Auslender,Teboulle(2003) asymptotic cones and functions in optimization and variational inequalities ()

Apivotal role in these developments has been played by convex analysis, a rich area covering a broad range of problems in mathematical sciencesand its applications.. Herewe develop many

Asymptotic Cones and

Functions in

Optimization and Variational Inequalities

Alfred Auslender

Marc Teboulle

Springer

This book is dedicated to

Martine, Fran¸cois, and J´erˆome Rachel, Yoav, Yael, and Keren

Nonlinear applied analysis and in particular the related fields of continuousoptimization and variational inequality problems have gone through majordevelopments over the last three decades and have reached maturity Apivotal role in these developments has been played by convex analysis,

a rich area covering a broad range of problems in mathematical sciencesand its applications Separation of convex sets and the Legendre–Fenchelconjugate transforms are fundamental notions that have laid the groundfor these fruitful developments Two other fundamental notions that havecontributed to making convex analysis a powerful analytical tool and thathave often been hidden in these developments are the notions of asymptoticsets and functions

The purpose of this book is to provide a systematic and comprehensiveaccount of asymptotic sets and functions, from which a broad and use-ful theory emerges in the areas of optimization and variational inequali-ties There is a variety of motivations that led mathematicians to studyquestions revolving around attaintment of the infimum in a minimizationproblem and its stability, duality and minmax theorems, convexification

of sets and functions, and maximal monotone maps In all these topics

we are faced with the central problem of handling unbounded situations.This is particularly true when standard compactness hypotheses are notpresent The appropriate concepts and tools needed to study such kinds ofproblems are vital not only in theory but also within the development ofnumerical methods For the latter, we need not only to prove that a se-quence generated by a given algorithm is well defined, namely an existence

result, but also to establish that the produced sequence remains bounded.One can seldom directly apply theorems of classical analysis to answer tosuch questions The notions of asymptotic cones and associated asymp-totic functions provide a natural and unifying framework to resolve thesetypes of problems These notions have been used mostly and traditionally

in convex analysis, with many results scattered in the literature Yet theseconcepts also have a prominent and independent role to play in both con-vex and nonconvex analysis This book presents the material reflecting thislast point with many parts, including new results and covering convex andnonconvex problems In particular, our aim is to demonstrate not only theinterplay between classical convex-analytic results and the asymptotic ma-chinery, but also the wide potential of the latter in analyzing variationalproblems

We expect that this book will be useful to graduate students at an advancedlevel as well as to researchers and practitioners in the fields of optimiza-tion theory, nonlinear programming, and applied mathematical sciences

We decided to use a style with detailed and often transparent proofs Thismight sometimes bore the more advanced reader, but should at least makethe reading of the book easier and hopefully even enjoyable The material

is presented within the finite-dimensional setting Our motivation for thischoice was to eliminate the obvious complications that would have emergedwithin a more general topological setting and would have obscured thestream of the main ideas and results For the more advanced reader, it isnoteworthy to realize that most of the notions and properties developedhere can be easily extended to reflexive Banach Spaces, assuming a sup-plementary condition with respect to weak convergence The extension tomore general arbitrary topological spaces is certainly not obvious, but thefinite-dimensional setting is rich enough to motivate the interested readertoward the development of corresponding results needed in areas such aspartial differential equations and probability analysis

Structure of the Book

In Chapter 1 we recall the basic mathematical background: elementary vex analysis and set-valued maps The results are presented without proofs.This material is classical and can be skipped by anyone who has had a stan-dard course in convex analysis None of this chapter’s results rely on anyasymptotic notions Chapter 2 is the heart of the book and gives the funda-mental results on asymptotic cones and functions The interplay betweengeometry and analysis is emphasized and will be followed consistently inthe remaining chapters Building on the concept of asymptotic cone of theepigraph of a function, the notion of asymptotic function emerges, andcalculus at infinity can be developed The role of asymptotic functions informulating general optimization problems is described Chapter 3 studiesthe existence of optimal solutions for general optimization problems andrelated stability results, and also demonstrates the power of the asymptotic

con-results developed in Chapter 2 Standard con-results under coercivity and weakcoercivity assumptions imply that the solution set is a nonempty compactset and the sum of a compact set with a linear space, respectively Here

we develop many new properties for the noncoercive and weakly coercivecases through the use of asymptotic sets to derive more general existenceresults with applications leading to some new theorems “`a la Helly” andfor the convex feasibility problems In Chapter 4 we study the subject ofminimizing stationary sequences and error bounds Both topics are central

in the study of numerical methods The concept of well-behaved asymptoticfunctions and the properties of such functions, which in turn is linked tothe problems of error bounds associated with a given subset of a Euclideanspace, are introduced A general framework is developed around these twothemes to characterize asymptotic optimality and error bounds for convexinequality systems Duality theory plays a fundamental role in optimiza-tion and is developed in Chapter 5 The abstract perturbational scheme,valid for any optimization problem, is the starting point of the analysis.Under a minimal set of assumptions and thanks to asymptotic calculus,

we derive key duality results, which are then applied to cover the classicalLagrange and Fenchel duality as well as minimax theorems, in a simple andunified way Chapter 6 provides a self-contained introduction to maximalmonotone maps and variational inequalities Solving a convex optimizationproblem is reduced to solving a generalized equation associated with thesubdifferential map In many areas of applied mathematics, game theory,and equilibrium problems in economy, generalized equations arise and aredescribed in terms of more general maps, in particular maximal monotonemaps The chapter covers the classical material together with some morerecent results, streamlining the role of asymptotic functions

Each chapter ends with some bibliographical notes and references We didnot attempt to give a complete bibliography on the covered topics, which

is rather large, and we apologize in advance for any omission in the citedreferences Yet, we have tried to cite all the sources that have been used

in this book as well as some significant original historical developments,together with more recent references in the field that should help to guideresearchers for further reading

The book can be used as a complementary text to graduate courses inapplied analysis and optimization theory It can also serve as a text for

a topics course at the graduate level, based, for example, on Chapters 2,

3, and 5, or as an introduction to variational inequality problems throughChapter 6, which is essentially self-contained

1 Convex Analysis and Set-Valued Maps: A Review 1

1.1 Convex Sets 1

1.2 Convex Functions 9

1.3 Support Functions 17

1.4 Set-Valued Maps 20

1.5 Notes and References 23

2 Asymptotic Cones and Functions 25 2.1 Definitions of Asymptotic Cones 25

2.2 Dual Characterization of Asymptotic Cones 31

2.3 Closedness Criteria 32

2.4 Continuous Convex Sets 44

2.5 Asymptotic Functions 47

2.6 Differential Calculus at Infinity 60

2.7 Application I: Semidefinite Optimization 66

2.8 Application II: Modeling and Smoothing Optimization Prob-lems 72

2.9 Notes and References 78

3 Existence and Stability in Optimization Problems 81 3.1 Coercive Problems 81

3.2 Weak Coercivity 85

3.3 Asymptotically Level Stable Functions 93

3.4 Existence of Optimal Solutions 96

3.5 Stability for Constrained Problems 100

3.6 Dual Operations and Subdifferential Calculus 107

3.7 Additional Results in the Convex Case 112

3.8 The Feasibility Problem 116

3.9 Notes and References 118

4 Minimizing and Stationary Sequences 119 4.1 Optimality Conditions in Convex Minimization 119

4.2 Asymptotically Well-Behaved Functions 124

4.3 Error Bounds for Convex Inequality Systems 133

4.4 Stationary Sequences in Constrained Minimization 140

4.5 Notes and References 143

5 Duality in Optimization Problems 145 5.1 Perturbational-Conjugate Duality 145

5.2 Fenchel Duality 154

5.3 Lagrangian Duality 157

5.4 Zero Duality Gap for Special Convex Programs 162

5.5 Duality and Asymptotic Functions 166

5.6 Lagrangians and Minimax Theory 170

5.7 Duality and Stationary Sequences 178

5.8 Notes and References 181

6 Maximal Monotone Maps and Variational Inequalities 183 6.1 Maximal Monotone Maps 183

6.2 Minty Theorem 186

6.3 Convex Functionals and Maximal Monotonicity 191

6.4 Domains and Ranges of Maximal Monotone Maps 195

6.5 Asymptotic Functionals of Maximal Monotone Maps 197

6.6 Further Properties of Maximal Monotone Maps 206

6.7 Variational Inequalities Problems 212

6.8 Existence Results for Variational Inequalities 214

6.9 Duality for Variational Inequalities 221

6.10 Notes and References 230

prob-1.1 Convex Sets

Throughout this book we will consider only finite-dimensional vector spaces

The n-dimensional real Euclidean vector space will be denoted byRn For

vectors x = (x1, , x n)∈ R n , y ∈ R n , the inner product between x and y

the above definition can also be written in the useful and compact notation

Hyperplanes, Convex and Affine Hulls

A hyperplane is a set H := {x ∈ R n |x, a = α}, where a ∈ R n , a

and α ∈ R A hyperplane divides the space R n into two closed half-spaces

H+ = {x ∈ R n |x, a ≥ α} and H − ={x ∈ R n |x, a ≤ α} Clearly, a

hyperplane and its associated half-spaces are convex sets

A set C is an affine manifold (or affine subspace) if

(a) C is an affine manifold;

(b) C = x+M = {y |y−x ∈ M}, where M is a subspace called the subspace parallel to M ;

(c) C = {x |Ax = b} for some A ∈ R m×n , b ∈ R m

For a set C ⊂ R n , the affine hull of C, denoted by aff C is the intersection of all affine manifolds containing C; the convex hull of C, denoted by conv C,

is the intersection of all convex sets containing C.

Proposition 1.1.2 For a set C ⊂ R n , the following properties hold: (a) conv C is the set of all convex combinations of elements of C, i.e.,

(c) The finite sum
m

i=1 C i is convex, with n i = n for all i.

(d) The image of a convex set under a linear mapping is convex.

A fundamental characterization of convex sets is provided by Carath´eodory’stheorem

Theorem 1.1.1 For any C ⊂ R n , any element of conv C can be sented as a convex combination of no more than (n + 1) elements of C.

repre-Topological Properties of Convex Sets

We now recall some basic topological concepts associated with convex sets

The closed unit ball in the n dimensional Euclidean space Rn will be noted by:

de-B = {x ∈ R n | x ≤ 1}.

The ball with center x0 and radius δ can thus be written as B(x0, δ) :=

x0+ δB Let C ⊂ R n be a convex set The interior and closure of C are

also convex sets defined respectively by

int C := {x ∈ R n | ∃ε > 0 such that x + εB ⊂ C},

cl C := 

ε>0

(C + εB).

The boundary of a set C ⊂ R n is bd C := cl C \ int C A point z belongs

to the boundary of C if and only if for any ε > 0 the ball B(z, ε) contains

a point of C as well as a point that is not in C.

For a nonempty convex set ofRn with int C n The

interior of a convex set C relative to its affine hull is called the relative interior of C and is defined by

ri C := {x ∈ aff C | ∃ε > 0 such that (x + εB) ∩ aff C ⊂ C}.

The relative interior clearly coincides with the interior when the affine hull

is the whole space Rn However, while for a nonempty convex set C we may have int C = ∅, in contrast, the relative interior ri C is not equal to

∅ This is an important property of the relative interior, which thus should

be used in place of the interior when int C = ∅.

The difference set cl C \ ri C is called the relative boundary of C and is

and thus ri C is convex Furthermore,

cl C = cl(ri C), ri C = ri(cl C).

Proposition 1.1.5 Let C ⊂ R n be nonempty and convex Then x ∈ ri C

if and only if for every y ∈ C there exists ε > 0 such that x + ε(y − x) ∈ C.

A useful consequence of this characterization is the following

Corollary 1.1.1 Let C ⊂ R n be convex Then x ∈ int C if and only if for every d ∈ R n there exists ε > 0 such that x + εd ∈ C.

Proposition 1.1.6 For a nonempty convex set C ⊂ R n one has

(a) ri C ⊂ C ⊂ cl C,

(b) cl cl C = cl C; ri(ri C)) = ri C,

(c) A(cl C) ⊂ cl A(C) and ri A(C) = A(ri C), where A : R n → R m is

a linear mapping Moreover, let A −1 (S) := {x ∈ R n | A(x) ∈ S} be the inverse image of A for any set S ⊂ R n Then, if A −1 (ri C)

ri(A −1 C) = A −1 (ri C), cl(A −1 C) = A −1 (cl C).

Proposition 1.1.7 For a family of convex sets C i ⊂ R n indexed by i ∈ I and such that ∩ i ∈I ri C i

cl

i∈I

C i=

i∈I clC i Furthermore, if I is a finite index set, then

ri

i ∈I

C i =

i ∈I riC i

As applications of these results one can obtain several other important rulesinvolving relative interiors The next one is frequently used

Proposition 1.1.8 For two convex sets C, D ⊂ R n and for any scalars

α, β ∈ R, one has

ri(αC + βD) = α ri C + β ri D.

Thus, in particular with α = −β = 1, we have

0

Proposition 1.1.9 Let C be a convex set in Rm+p For each y ∈ R m , let

C y:={z ∈ R p | (y, z) ∈ C} and let D := {y | C y

(y, z) ∈ ri C ⇐⇒ y ∈ ri D and z ∈ ri C

Separation of Convex Sets

Two fundamental results based on properties of closed convex sets andtheir interiors are the supporting hyperplane theorem and the separation

principle A hyperplane H := {x ∈ R n | x, a = α}, with a ∈ R n , a

0, α ∈ R, is said to separate two sets C1, C2 in Rn if C1 is contained in

the closed half-space H+ ={x ∈ R n | x, a ≥ α}, while C2 is contained

in the other The separation is called proper if the hyperplane itself does

not actually include both C1 and C2 When the sets C1, C2 lie in differenthalf-spaces {x ∈ R n | x, a ≥ α2}, {x ∈ R n | x, a ≤ α1}, α2> α1, then

one says that one has strong separation between C1 and C2

Proposition 1.1.10 Let C1 and C2 be nonempty sets in Rn Then there exists a hyperplane properly separating C1and C2if and only if there exists

a nonzero vector a ∈ R n such that

inf{a, x | x ∈ C1} ≥ sup{a, x | x ∈ C2} and

sup{a, x |x ∈ C1} > inf{a, x | x ∈ C2}.

The next two theorems provide the main conditions that guarantee ration between two convex sets

sepa-Proposition 1.1.11 (Proper separation)

Let C1 and C2 be nonempty convex sets in Rn Then there exists a plane that separates them properly if and only if ri C1 and ri C2 have no point in common.

hyper-Proposition 1.1.12 (Strong separation)

For two nonempty convex sets C1and C2inRn such that C1∩C2=∅, with

C1closed and C2 compact, there exists a hyperplane that strongly separates them; i.e., there exist a vector 0 n and a scalar α ∈ R such that

a, x1 ≤ α < a, x2 ∀x1∈ C1, x2∈ C2.

Cones and Polyhedral Sets

Cones are fundamental geometric objects associated with sets They play akey role in several aspects of mathematics, and will be used extensivelythroughout this book Here we recall some elementary properties, wellknown results, and examples Further properties and examples will be given

in Section 2.7, dealing with real symmetric matrices and related nite optimization problems

semidefi-Definition 1.1.2 A set K ⊂ R n is called a cone if tx ∈ K for all x ∈ K and for all t ≥ 0.

Examples of convex cones include linear subspaces of Rn and the ative orthant Rn

nonneg-+ := {x |x i ≥ 0, i = 1, , n} Other cones playing an

important role in convex optimization problems are the cone of symmetric

real positive semidefinite matrices of order n and the cone of Lorentz More

details are given in Chapter 2, Section 2.7

Proposition 1.1.13 For K ⊂ R n the following are equivalent:

(a) K is a convex cone.

(b) K is a cone such that K + K ⊂ K.

Definition 1.1.3 A cone K ⊂ R n is pointed if the equation x1+· · ·+x p=

0 has no solution with x i ∈ K unless x i = 0 for all i.

Pointedness of convex cones can be checked via the following test Let

K ⊂ R n be a convex cone Then

K is pointed ⇐⇒ K ∩ (−K) = {0}.

Given a cone K ⊂ R n , the polar of K is the cone defined by

K ∗:={y ∈ R n | y, x ≤ 0, ∀x ∈ K}.

Orthogonality of subspaces is a special case of polarity of cones If M is a

subspace of Rn, one has

M ∗ = M ⊥ ={y ∈ R n | y, x = 0, ∀x ∈ M}.

The bipolar is the cone K ∗∗ := (K ∗ ∗

Proposition 1.1.14 For a cone K ⊂ R n , the polar cone is closed and convex, and K ∗∗ = cl(conv K) If K is also closed and convex, one then

An important and useful object in variational problems is the normal cone

to a given convex set

Definition 1.1.4 Let C ⊂ R n be a nonempty convex set The normal cone

N C(¯x) to C at ¯ x ∈ C is defined by

N C(¯ := {v ∈ R n | v, x − ¯x ≤ 0 ∀x ∈ C}

= {v ∈ R n | v, ¯x = sup{v, x | x ∈ C}}.

For all x C (x) := ∅ and

x ∈ int C ⇐⇒ N C (x) = {0}.

Moreover, N C(¯x) is pointed if and only if int C

Furthermore, for a closed set C ⊂ R n , a boundary point of the set C can

be characterized through the normal cone of C In fact, one has

Another geometrical object associated with a convex set C ⊂ R n, and

closely related to the normal cone of C at ¯ x, is the tangent cone of C at ¯ x,

given by

T C(¯x) = cl {d ∈ R n | ∃t > 0 with ¯x + td ∈ C}.

For any ¯x ∈ C, the cones N C(¯x) and T C(¯x) are polar to each other.

Some useful operations on polar cones are summarized below

Proposition 1.1.16 For cones K i ofRn , i = 1, 2, one has:

(a) K1⊂ K2=⇒ K ∗

2 ⊂ K ∗

1 and K1∗∗ ⊂ K ∗∗

2 (b) K = K1+ K2=⇒ K ∗ = K ∗

1∩ K ∗

2 (c) K = K1∩ K2with K i closed = ⇒ K ∗ = cl(K ∗

1+ K2∗ ).

The closure operation can be removed if 0 ∈ int(K1− K2).

(d) For a family of cones {K i | i ∈ I} in R n ,

K = ∪ i ∈I K i =⇒ K ∗=∩ i ∈I K i ∗

(e) Let A :Rn → R m be a linear mapping and K a closed convex cone of

Rm Then {x | Ax ∈ K} ∗ = cl{A T y | y ∈ K} The closure operation can

be removed when 0 ∈ int(K − rge A), where rge A := {Ax |x ∈ R n }.

A cone K ⊂ R n is said to be finitely generated if it can be written as

For a given nonempty set C ⊂ R n, the smallest cone containing the set

C is called the positive hull (or conical hull) of C It is the smallest cone

containing the set C and is given by

Proposition 1.1.17 Let C1 and C2 be nonempty convex sets inRn , with

C1 being a polyhedral set Then C1∩ ri C2

hyperplane separating C1 and C2 properly and not containing C2.

Theorem 1.1.2 (Minkowski–Weyl Theorem)

A cone K is polyhedral if and only if it is finitely generated.

An important implication of this theorem is the following decompositionformula expressing a polyhedral set

Proposition 1.1.18 A set P is polyhedral if and only if there exist a

nonempty and finite collection of points {a1, , a p } and a finitely erated cone K such that

This kind of representation by convex hulls of minimal sets can be tended to general closed convex sets but requires the notion of extremepoint and extreme ray

ex-For any two points x and y, the closed (open) line segment joining x and

y defined for any t ∈ [0, 1] (t ∈ (0, 1)) by tx + (1 − t)y is denoted by [x, y]

(]x, y[).

Let C ⊂ R n be nonempty and convex A nonempty subset F ⊂ C is said

to be a face of C if

A point z ∈ C is an extreme point of C if {z} is a face, namely, z cannot

be written in the form z = λx + (1

λ ∈ (0, 1).The set of extreme points of C is denoted by ext C An extreme

ray of C is the direction of a half-line that is a face of C We denote by extray C the union of extreme rays of C.

Theorem 1.1.3 (Krein–Milman) Let C be a nonempty closed convex set

containing no lines Then C = conv(ext C ∪ extray C).

When C is a compact set, one has ext C

theorem implies the following,

Theorem 1.1.4 (Minkowski) Let C be a nonempty compact convex set in

Rn Then C = conv ext C.

1.2 Convex Functions

We denote byR := [−∞, +∞] the whole extended real line It is most

con-venient, in particular in the context of optimization problems, to workwith extended real-valued functions, i.e., functions that take values in

R ∪ {+∞} = (−∞, +∞], instead of just finite-valued functions, i.e., those

taking values inR = (−∞, +∞) Rules of arithmetic are thus extended to

include

∞ + ∞ = ∞, α · ∞ = ∞, ∀α ≥ 0, inf ∅ = ∞, sup ∅ = −∞.

Let f :Rn → R The effective domain of f is the set

dom f := {x ∈ R n |f(x) < +∞}.

A function is called proper if f (x) < ∞ for at least one x ∈ R n and

f (x) > −∞, ∀x ∈ R n otherwise, the function is called improper Two

important and useful geometrical objects associated with a function f are the epigraph and level set of f , defined, respectively by

epi f := {(x, α) ∈ R n × R |α ≥ f(x)},

lev(f, α) := {x ∈ R n |f(x) ≤ α}.

The epigraph is thus a subset of Rn+1 that consists of all points ofRn+1

lying on or above the graph of f From the above definitions one has

(x, α) ∈ epi f ⇐⇒ x ∈ lev(f, α).

For f :Rn → R, we write

inf f := inf{f(x) | x ∈ R n },

arg min f = arg min{f(x) | x ∈ R n } := {x ∈ R n | f(x) = inf f}.

An optimization problem can thus be expressed equivalently in terms of itsepigraph and level set as

inf f = inf {α | (x, α) ∈ epi f},

while the set of optimal solutions is

arg min f = lev(f, inf f ).

Some useful operations involving epigraphs and level sets are collected inthe next proposition

Proposition 1.2.1 Let f : R → R and let L : (x, α) → x be the projection

maps mapping from Rn+1 toRn The following hold:

(a) dom f = L(epi f ).

x →y f (x) = min {α ∈ R | ∃x n → y with f(x n)→ α}.

Note that one always has lim infx →y f (x) ≤ f(y).

Definition 1.2.1 The function f :Rn → R is lower semicontinuous (lsc)

at x if

f (x) = lim inf

y →x f (y), and lower semicontinuous on Rn if this holds for every x ∈ R n

Theorem 1.2.1 Let f :Rn → R The following statements are equivalent: (a) f is lsc at x.

(b) lim inf n→∞ f (x n)≥ f(x), ∀x n → x.

(c) For each α such that f (x) > α ∃δ > 0 such that f(y) > α, ∀y ∈ B(x, δ).

Lower semicontinuity onRn of a function can be characterized through itslevel set and epigraph

Theorem 1.2.2 Let f :Rn → R The following statements are equivalent: (a) f is lsc on Rn

(b) The epigraph epi f is closed on Rn × R.

(c) The level sets lev(f, α) are closed in Rn

When f is not lower semicontinuous, then its epigraph is not closed, but cl(epi f ) is closed and leads to the lower closure of f , denoted by cl f , and defined such that epi(cl f ) = cl(epi f ) In terms of f one then has

(cl f )(x) := lim inf

y →x f (y),

and it holds that cl f ≤ f This function is the greatest of all the lsc

functions g such that g ≤ f.

Definition 1.2.2 A function f :Rn

epi f is a nonempty convex set.

As a consequence, since the effective domain dom f is the projection of epi f , it follows that dom f is convex when f is convex.

Alternatively, for an extended real-valued function f :Rn → R ∪ {+∞}, f

is convex if and only if

f (tx + (1 − t)y) ≤ tf(x) + (1 − t)f(y), ∀x, y ∈ R n , ∀t ∈ (0, 1).

The function is called strictly convex if the above inequality is strict for all

x, y ∈ R n with x

A function f is called concave whenever −f is convex Convexity of f can

also be defined through Jensen’s inequality, namely, f is convex if and only

If f is convex, then lev(f, α) is a convex set for all α ∈ R (the converse

statement does not hold)

The indicator function of a set C ofRn , denoted by δ C, is defined by

The indicator function allows for recovering the definition of a convex

real-valued function defined on a convex set C Indeed, given g a real-real-valued function on a convex set C, then with f := g + δ C, Definition 1.2.2 becomes

g(tx + (1 − t)y) ≤ tg(x) + (1 − t)g(y), ∀x, y ∈ C, ∀t ∈ [0, 1].

The following inf-projection operation of a convex function is particularlyuseful

Proposition 1.2.2 Let Φ :Rn ×R m → R∪{+∞} be convex Then, ϕ(u) =

infx Φ(x, u) is convex onRm , and the set S(u) = arg min x Φ(x, u) is convex

func-Proposition 1.2.5 Let f : Rn → R ∪ {+∞} be a proper convex tion Then cl f is a lsc proper convex function, and dom(cl f ) and dom f have the same closure and relative interior, as well as the same dimension Furthermore, f = cl f on ri dom f

func-It is thus important to remark that a convex function that is lsc at somepoint and takes the value−∞ cannot take values other than +∞ and −∞.

Proposition 1.2.6 Let f :Rn → R ∪ {+∞} be a proper convex function Then

(cl f )(x) = lim

t→0+f (x + t(y − x)), ∀x ∈ R n , ∀y ∈ ri(dom f).

If in addition f is assumed lsc, then

f (x) = lim

t →0+f (x + t(y − x)), ∀x, ∀y ∈ dom f.

Proposition 1.2.7 Let f i:Rn → R ∪ {+∞}, i = 1, , p, be a collection

of proper convex functions If ∩ p

Theorem 1.2.3 Let f : Rn → R ∪ {+∞} be a convex function Then f

is relatively continuous on any relatively open convex set C contained in

dom f , in particular on ri(dom f ).

Recall that a function f :Rn → R∪{+∞} is called Lipschitz with constant

L on a subset C of dom f if there exists L ≥ 0 such that

|f(x) − f(y)| ≤ Lx − y, ∀x, y ∈ C,

and f is called locally Lipschitz on C if for every c ∈ C there exists a

neighborhood V of c such that f is Lipschitz on V ∩ C.

Theorem 1.2.4 Let f :Rn → R ∪ {+∞} be a proper convex function and let C be any compact subset of ri(dom f ).Then f is Lipschitz relative to C.

An important and useful class of convex functions is the class of polyhedralconvex functions

Definition 1.2.3 A polyhedral convex function is a convex function whose

epigraph is polyhedral, which holds, if and only if it is finitely generated Such a function, if proper, is lsc.

For a nonconvex function f there is a natural procedure to convexify it via the use of the convex hull of the epigraph of f

Definition 1.2.4 For a function f :Rn → R ∪ {+∞}, the convex hull of

f is denoted by conv f and is defined for each x ∈ R n by

conv f (x) := inf{r ∈ R | (x, r) ∈ conv(epi f)}

Similarly, for a collection of functions {f i |i ∈ I} on R n , where I is an

arbitrary index set, the convex hull is denoted by conv{f i | i ∈ I} and is

the greatest convex function h such that h(x) ≤ f i (x) for every x ∈ R nand

every i ∈ I Whenever the index set I := {1, , p} is finite, the convex

hull h of the finite family f i is then given by

combina-Legendre–Fenchel Transform and Conjugate Functions

Duality plays a fundamental role in optimization problems, convex andnonconvex ones A key player in any duality framework is the Legendre–Fenchel transform, also called the conjugate of a given function

Definition 1.2.5 For any function f : Rn → R ∪ {+∞}, the function

Whenever conv f is proper, one always has that both f ∗ and f ∗∗are proper,lsc, and convex, and the following relations hold:

f ∗∗ = cl(conv f ) and f ∗∗ ≤ cl f.

Theorem 1.2.5 (Fenchel–Moreau) Let f : Rn → R ∪ {+∞} be convex The conjugate function f ∗ is proper, lsc, and convex if and only if f is proper Moreover, (cl f ) ∗ = f ∗ and f ∗∗ = cl f

From the definition of the conjugate function, we immediately obtain Fenchel’sinequality:

x, y ≤ f(x) + f ∗ (y), ∀x ∈ dom f, y ∈ dom f ∗ .

Definition 1.2.6 Let f, g : Rn → R ∪ {+∞} be proper functions The infimal convolution of the function f with g is the function h defined by h(x) := (f2g)(x) = inf{f(x1) + g(x2)| x1+ x2= x }, ∀x ∈ R n

The above definition can be extended as well for a finite collection{f i | i =

1, , p } of proper functions, i.e.,

(f12 · · · 2f p )(x) := inf {f1(x1) +· · · + f p (x p)| x1+· · · x p = x }.

The next proposition gives some important results relating conjugates andtheir infimal convolutions

Proposition 1.2.10 Let f i:Rn → R∪{+∞}, i = 1, , p, be a collection

of proper and convex functions Then:

(a) (f12 · · · 2f p) = f1∗+· · · + f ∗

p , (b) (cl f1+· · · + cl f p) = cl(f1∗ 2 · · · 2f ∗

p ).

Polyhedrality is preserved under the conjugacy operation

Proposition 1.2.11 Let f, g be proper polyhedral convex Then:

(a) f + g is polyhedral.

(c) The conjugate of f is polyhedral.

Proposition 1.2.12 Let f i : Rn → R ∪ {+∞}, i ∈ I, be an arbitrary collection of functions with conjugates f i ∗ Then:

(a) (inf{f i |i ∈ I}) ∗= sup{f ∗

i | i ∈ I}.

(b) conv{f i | i ∈ I} ∗ = sup{f ∗

i | i ∈ I}, and sup{clf i | i ∈ I}) ∗ =

cl(conv{f ∗ | i ∈ I}), whenever the functions f i are proper convex.

Differentiability and Subdifferentiability of Convex Functions

Under differentiability assumptions, one can check the convexity of a

func-tion via the following useful tests For a funcfunc-tion f defined on Rn and

sufficiently smooth, the gradient of f at x, denoted by ∇f(x), is a vector in

Rn whose i-th component is the partial derivative of f with respect to x i,

while the Hessian of f at x, when f is twice differentiable, is a symmetric

n × n matrix, whose (i, j) element is the second-order partial derivative

∂2f (x)/∂x i ∂x j, and is denoted by ∇2f (x).

Theorem 1.2.6 Let f be a differentiable function on an open convex

sub-set S ofRn Each of the following conditions is necessary and sufficient for

A subgradient of an extended function ofRn at a point x with f (x) finite

is any vector g ∈ R n satisfying

f (y) ≥ f(x) + g, y − x, ∀y.

The set of all subgradients of f at x ∈ dom f is called the subdifferential

of the function f at x and is denoted by ∂f (x); that is,

∂f (x) := {g | f(y) ≥ f(x) + g, y − x, ∀y ∈ R n }.

When x

viewed as a set-valued map ∂f that assigns to each point x ∈ R n a certain

subset ∂f (x) of Rn, see Section 1.4 below for more on set-valued maps

In general, ∂f (x) may be empty When ∂f (x)

called subdifferentiable at x A remarkable and evident property is that

∂f (x) is a closed convex set Indeed, from its definition, the subdifferential

of f at x is nothing but an infinite intersection of closed half-spaces Let f :Rn → R∪{+∞} be a convex function, and let x be any point where

f is finite and d ∈ R n Then the function

Proposition 1.2.13 Let f :Rn → R∪{+∞} be convex and let x ∈ dom f Then g is a subgradient of f at x if and only if f  (x; d) ≥ g, d, ∀d ∈ R n

Proposition 1.2.14 Let f :Rn → R ∪ {+∞} be a proper convex function and let x ∈ dom f be such that there exists a nonzero vector of the form

d := y − x with y ∈ ri(dom f) and such that f  (x; d) >

Given subgradients of f one might identify the directional derivative In fact, the directional derivative and the subdifferential of f are related by

the following fundamental max-formula

Proposition 1.2.15 Let f :Rn → R∪{+∞} be a proper convex function Then for any x ∈ ri(dom f) and any direction d ∈ R n one has d → f  (x; d)

is lsc and proper as a function of d, and

f  (x; d) = max {g, d |g ∈ ∂f(x)}.

We have already observed that ∂f (x) is a closed convex set The next

result provides a simple criterion to determine when this set is nonemptyand bounded, and hence compact

Proposition 1.2.16 Let f :Rn → R∪{+∞} be a proper convex function Then ∂f (x) is nonempty and bounded if and only if x ∈ int dom f.

Let dom ∂f :=

necessarily convex However, from the above results it follows that for any

proper convex function f :Rn → R ∪ {+∞}, one has the relation

ri(dom f ) ⊂ dom ∂f ⊂ dom f. (1.1)

A proper convex function f is differentiable at x ∈ dom f if and only if the

subdifferential reduces to a singleton, which is the gradient of f , i.e., one has

∂f (x) = {∇f(x)} In that case, one has f  (x; d) = ∇f(x), d, ∀d ∈ R n

There exists an interesting and fundamental interplay between conjugatefunctions and subdifferentials of convex functions that is quite at the heart

of many significant results in convex optimization problems and that will

be used all along in this monograph We begin with a simple but most nificant result played by subgradients in convex optimization, which followsimmediately from the definition of the subdifferential

sig-Proposition 1.2.17 Let f :Rn → R∪{+∞} be a proper convex function.

A point x ∈ R n is a global minimizer of f if and only if 0 ∈ ∂f(x).

Using the definitions of the subdifferential and of the conjugate function,

it is possible to characterize subdifferentiabililty in the following useful andimportant way

Proposition 1.2.18 Let f :Rn → R ∪ {+∞} be a proper function Then

f (x) + f ∗ (y) ≥ x, y, ∀x, y, and equality holds if and only if y ∈ ∂f(x), or equivalently, if and only if

x ∈ ∂f ∗ (y) whenever f is in addition lsc and convex.

Proposition 1.2.19 Let f :Rn → R ∪ {+∞} and g : R n → R ∪ {+∞} be proper convex functions on Rn Then for any x ∈ R n ,

the set C; i.e.,

The normal cone of the level set of a convex function can be written plicitly:

ex-Proposition 1.2.22 Let f :Rn → R ∪ {+∞} be a proper convex function and let x ∈ ri(dom f) be such that f(x) is not the minimum Then for each

α ∈ R one has

N lev(f,α) (x) =

R+∂f (x) if f (x) = α, {0} if f (x) < α.

1.3 Support Functions

Support functions play an important role in convex analysis and tion In fact they allow a characterization of closed convex sets and of the

optimiza-position of a point relative to a set with powerful analytical tools This mits one to translate geometrical questions on convex sets into questions interms of convex functions, which often facilitate the way of handling thesesets.

per-Definition 1.3.1 Given a nonempty set C ofRn , the function σ C:Rn →

R ∪ {+∞} defined by

σ C (d) := sup {x, d | x ∈ C}

is called the support function of C The domain of σ C defined by

dom σ C:={x ∈ R n | σ C (x) < + ∞}

is called the barrier cone of C and is denoted by b(C).

One thus has

σ C (d) ≤ α ⇐⇒ C ⊂ {x | x, d ≤ α},

characterizing the closed half-space that contains C.

Definition 1.3.2 A function π : Rn → R ∪ {+∞} is positively neous if 0 ∈ dom π and π(λx) = λπ(x) for all x and all λ > 0 It is sublinear if in addition,

homoge-π(x + y) ≤ π(x) + π(y), ∀x, y ∈ R n

Examples of positively homogeneous functions that are also sublinear

in-clude norms and linear functions The indicator function δ C of a set C is positively homogeneous if and only if C is a cone.

Proposition 1.3.1 A function π : Rn → R ∪ {+∞} is convex and tively homogeneous if and only if epi π is a nonempty convex cone inRn ×R.

posi-From the definition of a support function we immediately obtain the lowing basic properties

fol-Proposition 1.3.2 For any nonempty set C of Rn , the support function

σ C satisfies the following:

(a) σ C is an lsc convex positively homogeneous function.

(b) The barrier cone dom σ C is a convex cone (not necessarily closed) (c) σ C = σ cl C = σ conv C = σ cl conv C

(d) σ C < +∞ if and only if C is bounded.

(e) If C is convex, then σ C = σ ri C

Example 1.3.1 (Examples of particular support functions)

(i) σRn = δ {0} , σ ∅=−∞, σ {0} ≡ 0.

(ii) σconv{a1, ,a p } (d) = max1≤i≤p d, a i .

(iii) max1≤j≤m d j= sup

The support functions possess a structure allowing the development ofcalculus rules We list below some important formulas for support func-tions and barrier cones associated with arbitrary (not necessarily convex)nonempty sets ofRn.

Proposition 1.3.3 Let C, D be nonempty sets ofRn

(a) C ⊂ D =⇒ dom σ D ⊂ dom σ C and σ C (d) ≤ σ D (d), ∀d ∈ R n (b) σ C+D (d) = σ C (d) + σ D (d), and dom σ C+D = dom σ C ∩ dom σ D

If D is a cone, then the formula above reduces to

σ C+D (d) =

σ C (d) if d ∈ D ∗ ,

+∞ otherwise, and dom σ C+D = dom σ C ∩ D ∗ .

(c) For any point y ∈ R n , σ C+y (d) = σ C (d) + d, y.

(d) For a family of nonempty sets {C} i∈I with C := cl conv(∪ i∈I C i ) one

cor-Theorem 1.3.1 Let C ⊂ R n The closed convex hull of C is given by

cl conv C = {x ∈ R n | x, d ≤ σ C (d), ∀d ∈ R n }.

As a result of this theorem a closed convex set is completely determined

by its support function Key properties can then be usefully expressed viasupport functions

Theorem 1.3.2 Let C be a nonempty convex set in Rn and define

B(d) := σ C (d) + σ C(−d), d ∈ R n Then B(d) ≥ 0 and:

(a) x ∈ aff C if and only if x, d = σ C (d), ∀d with B(d) = 0.

(b) x ∈ ri C if and only if x, d < σ C (d), ∀d with B(d) > 0.

(c) x ∈ int C if and only if x, d < σ C (d),

(d) x ∈ cl C if and only if x, d ≤ σ (d), ∀d.

Corollary 1.3.1 For any convex sets C, D in Rn one has

cl C ⊂ cl D if and only if σ C ≤ σ D

By its definition, the support function of a set C ⊂ R n is in fact the

conjugate function of the corresponding indicator function of that set, σ C=

δ ∗ C Therefore, one has the relation

convenient to characterize a set-valued map by its graph The graph of S

is the subset of X × Y defined by

gph S := {(x, y) | y ∈ S(x)}.

Conversely, every nonempty subset G of X ×Y is the graph of the set-valued

map S uniquely determined by

S(x) = {y ∈ Y | (x, y) ∈ G}.

We use the notation S : X ⇒ Y to quantify the multivalued aspect of

S whenever S(x) is a set containing more than one element When S is

single-valued, we mean that S(x) = {y} or simply S(x) = y, and in that

case we use the notation S : X → Y

We shall say that a set-valued map S is convex-valued or closed-valued, etc., when S(x) is convex or closed, etc.

The domain and range of S : X ⇒ Y are defined by the sets

dom S :=

rge S := {y | ∃x such that y ∈ S(x)} = 

x∈X S(x).

Thus, the domain and range of S are respectively the images of gph S under the projections (x, y) → x and (x, y) → y.

The inverse of any set-valued map always exists and is defined by

S −1 (y) := {x ∈ X | y ∈ S(x)},

One has the following relations:

dom S −1 = rge S, rge S −1 = dom S, (S −1)−1 = S.

In this book we will work with X, Y that are subsets of the finite-dimensional

spacesRn and Rm An easy way to extend a set-valued map to the wholespaceRn is formally achieved by defining the map

A single-valued map F : X → R m given on X ⊂ R n can be treated in

terms of the set-valued map S :Rn → R m by setting

to get the mapping T ◦ S : R n → R p

Semicontinuity and Continuity of Set-Valued Maps

Definition 1.4.1 A set-valued map S :Rn⇒ Rm is upper semicontinuous (usc) at ¯ x if for each open set N containing S(¯ x) there exists a neighborhood

U (¯ x) such that

x ∈ U(¯x) =⇒ S(x) ⊂ N.

Alternatively, upper semicontinuity of the set-valued map S at ¯ x can be

expressed as

∀ε > 0, ∃δ > 0 such that S(x) ⊂ B ε (S(¯ x)), ∀x ∈ B δ(¯x),

where we define for any set C, B ε (C) := {z | dist(z, C) < ε}.

Clearly, when S is a single-valued map on Rn, i.e., a function, then for

any x ∈ R n , S(x) reduces to a singleton, and thus the above formulation

coincides with the definition of continuous functions The latter formulationalso leads to the translation of upper semicontinuity in terms of infinitesequences Let{x k } ∈ dom S be a convergent sequence with limit ¯x Then,

the set-valued map is upper semicontinuous at ¯x if

S is called continuous at ¯ x if it is both lsc and usc at ¯ x We say that S is

lsc (usc) in Rn if it is lsc (usc) at every point ¯x ∈ R n, and continuous in

Rn if S is both lsc and usc inRn

Proposition 1.4.1 A set-valued map S :Rn⇒ Rm is lsc at ¯ x if and only

if for any sequence {x k } ∈ R n converging to ¯ x, and any ¯ y ∈ S(¯x), there exist k0∈ N and a sequence {y k } ∈ S(x k ) for k ≥ k0 that converges to ¯ y.

Proposition 1.4.2 Let Φ :Rn ×R m → R∪{+∞} and let S be a set-valued map S :Rn ⇒ Rm Define the function ϕ(x) = inf{Φ(x, u) |u ∈ S(x)} If

S is lsc at ¯ x and Φ is usc on {¯x} × S(¯x), then the function ϕ is usc at ¯x.

Definition 1.4.3 A set-valued map S :Rn⇒ Rm is closed at ¯ x if for any sequence {x k } ∈ R n and any sequence {y k } ∈ R m one has

x k → ¯x, y k → ¯y, y k ∈ S(x k) =⇒ ¯y ∈ S(¯x).

From this definition one can verify the following useful equivalence:

S : Rn ⇒ Rn is closed at ¯x if

U (¯ x), V (¯ y) such that x ∈ U(¯x) =⇒ S(x) ∩ V (¯y) = ∅.

Definition 1.4.4 A set-valued map S :Rn ⇒ Rm is locally bounded at ¯ x

if for some neighborhood N (¯ x) the set

x∈N(¯x) S(x) is bounded S is called locally bounded on Rn if this holds for every ¯ x ∈ R n It is bounded on Rn

if rge S is a bounded subset ofRm

From the above definition one clearly has that if S is locally bounded at ¯ x,

then the set S(¯ x) is bounded, and in particular that S is locally bounded

at any point ¯x

S :Rn ⇒ Rmis locally bounded if

{x k } is a bounded sequence, u k ∈ S(x k) =⇒ {u k } is a bounded sequence.

The next results give some important relations between upper uous and locally bounded set-valued maps

semicontin-Theorem 1.4.1 Suppose that the set-valued map S : Rn ⇒ Rm is usc at

Corollary 1.4.1 Let S be a closed set-valued map S :Rn → R m Then S

is locally bounded at x if and only if S(x) is bounded.

As already mentioned, the subdifferential map of a convex function can beseen as a set-valued map that enjoys the following important properties

Proposition 1.4.3 Let f :Rn → R ∪ {+∞} be a proper convex function Then ∂f is upper semicontinuous and locally bounded at every point x ∈

int(dom f ) Moreover, if f is assumed lsc, then ∂f is closed.

We end this section by recalling some useful and basic operations preservinglower/upper semicontinuity of set-valued maps

Proposition 1.4.4 Let {S i |i ∈ I} be a family of set-valued maps defined

in a finite-dimensional vector space with appropriate dimensions Then the following properties hold:

(a) The composition map S1◦ S2 of lsc (usc) maps S i , i = 1, 2 is lsc (usc) (b) The union of lsc maps ∪ i ∈I S i is an lsc map.

(c) The intersection of usc maps ∩ i∈I S i is a usc map.

(d) The Cartesian product of a finite number of lsc (usc) maps Π m i=1 S i is

an lsc (usc) map.

1.5 Notes and References

Convex analysis has emerged as the most powerful and elegant theory derlying the development of optimization and variational analysis Thenumber of references including research papers, books, and monographdealing with various aspects of convex analysis and its applications is todayvery large The results briefly outlined in this chapter are far from reflecting

un-all the convex-analytic machinery Thus this chapter is not an introduction

to convex analysis Rather, the writing of this chapter is intended to provide

to the reader a concise review of the elements of convex analysis that will beneeded and used throughout this monograph For details and proofs of allthe results summarized in this chapter, the reader is referred to the follow-ing short bibliography Standard reference on convex analysis include thebooks and monographs by J J Moreau,[103], which study mostly convexfunctions and their properties on arbitrary topological vector spaces R T.Rockafellar [119], is a comprehensive book on all aspects of convex analysis

in Rn Laurent [85], which is a book with fundamental results on the ory and applications of convex analysis to optimization and approximationtheory within the framework of abstract spaces Ekeland and Temam [67],which deals with variational and infinite-dimensional problems A recentconcise and accessible account of convex analysis is given in Borwein andLewis [38] Other classics and useful books studying mainly properties ofconvex sets are H G Eggleston [65] and F A Valentine [129]

Asymptotic Cones and Functions

This chapter provides the fundamental tools used throughout this graph For a given subset of Rn we are interested in studying its behavior

mono-at infinity This leads to the concepts of asymptotic cone and asymptoticfunction through its epigraph Using elementary real analysis and geomet-rical concepts we develop a complete mathematical treatment to handlethe asymptotic behavior of sets, functions, and other induced functionaloperations

2.1 Definitions of Asymptotic Cones

The set of natural numbers is denoted by N, so that k ∈ N means k =

1, 2, A sequence {x k } k∈N or simply {x k } in R n is said to converge to

x if x k − x → 0 as k → ∞, and this will be indicated by the notation

x k → x or x = lim k→∞ x k We say that x is a cluster point of {x k } if some

subsequence converges to x Recall that every bounded sequence inRn has

at least one cluster point A sequence inRn converges to x if and only if it

is bounded and has x as its unique cluster point.

Let{x k } be a sequence in R n We are interested in knowing how to handle

situations when the sequence{x k } ⊂ R nis unbounded To derive some

con-vergence properties, we are led to consider directions d k := x k x k  −1with

x k

implies that we can extract a convergent subsequence d = lim k ∈K d k , K ⊂

} ⊂ R n is such that

This leads us to introduce the following concepts.

Definition 2.1.1 A sequence {x k } ⊂ R n is said to converge to a direction

From the definition we immediately deduce the following elementary facts

Proposition 2.1.1 Let C ⊂ R n be nonempty Then:

(a) C ∞ is a closed cone.

(b) (cl C) ∞ = C ∞

(c) If C is a cone, then C ∞ = cl C.

The importance of the asymptotic cone is revealed by the following keyproperty, which is an immediate consequence of its definition

Proposition 2.1.2 A set C ⊂ R n is bounded if and only if C ∞={0}.

Proof It is clear that C ∞ cannot contain any nonzero direction if C

is bounded Conversely, if C is unbounded, then there exists a sequence

{x k } ⊂ C with x k k :=x k  → ∞ and thus the

vectors d k = t −1 k x k ∈ {d : d = 1} Therefore, we can extract a

subse-quence of{d k } such that lim k∈K d k = d, K ⊂ N, and with d = 1 This

nonzero vector d is an element of C ∞ by Definition 2.1.2, a contradiction

2

Associated with the asymptotic cone C ∞ is the following related concept,

which will help us in simplifying the definition of C ∞in the particular case

where C ⊂ R n is assumed convex

Definition 2.1.3 Let C ⊂ R n be nonempty and define

Proposition 2.1.3 Let C be a nonempty convex set in Rn Then C is asymptotically regular.

Proof The inclusion C1

∞ ⊂ C ∞ clearly holds from the definitions of C ∞1

and C ∞ , respectively Let d ∈ C ∞ Then∃{x k } ∈ C, ∃s k → ∞ such that

d = lim k→∞ s −1 k x k Let x ∈ C and define d k = s −1 k (x k − x) Then we have

d = lim k→∞ d k , x + s k d k ∈ C.

Now let {t k } be an arbitrary sequence such that lim k →∞ t k = +∞ For

any fixed m ∈ N, there exists k(m) with lim m →∞ k(m) = +∞ such that

t m ≤ s k(m) , and since C is convex, we have x  m := x + t m d k(m) ∈ C Hence,

d = lim m→∞ t −1 m x  m , showing that d ∈ C1

We note that a set can be nonconvex, yet asymptotically regular Indeed,

consider, for example, sets defined by C := S + K, with S compact and K

a closed convex cone Then clearly C is not necessarily convex, but it can

be easily seen that C ∞ = C1

∞.

Remark 2.1.1 Note that the definitions of C ∞ and C1

∞ are related to

the theory of set convergence of Painlev´e–Kuratowski Indeed, for a family

{C t } t>0 of susbsets ofRn , the outer limit as t → +∞ is the set

Proof Clearly, one always has pos C N ⊂ C ∞ Conversely, let 0 ∞.

Then there exists t k → ∞, x k ∈ C such that

Thus the sequence {t −1

k x k } is a nonnegative bounded sequence, and by

the Bolzano–Weierstrass theorem, there exists a subsequence{t −1

k x k } k ∈K

with K ⊂ N such that lim k∈K t −1 k x k  = λ ≥ 0, which means that d = λd N

We now turn to some useful formulations of the asymptotic cone for convexsets

Proposition 2.1.5 Let C be a nonempty convex set in Rn Then the asymptotic cone C ∞ is a closed convex cone Moreover, define the following sets:

Proof We already know that C ∞is a closed cone The convexity property

simply follows from Proposition 2.1.3, which ensures that C ∞1 = C ∞ We

now prove the three equivalent formulations Let d ∈ C ∞ , x ∈ C, t > 0.

Passing to the limit, we thus have x + td ∈ cl C, ∀t > 0, thus proving

the inclusion C ∞ ⊂ D(x), ∀x ∈ C Clearly, we also have the inclusion D(x) ⊂ E, ∀x ∈ C We now show that E ⊂ C ∞ For that, let d ∈ E

and x ∈ C be such that x(t) := x + td ∈ cl C, ∀t > 0 Then, since

d = lim t →∞ t −1 x(t) and x(t) ∈ cl C, we have d ∈ (cl C) ∞ = C ∞ (by

Proposition 2.1.1), which also proves that D(x) is in fact independent of

x, and we can write D(x) = D = C ∞ = E Finally, let d ∈ C ∞ Using

the representation C ∞ = D we thus have x + d ∈ cl C, ∀x ∈ C, and

hence d + cl C ⊂ cl C, proving the inclusion C ∞ ⊂ F Now, if d ∈ F , then

d + cl C ⊂ cl C, and thus

2d + cl C = d + (d + cl C) ⊂ d + cl C ⊂ cl C,

and by induction cl C + md ⊂ cl C, ∀m ∈ N Therefore, ∀x ∈ cl C, d m :=

x + md ∈ cl C and d = lim m →∞ m −1 d m , namely d ∈ C ∞, showing that

F ⊂ C ∞ , and hence the proof of the three equivalent formulations for C ∞