Farkas type results for nonconvex systems and applications to optimization = một số kết quả dạng farkas cho các hệ không lồi và áp dụng vào lý thuyết tối ưu

VIETNAM NATIONAL UNIVERSITY-HO CHI MINH CITY
UNIVERSITY OF SCIENCE

TRAN HONG MO

FARKAS-TYPE RESULTS FOR
NONCONVEX SYSTEMS AND
APPLICATIONS TO OPTIMIZATION
PhD Thesis: Optimization Theory
Code: 62 46 20 01

Reviewer 1: Assoc.Prof. Dr. Nguyen Dinh Huy
Reviewer 2: Assoc.Prof. Dr. Dinh Ngoc Thanh
Reviewer 3: Assoc.Prof. Dr. Pham Hoang Quan
Independent Reviewer 1: Dr. Bui Trong Kien
Independent Reviewer 2: Dr. Nguyen Xuan Hai
Supervisor: Assoc.Prof. Dr.Sc. NGUYEN DINH

Ho Chi Minh City - 2015


Originality statement
I hereby declare that this submission is my own work, done at the University of
Sciences - VNU HCMC under the supervision of Assoc.Prof. Dr.Sc. Nguyen Dinh,
International University - VNU HCMC, and, to the best of my knowledge, it contains
no materials previously published and written by another person.

Ho Chi Minh City, 2015
The author

Tran Hong Mo

i


Acknowledgements
First and foremost, I am deeply grateful to my supervisor, Assoc.Prof.Dr.Sc. Nguyen Dinh,
International University - VNU HCMC, for his patient guidance throughout the writing
of this thesis.
I also wish to express my thanks to Prof. Dr.Sc. Phan Quoc Khanh, University of
Sciences - VNU HCMC, for his valuable and insightful lectures in Optimization in my
doctorate programme.
Next, a very special thanks goes to the Management of the University of Sciences VNU HCMC, and the Office of Graduate Admission, for their support and assistance
all the time of my PhD programme and thesis project.
I will never forget how the Management of the University of Tien Giang, its Office
for Personnel and Administration, and Faculty of Basic science have been constantly
providing me with favourable conditions for the completion of this thesis.
My gratitude also goes to Assoc.Prof. Dr. Le Hoan Hoa, HCMC University of
Education, whose supervision of my Master’s thesis inspired and motivated me to
further and advance my research work.
In addition, I owe a lot to my friend Pham Duy Khanh, the University of Education,
who played a big role in persuading me to choose Optimization for my research project.
Finally, I am fully indebted to my family, especially my father and my wife for
nurturing my learning and supporting my dream.
I dedicate this thesis to the memory of my late mother. Without her, I would never
have pursued mathematical studies and become a PhD candidate.

Ho Chi Minh City, 2015
Tran Hong Mo

ii

Contents
Glossary of Notations

v

Introduction

vii

1 Notations and Preliminaries
2

1

Farkas-type results for systems involving composite functions
2.1 Dual qualification conditions and their relations . . . . . . . . . . . . .
2.1.1 Dual qualification conditions in purely algebraic setting . . . . .
2.1.2 Dual qualification conditions in convex setting . . . . . . . . . .
2.2 Characterizations of dual conditions–Generalized Moreau-Rockafellar results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Dual conditions characterizing generalized Moreau-Rockafellar
results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Nonconvex Farkas-type results . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Nonconvex Farkas-type results . . . . . . . . . . . . . . . . . . .
2.3.2 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Alternative-type theorems . . . . . . . . . . . . . . . . . . . . .
2.4.2 Set containments . . . . . . . . . . . . . . . . . . . . . . . . . .

2.4.3 Fenchel-Rockafellar duality formula . . . . . . . . . . . . . . . .

7
7
7
11
13
13
16
19
19
24
27
27
28
29

3 New versions of Farkas lemma and Hahn-Banach theorem under Slatertype conditions
32
3.1 New versions of the Farkas lemma under Slater-type conditions . . . . . 33
3.1.1 Farkas lemma for cone-convex systems . . . . . . . . . . . . . . 33
3.1.2 Farkas lemma for sublinear-convex systems . . . . . . . . . . . . 34
3.2 New versions of the Hahn-Banach theorem under Slater-type conditions 38
3.2.1 Extended Hahn-Banach-Lagrange theorem . . . . . . . . . . . . 38

iii


3.2.2

3.3

Extension of the Hahn-Banach theorem, the sandwich theorem,
and the Mazur-Orlicz theorem . . . . . . . . . . . . . . . . . . .
3.2.3 The equivalence between extended versions of the Farkas lemma
and the Hahn-Banach-Lagrange theorem . . . . . . . . . . . . .
Applications to optimization and convex analysis . . . . . . . . . . . .
3.3.1 Generalized optimization problems involving sublinear-convex mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 A Special case - Penalty problem in convex programming . . . .
3.3.3 Generalized Fenchel duality theorem and a separation theorem .
3.3.4 A conjugate formula for the supremum of a family of convex
functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 From Farkas lemma to Hahn-Banach theorem
4.1 Characterizing extended Farkas lemmas for cone-convex systems . . . .
4.2 Charactering extended Farkas lemmas for sublinear-convex systems . .
4.3 Characterizing extended Hahn-Banach theorems . . . . . . . . . . . . .
4.4 The equivalence between new versions of the Farkas lemma and HahnBanach-Lagrange theorem . . . . . . . . . . . . . . . . . . . . . . . . .

41
44
45
45
53
56
62
66
66
68
72

77

5 Sequential Farkas lemmas and approximate Hahn-Banach theorems 80
5.1 Sequential Farkas lemma for cone-convex systems . . . . . . . . . . . . 81
5.2 Sequential Farkas lemma for sublinear-convex systems . . . . . . . . . . 84
5.3 Approximate Hahn-Banach theorems . . . . . . . . . . . . . . . . . . . 87
5.4 Sublinear-convex optimization problems without any constraint qualification conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.5 An application: limiting conjugate formula for the supremum of a family
of convex functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
Conclusion and suggested further research

99

Author’s publications related to the thesis

101

Author’s conferences

102

Bibliography

103

iv


Glossary of Notations
Spaces and Sets

X, Y
X ∗, Y ∗
M⊥
cl A
int A
K+
lin D
.
R
|.|
∈
⊂
∅
∩, ∪, ×
lim inf ai

the locally convex topological spaces
the topological dual spaces of X, Y , respectively
the orthogonal subspace to M
the closure of the subset A
the interior of the subset A
the dual cone of K
the linear hull of the subset D of Y
the norm of a vector
the set of real numbers
the absolute value of a real number
the membership of an element in a set
the set inclusion
the empty set
intersection, union, Cartesian product

the limit inferior of a net (ai )i∈I

lim sup ai

the limit superior of a net (ai )i∈I

i∈I

i∈I

Functions
dom f
epi f
epis f
Γ(X)
f∗
f ∗∗
iA
∂ f (¯
x)
∂f (¯
x)

the
the
the
the
the
the
the

the
the

domain of the function f : X → R ∪ {+∞}
epigraph of the function f
strict epigraph of the function f
set of proper lower semi-continuous convex functions
conjugate function of f
biconjugate function of f
indicator function of the subset A
-subdifferential of f at x¯ in dom f
subdifferential of f at x¯ in dom f
v


NC (¯
x)
φ ψ
y∗, y
L(X, Y )
IdR
PrY

the
the
the
the
the
the

normal cone at x¯
infimal convolution of two function φ, ψ
image of y ∗ at y
set of all continuous linear operators from X to Y
identity function on R
function from X × Y into Y defined by PrY (x, y) = y

Partial orders
≤K
∞K
Y•
dom h
epiK h
h−1 (−K)
g◦h
≤S
K-convex
S-convex

the partial order on Y by a cone K in Y
the greatest element with respect to ≤K , i.e.,
y ≤K ∞K for all y ∈ Y
the space which we add ∞K to Y
the domain of the mapping h : X → Y • , i.e.,
the set {x ∈ X : h(x) ∈ Y }
the K-epigraph of the mapping h : X → Y • , i.e.,
the set epiK h := {(x, y) ∈ X × Y : y ∈ h(x) + K}
the set {x ∈ X : h(x) ∈ −K}
the composite function of g : Y → R ∪ {+∞}
and h : X → Y •

the partial order on Y by the sublinear function
S : Y → R ∪ {+∞}
convex with respect to a convex cone K
convex with respect to a sublinear function S

vi


Introduction
The statement of the original Farkas lemma was proposed by the Hungarian mathematician and physicist Gyula Farkas in 1894, inspired from his work concerning certain
equilibrium problems in mechanics. However, the first correct proof was published eight
years later, in 1902 [35]. The lemma states that given any vectors a1 , a2 , · · · , am and
c in Rn , the following statements are equivalent:
(i) x ∈ Rn , aTi x ≥ 0, i = 1, 2, . . . , m =⇒ cT x ≥ 0;
(ii) ∃λi ≥ 0, i = 1, 2, . . . , m, c =

m
i=1

λ i ai .

In the 1950s of the last century, after the time when Gale, Kuhn, and Tucker successfully applied this lemma to establish duality results and optimality conditions for linear
programming and nonlinear programming, the Farkas lemma had become one of the
well-known tools in optimization and in applied mathematics as well. Since then, many
efforts have been made to generalize this lemma, and as a result, many new extended
versions of the lemma have been discovered and many of their applications have been
found, not only in applied mathematics but also in other fields such as finance and
economics [33], [36].
In the theoretical aspect, it was proved in [38, Corollary 2, p.92] that the convex
extended version of the Farkas lemma, namely the so-called Farkas-Minkowski lemma,

is equivalent to the Hahn-Banach theorem. Moreover, the Farkas lemma is none other
than a “mathematical version” of the First Fundamental Principle of financial markets
[33]. Because of its importance, it was continuously extended in the last decades from
linear systems to convex systems, and also nonconvex systems; from finite to infinite
dimensional spaces and also, from systems involving single-valued functions to the ones
defined by multi-valued functions (see [9], [19], [23], [30], [31], [37], [41], [42], [50], [56],
and the recent survey paper [20] for more details).
It is worth noting that extended versions of the Farkas lemma for infinite dimensional spaces or for nonlinear systems often hold under some kinds of constraint qualification conditions such as the Slater condition or its generalizations (called interior-type
conditions) (see [5], [37], [38], and references therein). In the recent years, the authors
vii


V. Jeyakumar, N. Dinh, R. Burachik, R. I. Bot, G. Wanka introduced some kinds of
qualification conditions called “closedness conditions” which are much weaker than the
interior-type ones ( see [7], [9], [10], [11], [14], [15], [29], [31], and references therein).
Better still, in some of the works published by these authors, it was shown that the
latter type conditions characterize extended Farkas lemma in some concrete settings,
i.e., these conditions are necessary and sufficient conditions to ensure the validity of
extended Farkas lemmas, not only the sufficient conditions as usual (see, e.g., [42]).
All the extended versions of the Farkas lemma are known under the common name
“Farkas-type results” and have had many applications to the theory of optimization.
Specifically, versions of Farkas lemma have been applied to get duality results and
optimality conditions for cone-convex problems, DC (difference of convex functions)
problems, bilevel problems, variational inequalities, equilibrium problems, best approximation problems, etc (see [27], [28], [29], [30], [31], [42], [46], and references therein).
As mentioned above, many new extended versions of the Farkas lemma were proposed in the last decades with successful applications to optimization theory. However,
there still remain many problems and open questions not yet solved or answered, e.g.,
whether it is possible that some Farkas lemma version can be established for systems
involving composite functions? for systems involving vector functions? These expected
versions, if any, may help us a lot in studying classes of problems with composite functions or vector optimization problems. On the other hand, there appeared a few works
in recent years on some new versions of the Farkas lemma holding without any qualification conditions [21], [44]. Any more extended versions of the Farkas lemma holds

without any qualification conditions? Last but not least, are there any relation between
extended versions of Farkas lemma existing in the literature of fundamental mathematics? Motivated by these observations, we plan to study the following problems in this
thesis:
Problem 1. Farkas lemma for systems involving composite functions:
Studying and establishing some generalized versions of the Farkas lemma for systems
involving composite functions with/without convexity and lower semi-continuity from
the data. Applying the results obtained to convex/nonconvex composite optimization
problems.
Problem 2. Extended Farkas lemmas and fundamental mathematics: As
mentioned above, some earlier version of the Farkas lemma is equivalent to the HahnBanach theorem (see [38]). So, a natural question arises: Do there exist extended
versions of the Hahn-Banach theorem which are equivalent to recent/new extended
versions of the Farkas lemma?

viii


Problem 3. Sequential Farkas lemmas and approximate Hahn-Banach
theorems: Establishing new versions of the Farkas lemma without any qualification condition (called “sequential Farkas lemma”). Combining these versions with the
Problem 2, i.e., finding new versions of Hahn-Banach theorem (called “approximate
Hahn-Banach theorem”) holding without qualification condition that are equivalent to
the mentioned sequential Farkas lemmas.
During the years of studying to realize this thesis, we found part of the answers
for the Problems mentioned above. The results obtained will be presented in Chapters
2, 3, 4, and 5 of this thesis. Specifically, we get the following results concerning the
Problems 1,2, and 3:
Farkas-type results for systems involving composite functions (Chapter
2) We consider the functional inequality concerning composite functions of the form:
f (x) + g(x) + (k ◦ H)(x) ≥ h(x) ∀x ∈ X,

(1)

where X, Y are locally convex Hausdorff topological spaces, f, g, h : X → R ∪ {+∞}
are proper functions, H : dom H ⊂ X → Y is a mapping, and k : Y → R ∪ {+∞} is a
proper function.
We get the following main results:
• We introduce six dual conditions (of closedness-type) that will serve as qualification conditions in establishing versions of Farkas lemma concerning system (1). It
is worth mentioning that for these conditions, all the functions, mappings considered
are just proper but not necessarily convex nor lower semi-continuous.
• We establish characterizations of the inequality of the form (1). These characterizations are precisely Farkas-type results involving the inequality of the mentioned
form.
• Generalized Moreau-Rockafellar results involving composite functions are established. These results generalize the classical ones in three ways: the functions are
not necessarily convex, some assumptions are weakened, and the conditions are not
only sufficient but also necessary.
• Several alternative theorems, characterizations of set containments of a convex
set in a DC set or in a reverse convex set, and generalized Fenchel-Rockafellar duality
results are derived.
These results are presented in Chapter 2.
Concerning Problem 2, we get the following results, which will be divided into
two parts due to the qualification conditions considered and also the level of their
generalization. These results will be presented, one after another, in Chapter 3 and
Chapter 4.
ix


New versions of Farkas lemma and Hahn-Banach theorem under Slatertype conditions (Chapter 3).
• New versions of the Farkas lemma for cone-convex systems and for sublinearconvex systems are established under Slater-type constraint qualification conditions
and in the absence of the lower semi-continuity and the closedness of functions and
constrained sets involved.
• An extended version of Hahn-Banach-Lagrange theorem (that generalizes the
one in [62]) is established and it is shown to be equivalent to the extended Farkas

lemma for cone-convex systems and for sublinear-convex systems just obtained.
• Our results lead to extensions of other fundamental theorems such as the sandwich theorem, the Mazur-Orlicz theorem, and also the Hahn-Banach theorem itself
for the case involving extended sublinear functions (the situation where the celebrated
Hahn-Banach theorem failed).
• The results obtained are then applied to get duality results and optimality
conditions for a class of composite problems involving sublinear-convex mappings.
• As illustrative examples, we consider the penalty problems associated to convex programming problems, a formula for the conjugate of the supremum of a family
(possibly infinite, not lower semi-continuous) of convex functions, and a special class of
problems inspired in the Fenchel duality theorem. The duality results for the later class
of problems give rise to some generalized versions of the Fenchel duality theorem which
extends the one proposed recently by S. Simons in [62]. Moreover, the problems are
considered in normed spaces and in this setting, a duality result leads to a separation
theorem for convex sets in normed spaces.
These results are given in Chapter 3.
From Farkas lemma to Hahn-Banach theorem (Chapter 4).
The results in this part are extensions of the ones in the previous part (Chapter
3) and further development (such as stable versions of Farkas-type results). In a very
general setting, we introduce new qualification conditions (called closedness condition)
which are the weakest conditions that still ensure the validity of many new Farkastype results. It means that we give conditions that characterize (give necessary and
sufficient conditions for) Farkas-type results. We get the following results:
• We establish that some new closedness conditions characterize new versions
of the Farkas lemma for systems defined by convex cones and the ones defined by
sublinear-convex mappings. These results inspired us to obtain many versions of stable
Farkas lemmas which were given in [22].
• We show that the extended Farkas lemmas just obtained give rise to the characterization of an analytic version of the Hahn-Banach theorem, the analytic HahnBanach-Lagrange theorem, the analytic Mazur-Orlicz theorem for sublinear functions
x


which may take the value +∞ (the case where the classical Hahn-Banach theorem
fails).

• We show that the mentioned versions of the Farkas lemma and the analytic Hahn-Banach-Lagrange theorem established above are actually equivalent to each
other.
These results are provided in Chapter 4.
Sequential Farkas lemmas and approximate Hahn-Banach theorems (Chapter 5)
In this part, we focus on the development of new versions of Farkas lemma that
hold without any constraint qualification conditions. Since all the versions of this type
hold in the limits, they are often named as “sequential Farkas lemmas”. We also found
several new forms of “approximate Hahn-Banach theorem” that are equivalent to such
sequential Farkas lemmas.
• New versions of sequential Farkas lemma for cone-convex systems and sublinearconvex systems are established (without any constraint qualification condition).
• Several versions of approximate Hahn-Banach-Lagrange theorem are established.
• It is shown that sequential Farkas lemmas and approximate Hahn-BanachLagrange theorems are equivalent to each other. Versions of approximate Hahn-Banach
theorems and approximate sandwich theorems for extended sublinear functions are
derived from the results just obtained .
• The results are then applied to obtain approximate duality results and optimality conditions for optimization problem involving sublinear-convex mappings. As an
illustrative example, we provide an approximate conjugate formula for the supremum
of a family (possibly infinite) of convex functions.
Some notations and preliminaries are given in Chapter 1.
The results presented in this dissertation have been published in
• Vietnam Journal of Mathematics: ([24], Chapter 2, [25] for a part of Chapter 3);
• Taiwanese Journal of Mathematics: ([26] for a part of Chapter 3);
• SIAM Journal of Optimization: ([22] for Chapter 4).

xi


Chapter 1
Notations and Preliminaries
We now provide some definitions and preliminaries that will be used later on.
Let X and Y be locally convex Hausdorff topological vector spaces (l.c.H.t.v.s.). X ∗

and Y ∗ denote their topological dual spaces (respectively), endowed with the weak∗ topology. If M is a linear subspace of X, the orthogonal subspace to M is M ⊥ = x∗ ∈
X ∗ : x∗ , x = 0 for all x ∈ M . Let B, C be two subsets of some l.c.H.t.v.s.. We say
that B is closed regarding C if (clB) ∩ C = B ∩ C (see [5, p.56]). Given a set A
in one of the considered spaces, the closure of A and the interior of A will be denoted
simply by cl A and int A, respectively; the indicator function of A, iA , is defined by:
iA (x) := 0 if x ∈ A and iA (x) := +∞ if x ∈
/ A. The linear hull of the subset D of Y is
the linear subspace spanned by D
lin D :=

Y0 ⊂ Y : D ⊂ Y0 , Y0 linear subspace of Y .

Given f : X → R ∪ {+∞}, the domain of f , denoted by dom f , is given by
domf := {x ∈ X : f (x) < +∞}. The function f is said to be proper if dom f = ∅.
The conjugate function of f is the function f ∗ : X ∗ → R ∪ {±∞} defined by
f ∗ (x∗ ) = sup { x∗ , x − f (x)} , ∀x∗ ∈ X ∗ .
x∈X

Similarly, the conjugate function of f ∗ is the function f ∗∗ : X → R ∪ {±∞} defined by
f ∗∗ (x) = sup { x∗ , x − f ∗ (x∗ )} , ∀x ∈ X.
x∗ ∈X ∗

The set of proper lsc convex functions defined on X is denoted by Γ (X) . For any
proper f : X → R ∪ {+∞} one has
f ∈ Γ (X) ⇔ f = f ∗∗ .
The conjugate of iA is the support function of A, i.e., the function σ A : X ∗ → R∪{+∞}
such that σ A (x∗ ) := i∗A (x∗ ) = sup x∗ , x for any x∗ ∈ X ∗ .
x∈A

1

The epigraph of f is the set
epi f := {(x, r) ∈ X × R : x ∈ domf, f (x) ≤ r}.
The epigraph of the conjugate of f , f ∗ , will be defined in a similar way. The strict
epigraph of f is the set
epis f := {(x, r) ∈ X × R : x ∈ domf, f (x) < r}.
It is easy to check that
epis inf fi =
i∈I

epis fi

(1.1)

i∈I

It is worth mentioning that the function f is a proper lower semi-continuous (lsc)
convex function if epi f is a nonempty closed convex set.
Given the pair of proper functions f : X → R ∪ {+∞} and ψ : R →R ∪ {+∞}, we
define the function f : X × R →R ∪ {+∞}
f (x, α) := f (x) + ψ(α), ∀(x, α) ∈ X × R.

(1.2)

Then,
f ∗ (x∗ , γ) = f ∗ (x∗ ) + ψ ∗ (γ), ∀(x∗ , γ) ∈ X ∗ × R,
and
epi f ∗ = {(x∗ , 0, r) : (x∗ , r) ∈ epi f ∗ } + {(0X ∗ , γ, r) : (γ, r) ∈ epi ψ ∗ }.

(1.3)

For any nonnegative number , the -subdifferential of a proper function f at a
given point x¯ in domf is defined as the convex set
∂ f (¯
x) = {x∗ ∈ X ∗ | f (x) − f (¯
x) ≥ x∗ , x − x¯ −
∗

= {x ∈ X

∗

∗

∗

∗

∀x ∈ dom f }

| f (x ) + f (¯
x) − x , x¯ ≤ }.

(1.4)

then ∂ 1 f (¯
x) ⊂ ∂ 2 f (¯
x),

(1.5)

∂ 1 f (¯
x) + ∂ 2 g(¯
x) ⊂ ∂ 1 + 2 (f + g)(¯
x).

(1.6)

It is clear that
if 0 ≤
and for any

1, 2

1

<

2

≥ 0,

Note also that if f is convex and lsc at x¯ ∈ dom f and if > 0 then ∂ f (¯
x) = ∅.
Moreover, when = 0, -subdifferential of f collapses to the usual subdifferential in
the sense of convex analysis and will be denoted as ∂f , instead of ∂0 f (see, e.g., [47],
[66],...). In the case where C is a non-empty convex subset of X and x¯ ∈ C, ∂iC (¯
x) is
2

called the normal cone to C at x¯, denoted by NC (¯
x). If x¯ ∈
/ C then by convention, we
set NC (¯
x) = ∅. The Young-Fenchel inequality
f ∗ (x∗ ) ≥ x∗ , x¯ − f (¯
x)
always holds and further, the equality holds if and only if x∗ ∈ ∂f (¯
x) .
The infimal convolution of two proper functions φ, ψ : X ∗ → R ∪ {+∞} is the
function φ ψ defined by
(φ ψ) (x∗ ) = ∗inf ∗ {φ(u∗ ) + ψ(x∗ − u∗ )}
u ∈X

∀x∗ ∈ X ∗ .

Moreover, if φ and ψ are proper and convex then φ ψ is convex.
Obviously, one can also define the infimum convolution of φ, ψ : X → R ∪ {+∞}
in a similar way. Moreover, for any x∗ , u∗ ∈ X ∗ ,
(φ + ψ)∗ (x∗ ) ≤ φ∗ (u∗ ) + ψ ∗ (x∗ − u∗ ),
which leads to (by taking the infimum over u∗ ∈ X ∗ ),
(φ + ψ)∗ (x∗ ) ≤ (φ∗ ψ ∗ )(x∗ ) ∀x∗ ∈ X ∗
and, consequently,
epi φ∗ + epi ψ ∗ ⊂ epi(φ∗ ψ ∗ ) ⊂ epi(φ + ψ)∗ .

(1.7)

In general if f, g ∈ Γ (X) and (dom f ) ∩ (dom g) = ∅, then one has (see, e.g. [7,

Theorem 2.1])
epi(f + g)∗ = cl(epi f ∗ + epi g ∗ ),
(1.8)
where, as it was already stated, cl represents the closure with respect to the weak*topology. If one of the functions f or g is continuous at a point of the domain of the
other, then the closure cl can be removed from the right hand side of (1.8) (see, e.g.
[66, Theorem 2.8.7]).
For any cone K in Y , K + denotes the dual cone of K, defined by
K + := {y ∗ ∈ Y ∗ : y ∗ , y ≥ 0 for all y ∈ K}.
We define ≤K to be a partial order on Y by a closed convex cone K containing the
origin of Y (0Y ∈ Y ), i.e.,
y1 ≤K y2 if y2 − y1 ∈ K.
We add to Y a greatest element with respect to ≤K , denoted by ∞K which does
not belong to Y and let Y • = Y ∪ {∞K }. Then one has y ≤K ∞K for every y ∈ Y • .
3


Consider the following operations on Y • : y + ∞K = ∞K + y = ∞K for all y ∈ Y • ,
and α∞K = ∞K if α ≥ 0.
A function h : X → Y • , we call domain of h the set dom h = {x ∈ X : h(x) ∈ Y },
and we say that h is proper if dom h = ∅. The K-epigraph of h is the set
epiK h := {(x, y) ∈ X × Y : y ∈ h(x) + K}.
Moreover, for any y ∗ ∈ Y ∗ and h : X → Y • we define the composite function y ∗ ◦ h :
X → R ∪ {+∞} as follows:
(y ∗ ◦ h)(x) =

y ∗ , h(x) , if x ∈ domh,
+∞,
otherwise.

Definition 1.0.1. The function h : X → Y • is said to be K-convex if

x1 , x2 ∈ X, µ ∈ [0, 1] ⇒ h((1 − µ)x1 + µx2 ) ≤K (1 − µ)h(x1 ) + µh(x2 ),
where ≤K is the binary relation extended to Y • by setting y ≤K ∞K for all y ∈ Y • .
It is obvious that h is K-convex if and only if epiK h is convex.
Definition 1.0.2 ([51]). The function h : X → Y • is said to be K-epi closed if
epiK h is a closed set in the product space. Then, the cone K and the set h−1 (−K) are
both closed as h−1 (−K) × {0Y } = (epiK h) ∩ (X × {0Y }).
Definition 1.0.3. The function S : Y → R ∪ {+∞} is (extended) sublinear if
y1 , y2 ∈ Y ⇒ S(y1 + y2 ) ≤ S(y1 ) + S(y2 ),

(a)

y ∈ Y and α > 0 ⇒ S(αy) = αS(y).

(b)

and

We assume that S(0Y ) = 0 (this convention is appropriate to the assumption that
S is lsc). Such a function S can be extended to Y • by setting S(∞K ) = +∞. An
extended sublinear function S : Y → R ∪ {+∞} allows us to introduce in Y • a binary
relation which is reflexive and transitive:
y1 ≤S y2 if y1 ≤K y2 , where K := {y ∈ Y : S(−y) ≤ 0}.
This means that
y1 ≤S y2 ⇐⇒ S(y1 − y2 ) ≤ 0, ∀y1 , y2 ∈ Y.

(1.9)

S(y1 − y2 ) ≤ 0 ⇐⇒ S(y + y1 ) ≤ S(y + y2 ) ∀y ∈ Y.

(1.10)

Moreover,

4


Indeed, if S(y1 − y2 ) ≤ 0 then for any y ∈ Y , one has
S(y + y1 ) = S(y + y2 + y1 − y2 ) ≤ S(y + y2 ) + S(y1 − y2 ) ≤ S(y + y2 ).
Conversely, assume that S(y + y1 ) ≤ S(y + y2 ) for all y ∈ Y. Then by taking y = −y2 ,
one gets S(y1 − y2 ) ≤ S(−y2 + y2 ) = 0.
From (1.9) and (1.10), we have
y1 ≤S y2 ⇐⇒ (S(y + y1 ) ≤ S(y + y2 ) ∀y ∈ Y ) .

(1.11)

It is worth mentioning that the definition of the relation ≤S can be understood in the
extended sense of S : Y • → R ∪ {+∞}. We set y ≤S ∞K for all y ∈ Y • . Taking the
convention S(∞K ) = +∞ into account, the extension of the relation ≤S to Y • is in
accordance with (1.11).
With the extension of the relation ≤S to Y • , we can easily extend the concept of
S-convex w.r.t. a sublinear function S : Y → R introduced in [62] to the one of Sconvex function w.r.t. an extended sublinear function S : Y • → R ∪ {+∞} as follows
[22]
Definition 1.0.4. A mapping h : X →Y • is said to be (extended) S-convex if for
all x1 , x2 ∈ X, µ1 , µ2 > 0, µ1 + µ2 = 1, one has
h(µ1 x1 + µ2 x2 ) ≤S µ1 h(x1 ) + µ2 h(x2 ).
It is worth noting that, as mentioned in [61, Remark 1.10], ”S-convex can mean
different things under different circumstances”. For instance, when Y = R, if S(y) :=
|y|, S(y) := y, S(y) := −y, or S(y) = 0, respectively, then ”S-convex” means ”affine”,
”convex”, ”concave” or ”arbitrary”, respectively.
It can be easily verified that if h is S-convex then h is K-convex with K := {y ∈

Y : S(−y) ≤ 0}. Conversely, if h is K-convex with some convex cone K then h is
S-convex with S = i−K .
We consider the extended real line R ∪ {±∞} with the following conventions
+∞ + (+∞) = +∞ + (−∞) = +∞.
Definition 1.0.5. [1, p.32], [3, p.5], [54, p.217] Let (ai )i∈I be a net of extended real
numbers defined on a directed set (I, ). We define limit inferior of the net (ai )i∈I
as
lim inf ai := lim inf aj = sup inf aj .
i∈I

i∈I j i

i∈I j i

Similarly, we define limit superior of a net (ai )i∈I as
lim sup ai := lim sup aj = inf sup aj .
i∈I

i∈I j i

5

i∈I j i


We say that (ai )i∈I converges to a ∈ R, denoted by lim ai = a or ai −→ a, if for any
i∈I

> 0, there exists i0 ∈ I such that |ai − a| <

for all i

i0 .

The following properties were given in [3, p.9] and [54, p.221]
Lemma 1.0.1. Let (ai )i∈I and (bi )i∈I be nets of extended real numbers. Then the
following statements hold:
(i) lim sup(−ai ) = − lim inf ai and lim sup ai ≥ lim inf ai .
i∈I

i∈I

i∈I

i∈I

(ii) lim ai = a ∈ R if and only if lim inf ai = lim sup ai = a.
i∈I

i∈I

i∈I

(iii) If ai ≤ bi for all i ∈ I, then
lim inf ai ≤ lim inf bi and lim sup ai ≤ lim sup bi .
i∈I

i∈I

i∈I

i∈I

(iv) lim inf (ai +bi ) ≥ lim inf ai +lim inf bi , and lim sup(ai +bi ) ≤ lim sup ai +lim sup bi ,
i∈I

i∈I

i∈I

i∈I

i∈I

i∈I

provide that the right side of the inequalities are defined. Note that the equalities
hold whenever one of the nets is convergent.
Now let (u∗i )i∈I be a net in the topological space X ∗ . We say that the net (u∗i )i∈I
converges to u∗ ∈ X ∗ via the w∗ -topology if
lim u∗i , x = u∗ , x for all x ∈ X, and write u∗i −→∗ u∗ .
i∈I

To close this chapter, we now recall some results which will be used in this thesis.
The following result, Lemma 1.0.2, was proved in [44] where f is assumed to be convex
and lsc. However, it still holds without these assumptions.
Lemma 1.0.2. Let X be an l.c.H.t.v.s. and let f be a proper function on X. If
a ∈ dom f , then
epi f ∗ =

{(v ∗ , v ∗ , a + − f (a)) | v ∗ ∈ ∂ f (a)} .
≥0

6


Chapter 2
Farkas-type results for systems
involving composite functions
In this chapter, we firstly propose variants of closedness conditions involving the
inequality composite functions
f (x) + g(x) + (k ◦ H)(x) ≥ h(x),

∀x ∈ X,

in a general setting without convexity or lower semi-continuity of the functions and
mappings involved. Secondly, it is proved that under these new closedness conditions
(also called qualification conditions), several necessary and sufficient conditions for the
mentioned inequality are established. They are actually new Farkas-type results for the
setting in consideration (without convexity or lower semi-continuity). It turns out that
these qualification conditions are not only sufficient conditions but also necessary ones
for these Farkas-type results. The results extend or cover many known Farkas-type
results for convex systems or systems involving DC functions in the literature. These
results are then applied to obtain new results in convex analysis and optimization
such as: alternative-type theorems, characterizations of set containments, and FenchelRockafellar duality formula.

2.1

Dual qualification conditions and their relations

In this section we introduce several dual conditions in purely algebraic setting and
establish their relations. Some special cases are considered.

2.1.1

Dual qualification conditions in purely algebraic setting

Let X, Y be l.c.H.t.v.s. with its topological dual X ∗ , Y ∗ , respectively, f, g, h :
X → R ∪ {+∞} are proper functions, H : dom H ⊂ X → Y is a mapping, and
7


k : Y → R ∪ {+∞} is a proper function. Note that the functions f, g, h, λH (λ ∈ Y ∗ )
and k are proper functions but not necessarily convex nor lsc. Let
A := epi f ∗ + epi g ∗ +

epi(λH − k ∗ (λ))∗ ,
λ∈dom k∗

B := epi f ∗ +

epi(g + λH − k ∗ (λ))∗ ,
λ∈dom k∗

C := epi(f + g)∗ +

epi(λH − k ∗ (λ))∗ ,
λ∈dom k∗

D := epi g ∗ +

epi(f + λH − k ∗ (λ))∗ ,
λ∈dom k∗

epi(f + g + λH − k ∗ (λ))∗ ,

E :=
λ∈dom k∗
∗

F := epi f + epi(g + k ◦ H)∗ .
We start with the relations between the sets in consideration.
Lemma 2.1.1. With the previous notions, one gets
A ⊂ B ⊂ E,

A ⊂ C ⊂ E,

A ⊂ D ⊂ E,

A ⊂ B ⊂ F,

and
E ⊂ epi(f + g + k ◦ H)∗ , F ⊂ epi(f + g + k ◦ H)∗ .
In particular, if A = epi(f + g + k ◦ H)∗ then A = B = C = D = E = F.
Proof. The proof is easy, using mainly (1.7). We first observe that
epi g ∗ +

epi g ∗ + epi(λH − k ∗ (λ))∗

epi(λH − k ∗ (λ))∗ =

λ∈dom k∗

λ∈dom k∗

and, by (1.7) with φ = g and ψ = λH − k ∗ (λ), λ ∈ dom k ∗ , we have A ⊂ B. Applying
(1.7) one more time for φ = f and ψ = g + λH − k ∗ (λ), (λ ∈ dom k ∗ ), we get B ⊂ E.
Now, for any λ ∈ dom k ∗ , x ∈ dom H, we get
−k ∗ (λ) ≤ (k ◦ H)(x) − (λH)(x),
and so, λH − k ∗ (λ) ≤ k ◦ H, which yields f + g + λH − k ∗ (λ) ≤ f + g + k ◦ H. We
thus have, for all λ ∈ dom k ∗ ,
f + g + λH − k ∗ (λ)

∗

∗

≥ f + g + k ◦ H , g + λH − k ∗ (λ)

∗

∗

≥ g+k◦H ,

and therefore, it holds for all λ ∈ dom k ∗ ,
epi f +g +λH −k ∗ (λ)

∗

∗

⊂ epi f +g +k ◦H , epi g +λH −k ∗ (λ)
8

∗

∗

⊂ epi g +k ◦H ,


which, in turn, shows that
epi f + g + λH − k ∗ (λ)

E=

∗

⊂ epi(f + g + k ◦ H)∗ ,

λ∈dom k∗

and
epi f ∗ +

epi(g + λH − k ∗ (λ))∗ ⊂ epi f ∗ + epi (g + k ◦ H)∗ .
λ∈dom k∗

We have just proved that
A ⊂ B ⊂ E ⊂ epi(f + g + k ◦ H)∗

and B ⊂ F. The inclusion F ⊂ epi (f + g + k ◦ H)∗ follows from (1.7). Other inclusions
can be proved in the same way. The last assertion is obvious.
We now introduce the following dual conditions:
(CA)

A = epi(f + g + k ◦ H)∗ ,

(CB)

B = epi(f + g + k ◦ H)∗ ,

(CC)

C = epi(f + g + k ◦ H)∗ ,

(CD)

D = epi(f + g + k ◦ H)∗ ,

(CE)

E = epi(f + g + k ◦ H)∗ ,

(CF)

F = epi(f + g + k ◦ H)∗ .

The relations between these conditions are given in the next theorem.
Theorem 2.1.1. The following implications hold.
(CF)

(CA)

✦
✦
✜✔✔
✜
✜
✜

✓❏
✓
❏

✜
✜
✜ ✥
✥
✜
✚
✜
✓
✚✓
✜
✚
✜ ✚✚
✜
✚
✜ ✚
✚

✜
✚
✚
✚
✚
❛
❛
✦
✦
❩
❩❩
❩❩
❩❩
❩❩
❩❩❙
❵
❩
❵
❙

(CB)
❩
❩
❩
❩
❩
❩

(CC)
✚

✚✚
✚
✚
✚

❩
❩
❩
❩
❩
❩
❙
❵
❵
❙
❛
❛
✦
✦
✥
✥
✚✓
✓
✚
✚
✚
✚
✚

(CE)

(CD)
Here (A) =⇒ (B) means that condition (A) implies condition (B).

9


Proof. We observe that if (CA) holds, i.e.,
A = epi(f + g + k ◦ H)∗ ,
then by Lemma 2.1.1, we get A = B = E = epi(f + g + k ◦ H)∗ , A = C = E =
epi(f +g+k◦H)∗ , A = D = E = epi(f +g+k◦H)∗ , and A = B = F = epi(f +g+k◦H)∗ .
This means that if (CA) holds then (CB), (CC), (CD), (CE), and (CF) do, too. Also,
by the same argument using Lemma 2.1.1 we see that if one of (CB), (CC), (CD) holds
then (CE) holds, and if (CB) holds then (CF) holds. The proof is complete.
The conditions (CA) and (CB) were introduced in [32], where some Farkas-type
results were established under (CA).
We now consider a special case. Let C be a subset of X, K be a convex cone in Y .
Let g := iC , k := i−K and A := C ∩ H −1 (−K). Then it is easily seen that k ∗ = iK + ,
and hence, dom k ∗ = K + . Moreover, iA = iC + i−K ◦ H, where
(k ◦ H)(x) = (i−K ◦ H)(x) =

0
+∞

if H(x) ∈ −K,
otherwise.

Thus, the condition (CA) becomes
epi (λH)∗ + epi i∗C = epi (f + iA )∗ .

epi f ∗ +
λ∈K +

By the same procedure, we get other modifications of other conditions (CB), (CC),
(CD), (CE), and (CF). Specifically, we consider the following modifications of these
conditions:
(CA1 )

epi f ∗ +

epi (λH)∗ + epi i∗C = epi (f + iA )∗ ,
λ∈K +

(CB1 )

epi f ∗ +

epi (λH + iC )∗ = epi (f + iA )∗ ,
λ∈K +

(CC1 )

epi (f + iC )∗ +

epi (λH)∗ = epi (f + iA )∗ ,
λ∈K +

(CD1 )

epi i∗C +

epi (f + λH)∗ = epi (f + iA )∗ ,
λ∈K +

epi (f + λH + iC )∗ = epi (f + iA )∗ ,

(CE1 )
λ∈K +

(CF1 )

epi f ∗ + epi i∗A = epi (f + iA )∗ .

Then as a direct consequence of Theorem 2.1.1, we get

10


Corollary 2.1.1. The following relations hold.
(CF1 )
✦
✦
✜
✔✔
✜
✜
✜
✜
✜
✜

✥
✜ ✥
✚
✜
✓
✚✓
✜
✚
✚
✜
✜ ✚
✚
✜
✚
✜ ✚✚
✚
✚
✚
❛
❛
✦
(CA1 )
✦
❩
❩❩
❩❩
❩❩
❩❩
❩❩❙
❵

❩
❵
❙

2.1.2

✓❏
✓
❏

(CB1 )
❩
❩
❩
❩
❩
❩
❩
❩

(CC1 )

(CD1 )

✚
✚✚
✚

❩
❩

❩
❩
❙
❵
❵
❙
❛
❛
✦
✦
✥
✥
✚✓
✓
✚
✚
✚
✚
✚
✚
✚

(CE1 )

Dual qualification conditions in convex setting

We now give necessary and sufficient dual qualification conditions introduced in the
previous section in the cases where the convexity and lower semi-continuity of functions involved are assumed. Some sufficient conditions (also, necessary and sufficient
conditions) for the validity of (CA) were given in [32].
In this subsection we assume that f, g ∈ Γ(X), k ∈ Γ(Y ), λH ∈ Γ(X) for all

λ ∈ dom k ∗ , and dom(f + g + k ◦ H) = ∅.
Proposition 2.1.1. epi (f + g + k ◦ H)∗ = C if and only if C is w∗ -closed.
Proof. Let us introduce the extended real-valued function φ defined on X ∗ by
φ = (f + g)∗ ✷

inf

λ∈dom k∗

(λH − k ∗ (λ))∗ .

We firstly observe that φ is convex. Indeed, let ψ : dom k ∗ × X ∗ −→ R ∪ {+∞} be the
function defined by
ψ(λ, u) := (λH − k ∗ (λ))∗ (u).
It is easy to verify that ψ is convex. By Theorem 2.1.3(v) in [66], the function
inf

λ∈dom k∗

(λH − k ∗ (λ))∗ (.) =

inf

λ∈dom k∗

ψ(λ, .)

is convex and hence, φ is proper convex.
Now, from the assumption that f, g ∈ Γ(X), k ∈ Γ(Y ), and λH ∈ Γ(X) for all

11


λ ∈ dom k ∗ , one gets, for any x ∈ X,
φ∗ (x) =

x∗ , x − (f + g)∗ ✷

sup
x∗ ∈X ∗

=

x∗ , x −

sup
x∗ ∈X ∗

=
=

sup

sup

λ∈dom k∗

u,v∈X ∗

inf

inf

λ∈dom k∗

inf

λ∈dom k∗ x∗ =u+v

(λH − k ∗ (λ))∗

(x∗ )

(f + g)∗ (u) + (λH − k ∗ (λ))∗ (v)

u, x + v, x − (f + g)∗ (u) − (λH)∗ (v) − k ∗ (λ)

(f + g)(x) + (λH)(x) − k ∗ (λ)

sup
λ∈dom k∗

= f (x) + g(x) + (k ◦ H)(x)
and thus,
epi (f + g + k ◦ H)∗ = epi φ∗∗ = epi cl φ,

(2.1)

where cl φ is the lower semi-continuous regularization of φ (which is defined via the
equation epi cl φ = cl epi φ).

On the other hand, by Theorem 2.2(e) in [63], we get
epi cl φ = epi cl (f + g)∗ ✷

inf

λ∈dom k∗

(λH − k ∗ (λ))∗

= cl epi (f + g)∗ + epi [ inf

(λH − k ∗ (λ))∗ ]

= cl epi (f + g)∗ + epis [ inf

(λH − k ∗ (λ))∗ ]

λ∈dom k∗

λ∈dom k∗

epis [λH − k ∗ (λ)]∗

= cl epi (f + g)∗ +
λ∈dom k∗

epi [λH − k ∗ (λ)]∗

= cl epi (f + g)∗ +
λ∈dom k∗

= cl C.

(2.2)

Combining (2.1) and (2.2), we get epi (f +g +k ◦H)∗ = cl C. The proof is complete.
In the same way, we get
Proposition 2.1.2. The following statements are true:
(i) epi (f + g + k ◦ H)∗ = D if and only if D is w∗ -closed,
(ii) epi (f + g + k ◦ H)∗ = E if and only if E is w∗ -closed,
(iii) epi (f + g + k ◦ H)∗ = F if and only if F is w∗ -closed.
Proof. Similar to the proof of Proposition 2.1.1, using Theorem 2.2 in [63] and considering the following functions φ1 , φ2 , and φ3 for the case (i), (ii), and (iii), respectively:
φ1 = g ∗ ✷ inf λ∈dom k∗ (f + λH − k ∗ (λ))∗ , φ2 = inf λ∈dom k∗ (f + g + λH − k ∗ (λ))∗ , and
φ3 = f ∗ ✷(g + k ◦ H)∗ .
12


2.2

Characterizations of dual conditions–Generalized
Moreau-Rockafellar results

In this section we shall establish characterizations of the dual conditions introduced in Section 2.1, namely, (CA)–(CF). These characterizations at the same time
are variant versions of generalization of Moreau-Rockafellar results on the conjugates
and subdifferentials of a sum of a convex function with a composition of convex functions in Hausdorff locally convex spaces (see e.g., [2], [10], [66] for more details) to
nonconvex cases and with approximate subdifferentials. The results cover some recent
ones from [10] and generalize many other results of this type with the presence of convexity in the literature. It is worth mentioning that these results generalize classical
results in three aspects: The class of functions is broadened (to nonconvex ones), the
assumptions are weakened, and they supply necessary and sufficient conditions while
in most of the cases (even for the convex one) only the sufficient conditions are given in

the literature. In particular, they cover Fenchel duality, Moreau-Rockafellar theorems
concerning the sum of convex functions or of a convex function with a composite of a
convex function and a linear mapping (Corollary 2.2.2).

2.2.1

Dual conditions characterizing generalized Moreau-Rockafellar
results

Assume that the functions f, g, k, and λH (for arbitrary λ ∈ Y ∗ ) are proper
functions but not necessarily convex nor lower semicontinuous. Assume further that
dom(f + g + k ◦ H) = ∅. We start by establishing the characterizations of (CA).
Theorem 2.2.1. The following statements are equivalent:
(a) (CA) holds,
(b) For all x∗ ∈ X ∗ ,
(f + g + k ◦ H)∗ (x∗ )
=

min

λ∈dom k∗
u∈dom f ∗ , v∈dom g ∗

f ∗ (u) + g ∗ (v) + (λH)∗ (x∗ − u − v) + k ∗ (λ) ,

(c) For all x¯ ∈ dom(f + g + k ◦ H) and all

≥ 0,

∂ (f + g + k ◦ H)(¯

x)
=

∂ 1 f (¯
x) + ∂ 2 g(¯
x) + ∂ 3 (λH)(¯
x) .
λ∈dom k∗
1 + 2 + 3 +k

1 , 2 , 3 ≥0
∗ (λ)+(k◦H)(¯
x)=

+(λH)(¯
x)

13