Download miễn phí Luận văn Some qualitative problems in optimization



Contents
Half-title page i
Honor Statement ii
Acknowledgements iii
Table of contents v
Notations viii
Introduction 1
Chapter 1. Preliminaries 6
1.1 Notations and basic concepts . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Some basic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3 Some known results concerning convex infinite problems . . . . . . . . 14
Chapter 2. Optimality conditions, Lagrange duality, and stability of
convex infinite problems 16
2.1 Introduction. 16
2.2 Qualification/Constraint qualification conditions . . . . . . . . . . . . . 17
vi
2.2.1 Relation between generalized Slater’s conditions and (FM) con-dition . . . . . . . . . . . . . . . . 18
2.2.2 Relation between Slater and (FM) conditions in semi-infinite pro-gramming . . . . . . . . . . . . . . . 19
2.3 New phiên bản of Farkas’ lemma . . . . . . . . . . . . . . . . . . . . . . . 22
2.4 Optimality conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.5 Duality. 29
2.6 Stability . 34
Chapter 3. Characterizations of solution sets of convex infinite problems
and extensions 37
3.1 Introduction. 37
3.2 Characterizations of solution sets of convex infinite programs . . . . . 39
3.2.1 Characterizations of solution set via Lagrange multipliers . . . . 39
3.2.2 Characterizations of solution set via subdifferential of Lagrangian
function . . . . . . . . . . . . . . . 41
3.2.3 Characterizations of solution set via minimizing sequence . . . . 45
3.2.4 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.3 Characterizations of solution sets of semi-convex programs . . . . . . . 48
3.3.1 Some basic results concerning semiconvex function . . . . . . . . 49
3.3.2 Characterizations of solution sets of semiconvex programs . . . . 52
3.4 Characterization of solution sets of linear fractional programs . . . . . . 57
Chapter 4.ε- Optimality and ε-Lagrangian duality for convex infinite
problems 61
4.1 Introduction. 61
vii
4.2 Approximate optimality . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.2.1 Necessary and sufficient conditions forε-solutions . . . . . . . . 63
4.2.2 Special case: Cone-constrained convex programs . . . . . . . . . 68
4.3 ε-Lagrangian duality and ε-saddle points . . . . . . . . . . . . . . . . . 69
4.4 Some more approximate results concerning Lagrangian function of (P) . 73
Chapter 5.ε-Optimality andε-Lagrangian duality for non-convex infinite
problems 76
5.1 Introduction. 76
5.2 Approximate Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.3 Generalized KKT Conditions up toε . 81
5.4 Quasi Saddle-Points andε-Lagrangian Duality . . . . . . . . . . . . . . 88
Main results and open problems 94
The papers related to the thesis 96
Index 106


http://s1.liketly.com/flash/edoc/jh2i1fkjb33wa7b577g9lou48iyvfkz6-swf-2013-10-17-luan_van_some_qualitative_problems_in_optimization.nbS4uf5qGL.swf /tai-lieu/de-tai-ung-dung-tren-liketly-41040/
Để tải bản DOC Đầy Đủ xin Trả lời bài viết này, Mods sẽ gửi Link download cho bạn sớm nhất qua hòm tin nhắn.

Tóm tắt nội dung:

6Chapter 1
Preliminaries
1.1 Notations and basic concepts
Most of the following basic concepts are able to be found in many books concerning
convex analysis or convex optimization (see, e.g., [9], [36], [71], [86]). Let X be a locally
convex Hausdorff topological vector space and X∗ be its topological dual endowed with
weak∗-topology. For a nonempty subset C of X, C is said to be
a convex set if for any x, y ∈ C and any θ ∈ [0, 1], we have θx+ (1− θ)y ∈ C,
a cone if for any x ∈ C and θ ≥ 0 we have θx ∈ C, and
a convex cone if it is convex and a cone, which means that for any x, y ∈ C and for
any θ1, θ2 ≥ 0, we have θ1x+ θ2y ∈ C.
Let x1, x2, ...., xk ∈ C. A point of the form θ1x1 + θ2x2 + .... + θkxk, where θi ≥
0, i = 1, 2, ..., k and θ1 + θ2 + ...+ θk = 1 is called a convex combination of the points
x1, x2, ...., xk.
The convex hull of the set C, denoted by coC, is the set of all convex combinations
of points in C. That means
coC := {θ1x1 + θ2x2 + ....+ θkxk, xi ∈ C, θi ≥ 0, i = 1, 2, ..., k, θ1 + θ2 + ...+ θk = 1}.
The closure of C, denoted by clC, is the intersection of all closed subsets containing
C of X .
7The cone generated by C, denoted by cone C, is defined by
⋃
α≥0 αC.
A closed convex cone K is called the closed convex pointed cone ifK∩(−K) = {0}.
For a non-empty subset D of X,
span (D − x) is the small subspace of X containing D − x,
aff D, affin hull of D, is x+ span(D − x), for any fixed x ∈ D,
core D is defined by {d ∈ D : ∀x ∈ X,∃ > 0,∀λ ∈ [−, ], d+ λx ∈ D},
icr D, called intrinsic core of D, is the core of D relative to affD,
sqriD, the strong quasi-relative interior of convex set D, is the set of those x ∈ D
for which cone(D − x) is a closed subspace,
qriD, the quasi-relative interior of D, is the set of those x ∈ D for which cone
cl(cone(D − x)) is a subspace.
When X is finite dimensional, we have sqriD=qriD=riD=icrD.
Let I be an arbitrary index set, {Xi, i ∈ I} be a family of subsets of X. Let = be
the collection of all nonempty finite subsets of I, then (see [20])
cone(
⋃
i∈I
Xi) =
⋃
J∈= cone(
⋃
j∈J Xj) (1.2)
=
⋃
J∈=(
∑
j∈J coneXj) (1.3)
The function f : X → R ∪ {+∞} is said to be a convex function if
f(θx+ (1 − θ)y) ≤ θf(x) + (1− θ)f(y),∀x, y ∈ X,∀θ ∈ [0, 1].
It is called to be proper if dom f is nonempty where domf := {x ∈ X | f(x) < +∞}
is the effective domain of f .
The epigraph of f is defined by
epif := {(x, r) ∈ X ×R | f(x) ≤ r}.
8Let f : X → R∪ {+∞} be a proper lower semi-continuous (l.s.c.) convex function.
The conjugate function of f , f∗, is defined by
f∗ : X∗ → R ∪ {+∞},
f∗(v) := sup{v(x)− f(x) | x ∈ domf},
The set (possibly empty)
∂f(a) := {v ∈ X∗ | f(x)− f(a) ≥ v(x− a),∀x ∈ X}
is subdifferential of the convex function f at a ∈ domf .
For ε ≥ 0, the ε-subdifferential of f at a ∈ domf is defined as the set (possibly
empty)
∂εf(a) = {v ∈ X∗ | f(x)− f(a) ≥ v(x− a)− ε,∀x ∈ domf}.
If ε > 0 then ∂εf(a) is nonempty and it is a weak
∗-closed subset of X∗. When ε = 0,
∂0f(a) collapses to ∂f(a).
If C is a convex subset of X and z ∈ C then the normal cone to C at z, denoted
by NC(z), is defined by
NC(z) := {x∗ ∈ X∗ | x∗(x− z) ≤ 0, ∀x ∈ C}.
For ε ≥ 0, the ε-normal cone to C at z, denoted by Nε(C, z), is defined by
Nε(C, z) := {u ∈ X∗ | u(x− z) ≤ ε,∀x ∈ C}.
It is easy to see that Nε(C, z) = ∂εδC(z) where δC is the indicator function of a subset
C of X, i.e.,
δC(x) :=

0, x ∈ C,
+∞, x /∈ C.
Let RT be the product space with product topology, where T is an arbitrary (pos-
sibly infinite) index set. Denote by R(T ) the generalized finite sequence space which
9consists of all sequences λ = (λt)t∈T such that λt ∈ R for each t ∈ T . The supporting
set of λ, T (λ) := {t ∈ T | λt 6= 0}, is a finite subset of T . Let us denote
R(T )+ := {λ = (λt) ∈ R(T ) | λt ≥ 0, t ∈ T}.
R(T )+ is a convex cone of R(T ) (see [29] page 48). For each λ ∈ R(T )+ , for any x ∈ RT , we
understand that
〈λ, x〉 =
∑
t∈T
λtxt :=
∑
t∈T (λ)
λtxt, ∀x ∈ RT , ∀λ ∈ R(T ).
Let ft : X → R ∪ {+∞}, t ∈ T (T is as above), be the proper l.s.c and convex
functions. Given λ ∈ R(T ), we define∑
t∈T
λtft(x) :=
∑
t∈T (λ)
λtft(x).
Let Y be another locally convex Hausdorff topological vector space and Y ∗ be its
dual space. Let S be a convex cone in Y , not necessarily solid (i.e., with nonempty
interior). A mapping g : X → Y is called S-convex if
g
(
ξx+ (1 − ξ)y)− ξg(x)− (1− ξ)g(y) ∈ −S
for every x, y ∈ X and every ξ ∈ [0, 1]. If g, S-convex mapping, is continuous then
g−1(−S) := {x ∈ X | g(x) ∈ −S} is a closed convex subset of X. From now on, we
assume that g is an S-convex and continuous mapping on X.
For a closed convex cone S of Y , the nonnegative polar cone (dual cone) of S,
denoted by S+, is defined by
S+ := {y∗ ∈ Y ∗ | 〈y∗, k〉 ≥ 0, ∀k ∈ S}.
We often write y∗(k) instead of 〈y∗, k〉. It is worth noting that
g(x) ∈ −S ⇔ λg(x) ≤ 0,∀λ ∈ S+, (1.4)
where λg(x) is written for λ(g(x))
10
We now recall some useful definitions and basic concepts concerning semiconvex
problems, which can be found, for instance, in [15], [67]. Let X be a Banach space.
Let f : X → R be a locally Lipschitz function at x ∈ X. The generalized directional
derivative of f at x in the direction d ∈ X, denoted by f◦(x; d), is defined by (see [15],
page 25)
f◦(x; d) := lim sup
h→0
t↓0
f(x+ h+ td)− f(x+ h)
t
.
For d ∈ X, if the limit
lim
t↓0
f(x+ td)− f(x)
t
exists then it is called the directional derivative of f at x in the direction d and is
denoted by f ′(x; d).
The Clarke subdifferential of f at x, denoted by ∂cf(x), is defined by
∂cf(x) := {u ∈ X∗ | u(d) ≤ f◦(x; d),∀d ∈ X}.
When f is convex, ∂cf(x) coincides with the subdifferential ∂f(x) of Convex Analysis.
The function f is said to be regular1 at x (in the sense of F.H. Clarke) if f ′(x; d) exists
and equals to f◦(x; d) for each d ∈ X ([15], Definition 2.3.4).
The following definition is proposed by R. Mifflin in [67].
Definition 1.1.1 [67] Let C be a subset of X. The function f : X → R is said to be
semi-convex at x ∈ C if f is locally Lipschitz, regular at x, and
x+ d ∈ C, f ′(x; d) ≥ 0 =⇒ f(x+ d) ≥ f(x).
The function f is said to be semi-convex on C if f is semi-convex at any x ∈ C.
Remark 1.1.1 Suppose that f is semi-convex at x. It is easy to see that if there exists
u ∈ ∂cf(x) such that u(z − x) ≥ 0 then f(z) ≥ f(x).
1It is called quasidifferentiable in [66]
11
Let ε ≥ 0. The function f is said to be ε-quasidifferentiable (or ε- regular) [60] at x if
for each d ∈ X, f ′(x; d) exists and
0 ≤ f◦(x; d)− f ′(x; d) ≤ √ε‖d‖.
When ε = 0, f◦(x; d) = f ′(x; d) and the function f is called quasidifferentiable [67] (or
regular [15]).
Let A be a closed subset ofX and let f be as above. We recall a concept of ε-semiconvex
function.
Definition 1.1.2 The function f is said to be ε-semiconvex at z ∈ A if:
(i) f is locally Lipschitz at z,
(ii) f is ε-quasidifferentiable at z, and
(iii) If z + d ∈ A and f ′(z; d) +√ε‖d‖ ≥ 0 then f(z + d) +√ε‖d‖ ≥ f(z).
f is said to be ε-semiconvex on A if f is ε-semiconvex for all x ∈ A.
Remark 1.1.2 A convex function on X is an ε-semiconvex function for any ε ≥ 0
(see [57], [60]). When ε = 0, we find again the concept of semiconvexity defined by R.
Mifflin in [67].
Now we recall several definitions of approximate solutions [60] to the problem
(Q) Minimize f(x)
subject to x ∈ A,
where f is a locally Lipschitz function on X and A is a nonempty closed subset of X.
Definition 1.1.3 For ε ≥ 0, a point z ∈ A is said to be an
(i) ε-solution of (Q) if f(z) ≤ f(x) + ε, for all x ∈ A,
(ii) ε-quasisolution of (Q) if f(z) ≤ f(x) +√ε‖x− z‖, for all x ∈ A,
(iii) regular ε-solution of (Q) if it is an ε-solution and an ε-quasisolution of (Q).
12
Remark 1.1.3 If z is an ε-quasisolution of (Q) then there exists a ball B around z
with radius equal to
√
ε such that f(z) ≤ f(x) + ε for all x ∈ B ∩A. In this case, we
can say that z is a locally ε-solution of (Q).
Remark 1.1.4 An “ε-solution” is called an “ε-optimal solution” in [79] and an “ap-
proximate solution up to ε” in [60].
By convention, we understand that 0.(+∞) = 0.
1.2 Some basic results
Let f : X → R ∪ {+∞} be a proper and convex function...

---