and (c)

con-Although stability properties of (a)–(c) and related models have been ied intensively by many authors with various tools (see, e.g., [16, 22–24, 37,

stud-38, 62, 69–71, 74, 75]), the results obtained herein are new Namely, in thischapter we are able to establish the equivalence between the Lipschitz-likeproperty and the Robinson stability of the solution map of (2.1) and pro-vide a verifiable regularity condition which completely characterizes the twoproperties In addition, using the obtained results, we give necessary and suf-ficient conditions for the Lipschitz-like property and the Robinson stability

of the solution map of traditional generalized linear inequality systems undernonlinear perturbations By reducing linear complementarity problems tothe linear constraint system (2.1), we give a new result on the Lipschitz-likeproperty of their solution maps under nonlinear perturbations as well as tworelated local error bounds Similarly, at the end of the chapter, we showregularity conditions which guarantee two local error bounds and the solu-tion map of a broad class affine variational inequalities being Lipschitz-like atthe reference point when both the basic operator and the constraint systemundergo nonlinear perturbations

Since the linear complementarity problem is a type of affine variationalinequality and since the major applications of our theoretical results are re-lated to these models, we now give a brief survey of the preceding results onthe solution stability of affine variational inequalities

Dontchev and Rockafellar [18, Theorem 1] showed the equivalence betweenthe Lipschitz-like property of the solution map of a affine variational inequal-ity under canonical perturbation and others such as the semicontinuity prop-erty and the strong regularity Yao and Yen [85,86] studied the Lipschitz-likeproperty of the solution map of affine variational inequalities under linearperturbations Henrion, Mordukhovich and Nam [27] gave a comprehensivesecond-oder analysis of polyhedral systems in finite and infinite dimensionsand applications to robust stability of variational inequalities, including theaffine problems In a series of papers, Qui [65,67,70] investigated the stability

of the solution map of the parametric affine variational inequality with thematrix M being fixed Later, Qui [68] derived new results on solution stability

of parametric affine variational inequality under nonlinear perturbations

In a recent paper [26], Henrion et al have computed Fr´echet coderivative ofthe solution map of a parametric variational inequality, whose constraint set

is fixed Note that the constraint qualification condition, denoted by CRCQ(Constant Rank Constraint Qualification), automatically holds for inequalitysystems given by affine functions Hence, using [26, Theorem 3.2], one obtainsnecessary conditions for the Lipschitz-like property of the solution map of aparametric affine variational inequality with the constraint set being fixed.Sufficient conditions for the Lipschitz-like property of the solution map inquestion can be derived from the results of [18, 57].

Since we will focus mainly on parametric affine variational inequalities withthe constraint sets being perturbed, our results not only differ from those

of [18, 26, 57], but also from other existing results in [27, 65–68, 70, 85, 86]

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, weconsider 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

We will investigate the Lipschitz-like property of S around the point ( ¯w, ¯x)

in the graph of S and the Robinson stability of S at (¯x, ¯w, 0)

Since G : Rn × W → Rm is a continuously differentiable mapping, thecoderivative of G is the conjugate operator of its derivative by [50, Theo-rem 1.38, Vol 1, p 45] We can determine the operator

∇G(¯x, ¯w)∗ : Rm →Rn × W∗

by some arguments used in [40] (here we have W∗ = W ) Note that Leeand Yen [40] have considered the mapping G(x, w) = Ax, while here we haveG(x, w) = −Ax − b

Lemma 2.1 For any v0 = v10, , vm0
T

∈ Rm, it holds that

∇G(¯x, ¯w)∗(v0) = {− ¯ATv0} × {−(v0ix¯j)} × {−v0}, (2.3)where (vi0x¯j) is the m × n matrix whose (i, j)−th element is vi0x¯j

Proof The formula T (x, w) = −A¯x − ¯Ax − b, where x ∈ Rn and w = (A, b)belongs to W , defines a linear operator T : Rn × W → Rm We have

(2.4)

Since

kAxkkxk + kwk ≤

kAxkkAk ≤

kAkkxkkAk = kxkand x tends to 0, from (2.4) we can deduce that T is the Fr´echet derivative

T∗v0 = {− ¯ATv0} × {−(vi0x¯j)} × {−v0}

Since ∇G(¯x, ¯w) = T , this etablishes formula (2.3) 2Put ¯v = −G(¯x, ¯w) = ¯A¯x + ¯b Then, ¯x ∈ S( ¯w) if and only if ¯v ∈ K As aconsequence, ¯τ := (¯x, ¯w, ¯v) belongs to the graph of the constant multifunction

M : Rn × W ⇒ Rm Next, we will calculate the Mordukhovich coderivative

D∗M (¯τ ) :Rm

⇒ Rn× W∗ of M at ¯τ

Lemma 2.2 For any v0 ∈ Rm, it holds that

D∗M (¯τ )(v0) = {(x∗, w∗) ∈Rn× W∗ : (x∗, w∗, −v0) ∈ N (¯τ ; gph M )} (2.7)Since gph M = Rn × W × K, by [50, Proposition 1.2] we get

N (¯τ ; gph M ) = N ((¯x, ¯w);Rn × W ) × N (¯v; K) = {0Rn ×W} × N (¯v; K)

A criterion for the Lipschitz-like property of the solution map S of (2.1)around ( ¯w, ¯x) is formulated in the following theorem which is our first mainresult

Theorem 2.1 The mapping S is Lipschitz-like around ( ¯w, ¯x) if and only if

ker ¯AT

where ¯v = ¯A¯x + ¯b and ker ¯AT := {v0 ∈ Rm : ¯ATv0 = 0} is the kernel of ¯AT.Proof We first compute coderivative values of S Note that the multifunc-tion (x, w) 7→ M (x, w) has closed graph because G is a continuous single-f

valued map and K is closed In addition, ∇wG(¯x, ¯w) is a surjective operator.Therefore, by a result of Levy and Mordukhovich [41, Theorem 2.1], we obtain

n

{−(vi0x¯j)} × {−v0} : v0 ∈ −N (¯v; K) with 0 = ¯ATv0o = {0}

Example 2.1 Consider the case where n = m = 2,

ker ¯AT = {v0 = (v01, v20)T ∈ R2

: v20 = 0.1v10}and N (¯v; K) = {v0 = (v10, v20)T ∈ R2 : v20 = 0}, (2.8) is satisfied Hence S(·)

is Lipschitz-like around ( ¯w, ¯x) by Theorem 2.1

Example 2.2 Suppose that n, m, K are as in Example (2.1) and S : W ⇒ R2

is defined by S(w) = {x ∈ R2 | Ax + b ∈ K}, where w = (A, b) ∈ W Let

Choose ρ > 0 as small as ¯B(0, ρ) ⊂ V For w0 = ¯w and w = wε, by (2.12) weget

V ⊂ K + `k ¯w − wεk ¯BR2

or, equivalently,

V ⊂ K + `ε ¯BR2.Clearly, this conclusion is unavailable for any ε ∈ (0, 2`ρ) We have thus provedthat S(.) is not Lipschitz-like around ( ¯w, ¯x) Now, let us show that (2.8) isnot satisfied Since ¯v = ¯A¯x + ¯b = (0, 0)T, N (¯v; K) = K We have

¯

p := ( ¯A, ¯b) if kerD∗H(¯x|0) = {0} Since D∗H(¯x|0)(u) = {−ATu} for any

u ∈ −N ( ¯A¯x + ¯b; K) and D∗H(¯x|0)(u) = ∅ for any u /∈ −N ( ¯A¯x + ¯b; K), thelatter can be rewritten equivalently as (2.8) As ∇pf (¯x, ¯p)(A, b) = −A¯x − b,

we have rank∇pf (¯x, ¯p)(A, b) = m Hence, by the second assertion of Theorem3F.9 from [19, p 173], if the solution map (A, b) 7→ S(A, b) is Lipschitz-likearound (¯p, ¯x) then H is metrically regular at ¯x Therefore, the last twoproperties are equivalent, and they hold if and only if (2.8) is valid This isthe content of Theorem 2.1 The idea of Theorem 3F.9 from [19, p 173] is

to study the Lipschitz-like property of the solution map S(.) via the metricregularity of the multifunction

H(x) = f (¯x, ¯p) + ∇xf (¯x, ¯p)(x − ¯x) + F (x),which is an approximation of the map x 7→ f (x, ¯p) + F (x) The proof of thistheorem is rather complicated Our approach is simpler and more elementary.Namely, instead of using the approximation multifunction H(x), we use a

formula for the coderivative of implicit multifunctions from [41, Theorem2.1] Note that the latter was obtained just by invoking a basic sum rule and

a chain rule for coderivative

We now turn our attention to the Robinson stability of the solution map

S at a given point Firstly, using the specific structure of the constraintsystem (2.1), we will prove that Robinson stability implies the Lipschitz-likeproperty of S around the corresponding point

Lemma 2.3 If S has the Robinson stability at ω0 := ( ¯w, ¯x, 0), then it isLipschitz-like around ( ¯w, ¯x)

Proof Suppose S has the Robinson stability at ω0 It is easy to check thatthe multifunction M (x, w) = −Ax − b + K, where w = (A, b), has the innerf

semicontinuity property defined in (1.4) at the point ω0 = ( ¯w, ¯x, 0) ∈ gphM f

As observed in Sect 1.2, by the Robinson stability of S we can find a constant

r > 0, and bounded neighborhoods U of ¯w and V of ¯x, such that

d(x; S(w)) ≤ rd(0; F (x, w)) (2.13)for any w ∈ U and x ∈ V In particular, S(w) 6= ∅ for all w ∈ U

As V is bounded, we can choose α > 0 such that kxk ≤ α for all x ∈ V

To show that S is Lipschitz-like around ( ¯w, ¯x), take two distinct elements

w1 = (A1, b1), w2 = (A2, b2) in U , and suppose x1 ∈ S(w1) ∩ V is chosenarbitrarily Then, we have kx1k ≤ α and A1x1 + b1 ∈ K By (2.13),

and hence,

x1 ∈ S(w2) + ρkw2− w1k ¯BRn.Consequently, we get

kerD∗M (ωf 0) = {0}, (2.14)and the second is

n

w0 ∈ W∗ : ∃v0 ∈ Rm with (0, w0) ∈ D∗M (ωf 0)(v0)o = {0} (2.15)Due to the sum rule for the Mordukhovich coderivative [50, Theorem 1.62],for any v0 ∈ Rm, we have

ix¯j)} × {−v0} = 0 if andonly if v0 = 0 By (2.9), condition (2.15) means that (2.11) is satisfied In theproof of Theorem 2.1, we have shown that the latter is equivalent to (2.8)

In other words, condition (2.8) guarantees the Robinson stability of S at ω0

2

As shown by Jeyakumar and Yen [34, p 1119], the Lipschitz-like property

of an implicit multifunction doesn’t imply the Robinson stability, and versa But, for the solution map S of (2.1), these properties are equivalent.Let us explain why it happens so First, for a fixed x, the map

vice-w = (A, b) 7→ F (x, vice-w) = −Ax − b

is a linear operator In Example 3.6 from [34], which shows that the Robinsonstability does not yield the Lipschitz-like property, the corresponding map isnonlinear Second, for a fixed w = (A, b), the map

x 7→ F (x, w) = −Ax − b

is an affine operator Meanwhile, in Example 3.7 from [34], which showsthat the Lipschitz-like property does not imply the Robinson stability, thecorresponding map is again nonlinear Thus, the equivalence between thetwo properties in question is available for the solution map of (2.1) because,although the map (x, w) 7→ F (x, w) = −Ax − b with w := (A, b) is nonlinear,

it is bi-affine on the variables x and w

Our second main result is as follows

Theorem 2.2 The Lipschitz-like property of S around ( ¯w, ¯x) and its son stability at ω0 = ( ¯w, ¯x, 0) are equivalent Moreover, these propertiesappear if and only if condition (2.8) is satisfied

Robin-Proof The assertions follow from Theorem 2.1 and Lemmas 2.3 and 2.4 2Remark 2.2 The equivalence between the Robinson stability and the Lips-chitz-like property comes by dint of the intermediate condition (2.1) Wedon’t have any direct proof of the fact that the latter property also impliesthe former

Remark 2.3 Since the Mordukhovich coderivative of S at ( ¯w, ¯x) can be puted explicitly by (2.10), the exact Lipschitzian bound of S around ( ¯w, ¯x)can be estimated by invoking formula (4.5) from [50, Theorem 4.10] Then,the formula ` = max{r, rα}, which has been used in the proof of Lemma 2.3,gives us some idea about the relationships between the exact Lipschitzianbound of S around ( ¯w, ¯x) and the exact Robinson regularity bound of S at

com-ω0 = ( ¯w, ¯x, 0) A deeper analysis of the two bounds can lead us to an upperestimate for the latter

Condition (2.1) means that the conjugate operator ¯AT well interacts withthe normal cone N (¯v, K) Namely, it requires that the cone N (¯v, K) doesn’thave any non-zero common element with the linear subspace ker ¯AT ofRm Tocalculate ker ¯AT, we have to solve a homogeneous system of linear equations,which contains n equations and m variables If K is a polyhedral convex set,one has an explicit formula for N (¯v, K) If K is defined by finitely manysmooth equalities and inequalities, a formula for the cone N (¯v, K) is alsoavailable.

2.3 Stability Properties of Generalized Linear

Inequal-ity Systems

With K specially being a closed convex cone, applying the results of theprevious section to the generalized linear inequality system (2.2), we candescribe necessary and sufficient conditions for the Lipschitz-like propertyand the Robinson stability of the solution map S as follows

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

(a) S is Lipschitz-like around ( ¯w, ¯x);

(b) S has the Robinson stability at ω0 = ( ¯w, ¯x, 0);

(ker ¯AT) ∩ K∗ = {0};

(ker ¯AT) ∩ (¯v)⊥ = {0};

K∗ ∩ (¯v)⊥ = {0}

Proof The equivalence of (a), (b), and (c) follows from Theorem 2.2.

To show that (c) is equivalent to (d), we prove N (¯v; K) = K∗ ∩ (¯v)⊥.Once v0 ∈ K∗ and v0 ∈ (¯v)⊥, v0 ∈ N (¯v; K) is easy If v0 ∈ N (¯v; K) then

hv0, v − ¯vi ≤ 0 for all v ∈ K Substituting v = 0 and v = 2¯v in turn tothe last inequality, we have hv0, ¯vi ≥ 0 and hv0, ¯vi ≤ 0 Hence hv0, ¯vi = 0, or

v0 ∈ (¯v)⊥ In addition, for any v in K, replacing v by v + ¯v in this inequality,

we have hv0, vi ≤ 0; therefore, v0 belongs to K∗

To prove (c) implies (e), suppose on the contrary that (c) is valid, but

0 ∈ Rm is a boundary point of the convex set C := rge ¯A + K − ¯v Bythe separation theorem [78, Corollary 11.6.1, p 100] , we can find a nonzerovector x∗ ∈ Rm such that hx∗, yi ≤ 0 for every y ∈ C Substituting y = ¯Axinto the last inequality yields

h ¯ATx∗, xi = hx∗, ¯Axi ≤ 0 ∀x ∈ Rn

;hence x∗ ∈ ker ¯AT Now, for y = v − ¯v, v ∈ K, we get hx∗, v − ¯vi ≤ 0for all v ∈ K This means that x∗ ∈ N (¯v; K) Thus, the property x∗ 6= 0contradicts (c)

Since cone(rge ¯A + (K − ¯v)) = rge ¯A + cone(K − ¯v), (e) implies (f)

To complete the proof, we need to show that (f) yields (c) If x∗ belongs

to ker ¯AT

∩ N (¯v; K), then hx∗, ¯Axi = 0 for all x ∈ Rn and hx∗, v − ¯vi ≤ 0for all v ∈ K Therefore,

hx∗, ¯Ax + λ(v − ¯v)i ≤ 0 ∀x ∈ Rn, v ∈ K, λ ≥ 0

By (f), we can assert that x∗ = 0 Thus, (c) holds the true 

Remark 2.4 The equivalences among (b), (e), and (f) were established byRobinson [74] long time ago

Remark 2.5 We now look at the case where K is the second-order cone orthe positive semidefinite cone because this corresponds to the solution maps

of parametric second-order cone programming problems (see, e.g., [6]) andparametric semidefinite programming problems (see, e.g., [8, pp 470–496])which have numerous applications in engineering applications The second-order cone in Rm with m ≥ 3 (also called the Lorentz cone) is given by

Since K∗ = −K, the regularity condition (d) in Theorem 2.3 can be easilyverified Hence, the result can be applied to programming problems under theconstraint (2.1) with K being the second-order cone, or a product of severalsecond-order cones The latter situation was considered in [6, p 1732] Now,consider the linear semidefinite programming problem [8, p 471] of the formminn

Here K is also a closed convex cone It is not difficult to show that and

K∗ = −K and kerAT is the set of C ∈ Sm satisfying the following system

Mαβ indicates the αβ-element of M Hence the regularity condition (d) inTheorem 2.3 is verifiable

2.4 The Solution Maps of Linear Complementarity

Prob-lems

In this section, we apply results on the stability of solution map of metric linear constraint system (2.1) to investigate parametric linear comple-mentarity problems

para-Given a vector q inRn, and a matrix M inRn×n, the linear complementarityproblem (LCP) aims at finding a vector x in Rn such that

K =

(

uv

!

∈ Rn ×Rn : u ≥ 0, v ≥ 0, vTu = 0

)

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

!

,and note that ¯y ∈ K

Continuity properties of the solution map of LCP have been studied byseveral authors (see, e.g., [16, 22, 23, 37, 62] and the references therein) Inthis section, we obtain a new result on the Lipschitz-like property of thatsolution map, as well as two related local error bounds

For convenience, we present the above cone K in the form

K =

(

uv

Definition 2.1 We say that the solution map Sol(.) of LCP satisfies theuniform local error bound at (( ¯M , ¯q), ¯x) if there exist constants r > 0, δ > 0,and a neighborhood V of ¯x such that

for any x ∈ V and (M, q) satisfying kM − ¯M k + kq − ¯qk < δ

From (2.19) we can infer that

Let us consider a regularity condition: If u0 = (u01, , u0n)T ∈ Rn and if

The major result of this section reads as follows

Theorem 2.4 Suppose that ¯x ∈ Sol( ¯M , ¯q) If the regularity condition (2.21)

is satisfied, then the problem LCP has the uniform local error bound (2.19)and the traditional local error bound (2.20) at (( ¯M , ¯q), ¯x) and its solutionmap Sol(.) is Lipschitz-like around (( ¯M , ¯q), ¯x)

Proof According to Theorem 2.2, if (2.8) is fulfilled, then there exist aconstant r > 0, a neighborhood U of ¯w, and a bounded neighborhood V of

¯

x satisfying

d(x; S(w)) ≤ rd(Ax + b; K) (2.22)for all w = (A, b) ∈ U and x ∈ V Here we have

d(Ax + b; K) = d

M x + qx

!

; K

!

Recall that x ∈ S(w) if and only if x ∈ Sol(M, q) Hence, by (2.22) there is

δ > 0 such that, for any x ∈ V and (M, q) with kM − ¯M k + kq − ¯qk < δ, onehas

d(x; Sol(M, q)) ≤ rd

M x + qx

Moreover, similarly as in the proof of Lemma 2.3, from (2.23) we can derivethe existence of a constant ` = max{r, rα} such that

Sol(M1, q1) ∩ V ⊂ Sol(M2, q2) + `(kM2 − M1k + kq2− q1k) ¯BRn

for all the pairs (M1, q1) and (M2, q2) satisfying kM1− ¯M k + kq1− ¯qk < δ and

kM2 − ¯M k + kq2 − ¯qk < δ Thus, the solution map Sol(.) is Lipschitz-likearound (( ¯M , ¯q), ¯x)

To complete the proof, it suffices to show that (2.8) is equivalent to (2.21).Let

Meanwhile, using the presentation (2.17) for K where the cones Ki are given

in (2.18), we can assert by [50, Proposition 1.2] that

− if i ∈ I3.(Here and in the sequel, Rn

− denotes the nonpositive orthant in Rn.) Thus, byvirtue of (2.24), we see that the regularity condition (2.21) can be rewritten

Remark 2.6 If I1 = I, i.e., ¯x = 0, ¯M ¯x+ ¯q > 0, the regularity condition (2.21)holds automatically.

Remark 2.7 If I2 = I, i.e., ¯x > 0, ¯M ¯x + ¯q = 0, condition (2.21) is equivalentto

¯

MTu0 = 0 =⇒ u0 = 0 (∀u0 ∈ Rn

),which means that ¯M is nonsingular

We now give a complete geometrical description of the regularity tion (2.21)

of finitely many closed cones In this way, (2.21) is reformed as

u0 ∈ L1 ∩ L2 ∩ L3 =⇒ u0 = 0 (∀u0 ∈ Rn

)

Let us consider two illustrative examples

Example 2.3 Choose n = 2, and ¯M =

u0 = 0

In each one of these situations, by Theorem 2.4 and the above remarks,the solution map Sol(.) of LCP is Lipschitz-like around (( ¯M , ¯q), ¯x) and it hasthe uniform local error bound (2.19) at (( ¯M , ¯q), ¯x)

noth-Based on a recent study of Huyen and Yao [28], we now give some remarks

on the observation from one referee who says that the sufficient condition(2.21) for the local error bounds (2.19) and (2.20) might be related to thedefinition of P -matrix

Recall [16, Definition 3.3.1] that M ∈Rn×n is a P -matrix if all its principalminors are positive The class of such matrices is denoted P For a subset

α ⊂ I, the submatrix (mij)i∈α,j∈α of an (n × n) matrix M = (mij) is denoted

by Mαα Given three subsets α, β, γ of I, the 9-block matrix

to the local error bounds for affine variational inequalities, which will be