DSpace at VNU: Functional inequalities and theorems of the alternative involving composite functions

DSpace at VNU: Functional inequalities and theorems of the alternative involving composite functions tài liệu, giáo án,...

DOI 10.1007/s10898-013-0100-z

Functional inequalities and theorems of the alternative

involving composite functions

N Dinh · G Vallet · M Volle

Received: 30 June 2012 / Accepted: 5 August 2013

© Springer Science+Business Media New York 2013

Abstract We propose variants of non-asymptotic dual transcriptions for the functional

inequality of the form f + g +k ◦ H ≥ h The main tool we used consists in purely algebraic

formulas on the epigraph of the Legendre-Fenchel transform of the function f + g + k ◦ H

that are satisfied in various favorable circumstances The results are then applied to the texts of alternative type theorems involving composite and DC functions The results coverseveral Farkas-type results for convex or DC systems and are general enough to face withunpublished situations As applications of these results, nonconvex optimization problemswith composite functions, convex composite problems with conic constraints are examined

con-at the end of the paper There, strong duality, stable strong duality results for these classes

of problems are established Farkas-type results and stable form of these results for thecorresponding systems involving composite functions are derived as well

Keywords Functional inequalities· Alternative type theorems · Farkas-type results ·

Stable strong duality· Stable Farkas lemma · Set containments · Nonconvex composite

optimization problems· Convex composite problems with conic constraints

Mathematics Subject Classification 39B62· 49J52 · 46N10

N Dinh (B)

Department of Mathematics, International University, VNU-HCM, Thuduc district,

Ho Chi Minh City, Vietnam

e-mail: ndinh@hcmiu.edu.vn

G Vallet

Laboratory of Applied Mathematics, UMR-CNRS 5142, University of Pau, IPRA, BP 1155,

64013 Pau Cedex, France

1 Introduction

Given two proper convex functions a , b : X →R∪ {+∞} on a locally convex Hausdorff

topological vector space (l.c.H.t.v.s.) X , the relation

(a + b)∗(x∗) = min

y∗∈X∗



a∗(x∗− y∗) + b∗(y∗) (1)has been used in [10] for the usual formula on the-subdifferential of the sum of the proper convex function a + b It should be emphasized that a classical sufficient condition for the

validity of (1) is that a is finite and continuous at a point of dom b (see [24, Theorem 2.8.7])

It is easy to see that (1) is equivalent to

epi(a + b)∗= epi a∗+ epi b∗. (2)

In the case when a , b ∈ Γ (X) and a + b is proper, it was observed in [3, Theorem 2.1]that

epi(a + b)∗= w∗− cl(epi a∗+ epi b∗)

and, consequently, that (2) is equivalent to

epi a∗+ epi b∗isw∗-closed.

A generalization of the previous result has been established in [2, Theorem 9.2] which

reads as follows: Given a1, , a n ∈ Γ (X) such that ∩ n

i=1dom a i subset of X∗, the following statements are equivalent:

i=1epi a i∗inw∗-closed regarding the setΔ ×R

Let us quote that condition(i) above amounts to saying that

(I) f (x) + g(x) + (k ◦ H)(x) ≥ h(x), ∀x ∈ X where f, g, h : X →R∪{+∞} are extended real-valued functions defined on the l.c.H.t.v.s

X , H : dom H ⊂ X → Z is a mapping defined on a non-void subset dom H of X, with values in another l.c.H.t.v.s Z , and k : Z →R∪ {+∞} is an extended real-valued

function One may consider the mapping H defined on the whole space X with values on

Z•:= Z ∪{∞ Z } and H(x) = ∞ Z when x Zdoes not belong

to Z We then show that relations like (2) or (3) involving composite functions are necessaryand sufficient conditions for such transcriptions It is worth mentioning that several of thesecharacterizations hold without any convexity and lower semi-continuity assumptions.The reason why we study relations like(I ) is that this frame encompasses most of gen-

eralized versions of the Farkas Lemma (see, e.g., [2,8,11]), together with convex/concavecontainments (see, e.g., [12,17]) In fact, the relation(I ) is sufficiently general to face also

with unpublished circumstances (see Sect.6)

Fig 1 Double composite model

The recent papers [4,6] partially devoted to asymptotic dual transcriptions of such a tional inequality with or without convexity, lower semicontinuity assumptions on the func-tions involved, and without any constraint qualification conditions There, the results werethen applied to derive sequential Lagrange multipliers, duality results for general optimiza-tion problems without convexity nor constraint qualification conditions, asymptotic Farkaslemmas for nonconvex systems, for linear infinite systems without qualification conditions.The abilities of applying these results to other optimization problems such as variationalinequalities, equilibrium problems were shown in [4,6] as well

func-The aim of the present paper is twofold We first provide purely algebraic necessaryand sufficient conditions for the validity of non-asymptotic dual transcriptions (Theorems

2 3) The counter part of such results are obtained in the presence of convexity and lowersemicontinuity was given in Sect.4which gives rise to variants of characterizations of Farkas-type results (Theorem5, Corollary3) and strong duality and stable strong duality for convexand DC optimization problems and also stable Farkas-type results for systems involvingconvex systems or systems with DC functions (Corollaries4 5) The second purpose of thepaper is to apply our approach to the context of the alternative theorems (Theorems10–14,Corollaries6 7, Theorem15), to convex/concave containments (Proposition2) These aregiven in Sect.5 To show the possibilities of using the results obtained to get various newresults and applications, we consider the use of “double composite model” of the form

corresponding to the diagram shown in Fig.1 Where U , V are l.c.H.t.v.s with U•:= U ∪

{∞U }, V•:= V ∪ {∞ V}, and ∞U , ∞ V are elements that do not belong to U and V (resp.) while U+and V+are closed convex cones in U and V , respectively If we set H := (M, N)

and k(u, v) = p(u) + q(v) then (4) is nothing but(I ) This will be realized in Sect.6.There, some special cases will be examined to get strong duality, stable strong duality fornonconvex and convex optimization problems involving composite functions New versions

of Farkas-type results and their stable versions are proposed as well

2 Notations and preliminary results

Throughout this paper f, g, h : X →R∪ {+∞} are extended real-valued functions defined

on the locally convex Hausdorff topological vector space X with its topological dual X∗

equipped with the weak∗-topology, H : dom H ⊂ X → Z is a mapping defined on a

non-void subset dom H of X , with values in an other locally convex Hausdorff topological vector space Z , and k : Z →R∪ {+∞} is an extended real-valued function

Let us now introduce some usual notations For a subset B of X , the indicator function

of B, denoted by i B , is defined by i B (x) = 0 if x ∈ B and i B

For any functionϕ : X →R∪ {+∞}, the epigraph of ϕ is defined as

epiϕ := {(w, r) ∈ X ×R| ϕ(w) ≤ r}

while the strict epigraph ofϕ is

epis ϕ := {(w, r) ∈ X ×R| ϕ(w) < r}.

Epigraphs of extended real-valued functions defined on the dual space X∗will be defined in

the same way

Let us recall that the Legendre-Fenchel transform of a given a : X →R∪ {+∞} is

defined as (see, e.g., [24])

It is worth noticing that if k is a proper function on X, k∗does not take the value−∞ and

so, for anyλ ∈ dom k∗, k∗(λ) ∈R Moreover, if k is a proper lower semi-continuous (lsc) convex function, in other words k belongs to the set Γ (X) of extended real-valued proper, lower semi-continuous convex functions on X , then k coincides with its Legendre-Fenchel biconjugate, i.e., k = k∗∗.

The infimal convolutionϕ2ψ of the functions ϕ, ψ : X∗→Ris given by

(ϕ2ψ)(w) := inf

v∈X∗(ϕ(w − v) + ψ(v)) , ∀w ∈ X∗.

We will use the classical properties below: for any a , b : X −→R

and dom b

epi a∗+ epi b∗⊂ epi (a∗2b∗) ⊂ epi (a∗∗+ b∗∗)∗⊂ epi (a + b)∗. (5)

Let S be a convex cone in Z , i.e., S is a convex set and

If H has closed S-epigraph then it is called S-epi-closed The mapping H is said to be S-convex if dom H is convex and

H (tu + (1 − t)z) ≤ S t H(u) + (1 − t)H(z), ∀u, z ∈ dom H, ∀t ∈ [0, 1] Moreover, a function k : Z →R∪ {+∞} is called increasing with respect to S (or,

briefly S-increasing) if u, z ∈ dom k, u ≤ S z implies k(u) ≤ k(z).

For a mapping H : dom H ⊂ X → Z and a function k : Z →R∪ {+∞}, let us set

(k ◦ H)(x) = k(H(x)) if x ∈ dom H+∞ otherwise,

and for the sake of convenience, we writeλH instead of λ ◦ H for any λ ∈ Z∗,

(λH)(x) := λ, H(x) if x ∈ dom H,+∞ else,

where Z∗is the topological dual of the space Z

The following classical theorem will be used in the sequel

Theorem 1 [24, Theorem 2.8.10] Let a : X −→ R∪ {+∞}, k : Z −→R∪ {+∞} be

proper convex functions and let H : dom H ⊂ X → Z be S-convex Assume that k is finite

and continuous at a point of H (dom H ∩ dom a) and k is increasing with respect to S on

H (dom H ∩ dom a) + S We then have, for all w ∈ X∗,

3 The use of the Legendre-Fenchel transform in a purely algebraic setting

Although the functions f, g, λH (λ ∈ Z∗) are not necessarily convex, the use of their

con-jugates remain relevant, as shown by the next lemma

Lemma 1 One always has

A := epi f∗+ epi g∗+ 

λ∈dom k∗epi 

λH − k∗(λ)∗

⊂ B := epi f∗+ 

λ∈dom k∗epi

g + λH − k∗(λ)∗

⊂ epi ( f + g + k ◦ H)∗.

So, if A = epi ( f + g + k ◦ H)∗then

( f + g + k ◦ H)∗=f∗∗+ g∗∗+ k∗∗◦ H∗.

Proof • A⊂B One has

Applying (5) with a = f and b = g + k ◦ H we get

B ⊂ epi ( f + g + k ◦ H)∗.

For the last assertion, let us first observe thatA does not change when replacing f, g, k by

f∗∗, g∗∗, and k∗∗ Hence, we have

A⊂ epif∗∗+ g∗∗+ k∗∗◦ H∗⊂ epi ( f + g + k ◦ H)∗=A.

Consequently, the function( f∗∗+ g∗∗+ k∗∗◦ H)∗and( f + g + k ◦ H)∗have the same

The next proposition gives two sufficient conditions for having (I) They hold without any

convexity, nor lower semi-continuity assumption on f, g, λH, and k ◦ H.

Proposition 1 Given h ∈ Γ (X), let us consider the statements

(I I ) ∀w ∈ dom h∗, ∃v ∈ dom f∗, ∃v1∈ dom g∗, ∃λ ∈ dom k∗,

• (I I I ) ⇒ (I ) Let x ∈ A = dom f ∩ dom g ∩ H−1(dom k) For any w ∈ dom h∗, by

(III), one has

x, w − h∗(w) ≤ x, w − f∗(v) − (g + λH)∗(w − v) − k∗(λ)

≤ x, w − x, v + f (x) − x, w − v

+g(x) + (λH)(x) − λ, H(x) + k(H(x))

= f (x) + g(x) + (k ◦ H)(x).

Taking the supremum overw ∈ dom h∗, we get

h(x) = h∗∗(x) ≤ f (x) + g(x) + (k ◦ H)(x)

We now provide necessary and sufficient conditions ensuring respectively that the alence holds between(I ) and (I I ) and between (I ) and (III) In the sequel we just assume that f + g + k ◦ H is proper We begin with a lemma.

equiv-Lemma 2 For any nonempty subset V of X∗, the following statements are equivalent:

Proof (ii) means exactly that (w, r) ∈ (V ×R) ∩ epi ( f + g + k ◦ H)∗if and only ifw ∈ V

and there exist(λ, v, v1) ∈ dom k∗× dom f∗× dom g∗such that

Analogously, we can state

Lemma 3 For any nonempty subset V of X∗, the following statements are equivalent:

results and also their stability under linear perturbations They serve as the main tools forestablishing alternative type theorems, set containments (Sect.5), and for strong duality orstable strong duality for optimization problems (see Sect.6for concrete examples)

Theorem 2 The following assertions are equivalent

(i) epi ( f + g + k ◦ H)∗=A,

(ii) For any h ∈ Γ (X), (I ) ⇐⇒ (I I ),

(iii) For any x∗∈ X∗, any γ ∈R,

f + g + k ◦ H − x∗≥ γ on X



∃v ∈ dom f∗, v1∈ dom g∗, ∃λ ∈ dom k∗,

f∗(v) + g∗(v1) + (λH)∗(x∗− v − v1) + k∗(λ) ≤ −γ Proof (i) ⇒ (ii) By Proposition1, it is sufficient to check that(I ) ⇒ (I I ) So, let

w ∈ dom h∗ Observe firstly that if(i) holds then (i) in Lemma2holds with V := dom h∗.

Secondly, if(I ) holds then one has ( f + g + k ◦ H)∗(w) ≤ h∗(w) and hence, (w, h∗(w)) ∈ (V ×R) ∩ epi ( f + g + k ◦ H)∗ = (V ×R) ∩ A By(ii) of Lemma2there existv ∈ dom f∗, v1∈ dom g∗, λ ∈ dom k∗such that

f∗(v) + g∗(v1) + (λH)∗(w − v − v1) + k∗(λ) ≤ h∗(w),

and(I I ) holds.

(ii) ⇒ (iii) For any x∗∈ X∗and anyγ ∈R, take h (x) = x∗, x + γ for all x ∈ X in (ii) and then (iii) follows.

(iii) ⇒ (i) By Lemma1we have just to check that the inclusion “⊂” holds in (i) Assume

that( f + g + k ◦ H)∗(x∗) ≤ r We thus have f + g + k ◦ H ≥ x∗− r on X Since (iii)

holds, then there existv ∈ dom f∗, v1∈ dom g∗, and λ ∈ dom k∗such that

f∗(v) + g∗(v1) + (λH)∗(x∗− v − v1) + k∗(λ) ≤ r,

or equivalently,

f∗(v) + g∗(v1) + (λH − k∗(λ))∗(x∗− v − v1) ≤ r, which yields x∗− v − v1∈ dom (λH − k∗(λ))∗ Lettingα := r − f∗(v) − g∗(v1) − (λH −

k∗(λ))∗(x∗− v − v1) ≥ 0, we get

(x∗, r) = (v, f∗(v)) + (v1, g∗(v1)) +x∗− v − v1, (λH − k∗(λ))∗(x∗− v − v1)+ (0, α)

∈ epi f∗+ epi g∗+ epi (λH − k∗(λ))∗+ (0, α) ⊂ A,

By the same way, using Lemma3instead of Lemma2we can state

Theorem 3 The following assertions are equivalent

(i) epi ( f + g + k ◦ H)∗=B,

(ii) For any h ∈ Γ (X), (I ) ⇐⇒ (I I I ),

(iii) For any x∗∈ X∗, any γ ∈R,

f +g+k ◦ H −x∗≥γ on X ⇐⇒ ∃v ∈ dom f∗, ∃λ ∈ dom k∗,

f∗(v) + (g + λH)∗(x∗− v) + k∗(λ)≤−γ . When h is a constant function, we get the following corollaries from Theorems2 3andLemmas2 3whose proofs are similar and only one of them is given

Corollary 1 The following statements are equivalent:

(i)

{0X∗} ×R epi( f + g + k ◦ H)∗= {0X∗} ×R A,

(ii) For any γ ∈R,

−γ ≥ min λ∈domk∗

v∈dom f∗

f∗(v) + (g + λH)∗(−v) + k∗(λ) ,

and(ii) follows.

(ii) ⇒ (i) Similar to the proof of (iii) ⇒ (i) in Theorem2with h(x) = γ for all x ∈ X.



4 On the relations epi( f + g + k ◦ H)∗ =AorB: the use of convexity and lower semicontinuity

In this section, we will establish necessary and sufficient conditions for the relation epi( f +

g + k ◦ H)∗ = AorB and some new transcriptions of the functional inequality(I ) in

the presence of convexity (resp convexity and lower semicontinuity) As an application ofthese transcriptions, we consider some concrete situation where these conditions characterizeFarkas-type results and/or stable form of these results, stable Lagrange duality for convexoptimization problems with convex conic constraints (Corollaries4,5)

Through out this section, we assume that f , g, k are proper convex functions.

4.1 A first approach

Assume that f, g ∈ Γ (X), k ∈ Γ (Z), and

Observe that (7) is in particular satisfied if k is S-increasing, H is S-convex and star-S-lower

semicontinuity (i.e.,λH is lsc for all λ ∈ S∗).

Let us now introduce the extended real-valued functionsϕ and ψ defined on X∗by

ϕ = inf

λ∈dom k∗ f∗2g∗2 [λH − k∗(λ)]∗,

ψ = inf

λ∈dom k∗ f∗2 [g + λH − k∗(λ)]∗.

By using classical arguments, one easily sees thatϕ and ψ are convex and

ϕ∗= f + g + k ◦ H = ψ∗ Assuming moreover f + g + k ◦ H is proper (i.e., dom f ∩ dom g ∩ H−1

have

w∗− cl epi ϕ = epi ( f + g + k ◦ H)∗= w∗− cl epi ψ.

Denoting by epis ϕ the strict epigraph of ϕ, one also has

epis ϕ = epi s f∗+ epis g∗+ 

λ∈dom k∗epis [λH − k∗(λ)]∗,

epis ψ = epi s f∗+ 

λ∈dom k∗epis [g + λH − k∗(λ)]∗.

Taking thew∗-closure in the above relations we get easily

w∗− clA = w∗− cl epi ϕ = w∗− cl epi ψ = w∗− clB,

and we can state that:

Theorem 4 Assume that f + g + k ◦ H is proper Then epi ( f + g + k ◦ H)∗=A (resp B) if and only if A (resp B) is w∗-closed.

4.2 Using Moreau-Rockafellar theorem

In this subsection we assume that k is S-increasing on H (dom H) + S and H is S-convex.

Let us slightly modify the setAandBby setting

(I I ) ∀w ∈ dom h∗, ∃v ∈ dom f∗, ∃v1∈ dom g∗, ∃λ ∈ dom k∗∩ S∗,

f∗(v) + g∗(v1) + (λH)∗(w − v − v1) + k∗(λ) ≤ h∗(w).

(I I I ) ∀w ∈ dom h f∗(v) + (g + λH)∗, ∃v ∈ dom f∗(w − v) + k∗, ∃λ ∈ dom k∗(λ) ≤ h∗(w).∗∩ S∗,

It is worth noticing that if k is S-increasing on the whole Z , then dom k∗ ⊂ S∗, A=

A, B = B, and(I I ) ≡ (I I ), (I I I ) ≡ (I I I ) As repetitions of Theorems2,3we can

state:

Theorem 5 The following assertions hold:

epi( f + g + k ◦ H)∗=A ⇐⇒ (I ) ⇔ (I I ), ∀h ∈ Γ (X) ,

epi( f + g + k ◦ H)∗=B ⇐⇒ (I ) ⇔ (I I I ), ∀h ∈ Γ (X)

Remark 1 It should be emphasized that, by a reasoning similar to that of Theorem3, Theorem

5still holds when replacing “∀h ∈ Γ (X)” by “∀x∗∈ X∗, ∀γ ∈R” In the same way, when

h are constant functions, the following corollary is at hand, whose proof is similar to those

of Corollaries1 2and will be omitted

Corollary 3 The following assertions hold:

(0 X∗×R) ∩ epi ( f + g + k ◦ H)∗=(0 X∗×R) ∩ B ⇐⇒ (I )⇔(I I I ), ∀h ≡γ ∈R .

Let us now compute epi( f + g + k ◦ H)∗in favorable circumstances.

Theorem 6 Assume that there exist x0 ∈ dom f ∩ dom g ∩ H−1(dom k) and x1∈ dom f ∩

dom g ∩ dom H such that k is continuous at H(x0) and f is continuous at x1 Then

epi( f + g + k ◦ H)∗=B Proof From Theorem1with a = f + g we have

epi( f + g + k ◦ H)∗= 

λ∈dom k∗∩S∗

epi[ f + g + λH − k∗(λ)]∗.

Applying [24, Theorem 2.8.7] (see (1)) with a = f and b = g + λH − k∗(λ) we have

epi[ f + g + λH − k∗(λ)]∗= epi f∗+ epi [g + λH − k∗(λ)]∗for anyλ ∈ dom k∗∩ S∗ It

Theorem 7 Assume that there exist x0 ∈ dom f ∩ dom g ∩ H−1(dom k) and x1∈ dom f ∩

dom g ∩ dom H such that k is continuous at H(x0) and f, g is continuous at x1 Then

epi( f + g + k ◦ H)∗=A Proof Given λ ∈ dom k∗∩ S∗, apply Theorem1with a = g and b = λH − k∗(λ) we

get epi[g + λH − k∗(λ)]∗ = epi g∗+ epi [λH − k∗(λ)]∗ The conclusion follows from

4.3 Using closedness regarding a set

We now assume that H is S-convex and S-epi-closed (see Sect. 2), f , g ∈ Γ (X), k ∈

Γ (Z), k is S-increasing on H(dom H) + S, and f + g + k ◦ H is proper Recall that a set

U ⊂ X∗×Ris said to bew∗-closed regarding a set V ⊂ X∗×Rif

Proof Define F1(x, z) := f (x) + k(z) and

(w, −λ, r)|(w, r)∈epi (λH)∗

is w∗-closed regarding to the set Δ × {0 Z∗} ×R.

Proof Define F1(x, z) := f (x) + k(z), F2(x, z) := g(x), and

F3(x, z) = 0+∞ otherwise. if x ∈ dom H and z − H(x) ∈ S,

Applying [2, Theorem 9.2] with n = 3, a i = F i, we obtain that(ii) is equivalent to (for all

It follows that(w, r) belongs toΔ ×R

∩ epi ( f + g + k ◦ H)∗if and only if there exist

v ∈ dom f∗, v1∈ dom g∗, λ ∈ dom k∗∩ S∗, s, s1, t ∈Rsuch that

f∗(v) ≤ s, g∗(v1) ≤ s1, (λH)∗(w − v − v1) ≤ t,

s + s1 + t = r, or, equivalently, (v, s) ∈ epi f∗, (v1, s1) ∈ epi g∗, (w − v − v1, t) ∈

epi(λH − k∗(λ)) , s + s1+ t = r Hence, (ii) amounts to

char-Let us take a closer look to the general convex conic constraint problem:

x ∈C,H(x)∈−S f0(x) and its perturbed version (for any x∗∈ X∗)

(PC x∗) inf

x ∈C,H(x)∈−S



f0(x) − x∗, x, where H is S-convex and S-epi-closed, f0∈ Γ (X), and C is a closed convex subset of X.

Assume that dom f0∩ C ∩ H−1

Let us introduce the mapping F(x, z) := f0(x) + i C (x) + i epi H (x, z) One has F ∈

Γ (X × Z) and

F∗(w, −λ) = ( f0+ i C + λH)∗(w) if λ ∈ S∗,

Corollary The following statements are equivalent:... AorB and some new transcriptions of the functional inequality(I ) in

the presence of convexity (resp convexity and lower semicontinuity) As an application ofthese transcriptions,... ,

and< i>(ii) follows.

(ii) ⇒ (i) Similar to the proof of (iii) ⇒ (i) in Theorem2with h(x) = γ for all x ∈ X.



4 On the relations epi(