DSpace at VNU: An approximate Hahn-Banach theorem for positively homogeneous functions

Volled a Department of Mathematics, International University, Vietnam National University, Ho Chi Minh City, Vietnam; b UMR7353, Aix-Marseille University, Marseille, France; c Faculty of

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An approximate Hahn–Banach theorem for positively homogeneous functions

N Dinha, E Ernstb, M.A Lópezc & M Volled a

Department of Mathematics, International University, Vietnam National University, Ho Chi Minh City, Vietnam

b UMR7353, Aix-Marseille University, Marseille, France

c Faculty of Sciences, Department of Statistics and Operations Research, Alicante University, Alicante, Spain

d EA2151, Université d’Avignon et des Pays de Vaucluse, Avignon Cedex 1, France

Published online: 17 Dec 2013

To cite this article: N Dinh, E Ernst, M.A López & M Volle (2015) An approximate Hahn–Banach

theorem for positively homogeneous functions, Optimization: A Journal of Mathematical

Programming and Operations Research, 64:5, 1321-1328, DOI: 10.1080/02331934.2013.864290

To link to this article: http://dx.doi.org/10.1080/02331934.2013.864290

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Optimization, 2015

Vol 64, No 5, 1321–1328, http://dx.doi.org/10.1080/02331934.2013.864290

An approximate Hahn–Banach theorem for positively homogeneous

functions

N Dinha, E Ernstb ∗, M.A Lópezcand M Volled

a Department of Mathematics, International University, Vietnam National University, Ho Chi Minh City, Vietnam; b UMR7353, Aix-Marseille University, Marseille, France; c Faculty of Sciences, Department of Statistics and Operations Research, Alicante University, Alicante, Spain;

d EA2151, Université d’Avignon et des Pays de Vaucluse, Avignon Cedex 1, France

(Received 10 April 2013; accepted 24 October 2013)

This note provides an approximate version of the Hahn–Banach theorem for non-necessarily convex extended-real valued positively homogeneous functions

of degree one Given p : X → R∪{+∞} such a function defined on the real vector space X , and a linear function  defined on a subspace V of X and dominated by

p (i.e (x) ≤ p(x) for all x ∈ V ), we say that  can approximately be p-extended

to X , if  is the pointwise limit of a net of linear functions on V , every one of

which can be extended to a linear function defined on X and dominated by p.

The main result of this note proves that can approximately be p-extended to X

if and only if is dominated by p∗∗, the pointwise supremum over the family of

all the linear functions on X which are dominated by p.

Keywords: approximate Hahn–Banach theorem; non-convex Hahn–Banach

theorem; Fenchel–Legendre conjugate; positively homogeneous functions of degree one

AMS Subject Classifications: 46A22; 46A20

1 Introduction

Let us consider X , a non-zero real vector space, and p : X → R ∪ {+∞}, an extended-real valued function which is positively homogeneous of degree one (in short, a ph-function) Given V , a linear subspace of X , we call the linear function  : V → R such that

(x) ≤ p(x) for all x ∈ V , a p-dominated linear function As customary, we say that

a p-dominated linear function  : V → R can be p-extended to X if there is a p-dominated

linear function g : X → R such that g(x) = (x) for all x ∈ V

The Hahn–Banach theorem (the Crown Jewel of Functional Analysis, as called in [1]),

simply says that, provided that the ph-function p is real-defined and convex (in other words, sublinear), then any p-dominated linear function  : V → R can be p-extended

to X , generally in more than one way.

A surprising fact occurs when, as requested in many constraint optimization problem,

p is allowed to take the value +∞ When the dimension of the underlying space X is

∗Corresponding author Email: emil.ernst@univ-amu.fr

1322 N Dinh et al.

infinite, S Simons provided (see the paragraph of the article [2] entitled ‘Counterexample

to 4’, at p.114) a highly counterintuitive example: a hypolinear function (that is a convex

ph-function) p : X → R ∪ {+∞} such that the inequality g ≤ p is false for all the linear mappings g : X → R.

Arguably, the better illustration of the difficulty to address this case is the unusually large number of flawed Hahn–Banach type theorems for hypolinear functions which can be found in the mathematical literature; in the articles,[3 5] the reader can find examples and criticism of as much as seven such incorrect results published between 1969 and 2005

The first correct Hahn–Banach theorem for hypolinear functions have been achieved by Anger and Lembcke, [3, Theorem 2.4, p.135] (result generalized for multifunctions in [6]):

the authors prove that a p-dominated linear function  : V → R can be p-extended to X if

and only if the largest hypolinear minorant of both the linear and the hypolinear functions, namely

p  : X → R ∪ {+∞}, p  (x) = inf {p(x + y) − (y) : y ∈ V },

is lower semi-continuous at 0 with respect to the finest locally convex topology on X

However, as remarked by the authors themselves ([7, p.251]): ‘[not even the] continuity

of the linear form and lower semicontinuity of the hypolinear domination functional […] guarantee the existence of a dominated […] linear extension, even in a finite dimensional setting’

In order to overcome this difficulty and achieve a result easier to use in applications,

we address, in very much the same spirit as theε-strategies in mathematics analysed in [8],

an approximate version of the Hahn–Banach theorem for extended-real valued functions which are positively homogeneous of degree one but non-necessarily convex Namely, given

 : V → R, a p-dominated linear function, we say that  can approximately be p-extended

to X , if  is the pointwise limit of a net ( i ) i ∈I of p-dominated linear functions on V , every

one of which can be p-extended to X A simple remark states that if a p-dominated linear

function : V → R can approximately be p-extended to X, then  must be dominated not

only by p, but also by p∗∗, where p∗∗: X → R ∪ {+∞} is the pointwise supremum over the family of p-dominated linear functions on X (equivalently, p∗∗is the bi-conjugate of

p with respect to the finest locally convex topology defined on X ).

Our main result, Theorem2.3, proves that the simple necessary condition previously

defined is sufficient for approximately extending a p-dominated linear function Namely,

we prove that given p : X → R ∪ {+∞}, a (non-necessarily convex) ph-function and V

a linear subspace of X , then a p-dominated linear function  : V → R can approximately

be p-extended to X if and only if  is also dominated by p∗∗.

On this ground, we address the class of ph-functions with the V -approximate extension

property in the setting of a Hausdorff locally convex space X : Proposition 3.1 proves

that, given a ph-function p, and a closed subspace V of X , any p-dominated linear

function ∈ V∗can approximately be p-extended to X , if and only if p∗∗|V = (p| V )∗∗.

Finally, Theorem 4.1 addresses the case of ph-functions with the approximate extension property, and proves that, given a ph-function p, then, for any closed subspace V of X , any

p-dominated linear function  : V → R can approximately be p-extended to X, if and only

if p and p∗∗coincide.

Optimization 1323

2 The main result

Let X be a non-zero real vector space endowed with a locally convex topology τ; its

continuous dual, X∗, is the vector space of all the linear and continuous functions on X

Of a particular interest for our study is theσ (X∗, X)-topology on X∗(the weak topology),

customary defined as the coarsest locally convex topology on X∗which makes continuous

all the mappings

T x : X∗→ R, T x (g) = g(x), g ∈ X∗, x ∈ X.

Any closed linear subspace V of X is tacitly considered as endowed with the topology

induced byτ; similarly, we define V∗, the continuous dual of V , and endow it with the

σ (V∗, V )-topology ([9, Chapter 6] provides an excellent short introduction to this topic)

Finally, let us pick p : X → R ∪ {+∞}, a ph-function, that is an extended-real-valued

function which fulfills the relation

p (s x) = sp(x) ∀s ≥ 0, x ∈ X;

we make the convention that 0· ∞ = 0 So any ph-function vanishes at 0.

Definition 2.1 Let ∈ V∗be a p-dominated continuous linear function We say that  can

approximatively be p-extended to X with respect to τ, if  is the σ (V∗, V )-limit of a net ( i ) i ∈I ⊂ V∗of p-dominated linear functions on V , every one of which can be p-extended

to X by an element L i in X∗.

Remark 1 Let us consider the particular case when V = {0} and the linear function

 : V → R is defined by (0) = 0 As  is the only linear function on V , saying that  can

approximately be p-extended to X with respect to τ simply means that  can be p-extended

to X (any of the elements of the net ( i ) i ∈I whichσ (V∗, V )-converges to  is necessarily

equal to).

Moreover, as any linear and continuous function on X is an extension of , we may

conclude that = 0 can approximately be p-extended to X with respect to τ if and only if

p dominates at least one element of X∗.

Before proceeding to our main technical result, we recall the definition of the Fenchel–

Legendre conjugate for functions defined on X and on X∗ Let us consider h : X →

R ∪ {+∞} and j : X∗→ R ∪ {+∞}, two extended-real-valued function on X and X∗ The

conjugate of h is defined as

h∗: X∗→ R ∪ {−∞, +∞}, h∗( f ) = sup{ f (x) − h(x) : x ∈ X},

while relation

j∗: X → R ∪ {−∞, +∞}, j∗(x) = sup{ f (x) − j( f ) : f ∈ X∗}

defines the conjugate of j

Pr o p o s it io n 2.2 Let us consider (X, τ), a non-zero Hausdorff locally convex real vector space, p : X → R ∪ {+∞}, a ph-function, V , a closed linear subspace of X and 0∈ V∗,

a p-dominated continuous linear function The two following sentences are equivalent:

1324 N Dinh et al.

(i) 0can approximately be p-extended to X with respect to τ,

(ii) 0is dominated by p∗∗, the bi-conjugate of p.

Proof Let us first remark that p∗ : X∗ → R ∪ {+∞} is the conjugate of a ph-function,

so it is a convex function which can only take two values: 0 and+∞ Moreover, relation

p∗(g) = 0 holds if and only if the function g ∈ X∗is p-dominated.

Let us define the mappingv : V∗→ R ∪ {+∞} as

v() = inf {p∗(g) : g ∈ X∗, g |V = }.

It is straightforward that v is also a convex function which can take only the values 0

and+∞; moreover, the definition of v implies that the linear function  ∈ V∗ can be

p-extended to X by an element in X∗if and only ifv() = 0.

Set now A = { ∈ V∗ : v() = 0}; Definition2.1reads that0can approximatively

be p-extended to X with respect to τ if and only if it lies in the σ (V∗, V )-closure of the

set A But v is the indicator function of the set A,

v() = ι A () =



so, by virtue of a well-known convex analysis result (see for instance [9, p.3, below Proposition 6.2]), it follows that its bi-conjugate is the indicator function of theσ (V∗, V

)-closure of the set A: v∗∗= ι A σ(V ∗,V )

To the purpose of computingv∗∗, let us consider x ∈ V Relation g |V =  means that

(x) = g(x); it results that

v∗(x) = sup

∈V∗((x) − v()) = sup

∈V∗



(x) − inf

g ∈X∗, g|V = p

∗(g)



= sup

∈V∗



(x) + sup

g ∈X∗, g|V = (−p∗(g))



= sup

g ∈X∗



g (x) − p∗(g)= p∗∗(x).

Accordingly,v∗∗=(p∗∗) |V∗, that isι A σ(V ∗,V ) =(p∗∗) |V∗

We have thus proved the following facts: first, that the linear and continuous function

0: V → R can approximatively be p-extended to X with respect to τ if and only if 0

belongs to theσ (V∗, V )-closure of the set A, second, that 0belongs to A σ(V∗,V )

if and only

if

(p∗∗) |V∗(0) = 0, and third, that(p∗∗) |V∗(0) = 0 if and only if 0is dominated by

the hypolinear function p∗∗ The conclusion of Proposition2.2follows by combining these

Let us consider the case whenτ is the finest locally convex topology defined on X Then,

every linear subspace V of X is closed, and any linear function  : V → R is continuous.

Moreover, as for any locally convex topologyτ, and for any ph-function p : X → R∪{+∞}

it holds that

p∗∗(x) = sup{g(x) : g ∈ X∗, g ≤ p}

Optimization 1325 (see for instance [9, Proposition 6.1]), we deduce that, for the finest locally convex topology

on X , the bi-conjugate of the function p is the pointwise supremum over the family of

p-dominated linear functions g : X → R.

We conclude that, when specified to the case of the finest locally convex topology, Proposition2.2yields the following approximate version of the Hahn–Banach theorem

Th e o r e m 2.3 Let us consider X , a non-zero real vector space, together with p : X →

R ∪ {+∞}, a ph-function, V , a linear subspace of X, and  : V → R, a p-dominated

linear function The two following sentences are equivalent:

(i)  can approximatively be p-extended to X and

(ii)  is dominated by the pointwise supremum over the family of linear functions on X which are p-dominated.

3 Positively homogeneous functions of degree one with the V -approximate extension

property

The main result of this section uses Proposition2.2in order to provide an approximate version of the Hahn–Banach theorem valid for all the linear functions defined on some

subspace V of X

Pr o p o s it io n 3.1 Let us consider (X, τ), a locally convex real vector space, p : X →

R ∪ {+∞}, a ph-function and V , a closed linear subspace of X The two following

sentences are equivalent:

(i) any linear, continuous and p-dominated function  : V → R can approximately be p-extended to X with respect to τ and

(ii) p∗∗|V = (p| V )∗∗.

Proof (i) ⇒ (ii) Let P(V ) = { ∈ V∗ :  ≤ p} be the set of all the linear and continuous functions defined on V which are dominated by p As already noticed, for any

x ∈ V it holds that

(p| V )∗∗(x) = sup{(x) :  ∈ P(V )};

moreover, statement (i) says that any ∈ P(V ) can approximately be p-extended to X with

respect toτ, so, in virtue of Proposition2.2,

(x) ≤ p∗∗(x) ∀x ∈ V,  ∈ P(V ).

Accordingly,

(p| V )∗∗(x) ≤ p∗∗(x) ∀x ∈ V ;

since the opposite inequality is clear, statement (ii) follows

(ii) ⇒ (i) It is well-known that the bi-conjugate of an extended-real valued function

defined on X which is not identically equal to+∞, either takes only the value −∞, or is

an extended-real valued lower semi-continuous and convex function which takes at least one real value (the class of those functions is usually called0(X)).

1326 N Dinh et al.

If p∗∗ amounts to −∞X , the constant function on X taking the value −∞, relation (ii) implies that the same holds for(p| V )∗∗; thus, the setP(V ) is void, so statement

(i) obviously holds

Let us now assume now that p∗∗belongs to0(X), and let  ∈ V∗be such that ≤ p

on V By Proposition2.2one has to check that ≤ p∗∗on V

Now, ≤ p| V, a fact which implies that ≤ (p| V )∗∗ By relation (ii), we know that

(p| V )∗∗= (p∗∗) | V, hence ≤ (p∗∗) | V, that is ≤ p∗∗on V 

Remark 1 Recently, an analytical result involving affine functions dominated by an extended-real valued mapping and their approximate extensions, has been used by Dinh

et al [10] to derive a series of subdifferential calculus rules, different generalizations of Farkas lemma for non-necessarily convex systems, optimality conditions and duality theory for infinite optimization problems

Theorem 1, the main result from the article [10] (for a convex version, the reader is refereed to [11, Proposition 1]), reads as follows:

[10, Theorem 1]: Let U and X be two locally convex spaces, and consider a function

F : U × X → R ∪ {+∞} such that F is finite at some point of 0 × X, and F∗is finite at

some point of U∗× X∗ Then, the following statements are equivalent:

(a) F∗∗(0, ·) = (F(0, ·))∗∗.

(b) For any h : X → R ∪ {+∞}, convex and lower semi-continuous function, the two

following statements are equivalent:

(b1) F (0, x) ≥ h(x), ∀x ∈ X

(b2) if, for some x∗∈ X∗, h∗(x∗) is finite, then there exists a net (u∗

i , x∗

i , ε i ) i ∈I ⊂

U∗ × X∗ × R+ such that F∗(u∗

i , x∗

i ) ≤ h∗(x∗) + ε i , ∀i ∈ I , and

limi ∈I (x∗

i , ε i ) = (x∗, 0).

This result may be used in order to give an alternative proof of the implication(ii) ⇒ (i)

of Proposition3.1, by setting U = X,

F (u, x) := p(x + u) + ι V (x), (u, x) ∈ X × X,

h (x) =



(x) x ∈ V

+∞ x ∈ X \ V ,

and by noticing that statements(a) from [10, Theorem 1] and (ii) from Proposition3.1are equivalent, while the statement (i) from Proposition3.1is implied by statement(b2) from

[10, Theorem 1]

4 Positively homogeneous functions of degree one with the approximate extension property

The following result characterizes the class of hypolinear functions with the approximate

extension property, that is hypolinear functions p such that, for any subspace V of X , any p-dominated linear function  : V → R can approximately be p-extended to X.

Optimization 1327

Th e o r e m 4.1 Let us consider (X, τ), a Hausdorff locally convex real vector space, and

p : X → R ∪ {+∞}, a positively homogeneous function of degree one The two following

sentences are equivalent:

(i) p is lower semi-continuous and convex,

(ii) for any closed subspace V of X , any p-dominated continuous linear function  ∈ V∗

can approximately be p-extended to X with respect to τ.

Proof (i) ⇒ (ii) Any lower semi-continuous and convex function which takes at least one

real value coincides with its bi-conjugate Accordingly, if p ∈ 0(X), then any p-dominated

function is also p∗∗-dominated, and Proposition2.2proves the desired implication.

(ii) ⇒ (i) Let us consider p : X → R ∪ {+∞}, a ph-function with the approximate

extension property Then it is impossible for p∗∗ to take the value −∞ (proven as in Remark2, by taking V := {0}) To the aim of achieving a contradiction, let us assume that

there exists x ∈ X such that p∗∗(x) < p(x), and let us pick α such that p∗∗(x) < α ≤ p(x).

Let us first notice that −α ≤ p(−x); indeed, if −α > p(−x), then p∗∗(−x) ≤

p (−x) < −α, whence the contradiction

0= p∗∗(0) ≤ p∗∗(x) + p∗∗(−x) < α − α = 0.

Take V := R x (a closed linear subspace of X because τ is Hausdorff) and 0: V → R

defined by0(t x) := t α Clearly, 0(y) ≤ p(y) for every y ∈ V (this is because p is

positively homogeneous,α ≤ p(x), and −α ≤ p(−x)) By Proposition2.2, we have that

0≤ p∗∗; in particular,α = 0(x) ≤ p∗∗(x), a contradiction. 

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