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On the Sub Convexlike Optimization Problems

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A peer-reviewed article of this preprint also exists.

Renying Zeng^*

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Abstract

In this paper, we prove that the sub convexlikeness introduced by V. Jeyakumar [1], and the subconvexlikeness defined in V. Jeyakumar [2] are equivalent in loccallly convex topological spaces. And then, we work with set-valued vector optimization problems and obtain some vector saddle-point theorems and vector Lagrangian theorems.

Keywords:

Subject: Computer Science and Mathematics - Applied Mathematics

0. Introduction

Generalized convex optimization are very well studied branches of mathematics. There are many very meaningful and useful definitions of generalized convexities. Let X be a normed space, and X₊ a convex cone of X. K. Fan [3] introduced the definition of X₊-convexlike function. Jeyakumar [1] introduced the definition of X₊-sub convexlike function. And, Jeyakumar defined X₊-subconvexlike function in [2]. There are plenty of research articles discussing subconvexlike optimization problems, e.g., see [4,5,6,7,8,9,10,11,12]. In this paper, we show that the sub convexlikeness introduced in [1] and the subconvexlikeness in [2] are actually equivalent in locally convex topological spaces (including normed linear spaces).

Most of papers in set-valued optimization studied the problem with inequality constraint and the abstract constraint. In this paper we consider the set-valued optimization problem with not only inequality, abstract but also equality constraints. The explicit statement of the equality constraint would be very convenient in applications. For example, recently, the mathematical programs with equilibrium constraints has received a lot of attentions from the optimization community. The mathematical programs with equilibrium constraints are a class of optimization problem with variational inequality constraints. By representing the variational inequality as a generalized equation, e.g., [8, 13, 14], a mathematical program with equilibrium constraints can be reformulated as an optimization problem with an equality constraint. This paper deals with the set-valued optimization problem with inequality, equality as well as abstract constraints, and obtains some vector saddle-point theorems and vector Lagrangian theorems.

1. Preliminary

Let X be a real topological vector space, a subset X₊ of X is said to be a convex cone if

α x^{1} + β x^{2} \in X_{+}, \forall x^{1}, x^{2} \in X_{+}, \forall α, β \geq 0 .

We denote by

0_{X}

the zero element in the topological space X and simply by 0 if there is no confusion.

A convex cone X₊ of X is called a pointed cone if

X_{+} \cap (- X_{+}) = {0}

A real topological vector space X with a pointed cone is said to be an ordered topological liner space. We denote intX₊ the topological interior of X_{+ .} The partial order on X is defined by

x^{1} ≺_{X_{+}} x^{2}, if x^{1} - x^{2} \in X_{+}, x^{1} ≺ ≺_{X_{+}} x^{2}, if x^{1} - x^{2} \in i n t X_{+},

Or, if there is no confusion, they may just be denoted by

x^{1} ≺ x^{2}, if x^{1} - x^{2} \in X_{+}, x^{1} ≺ ≺ x^{2}, if x^{1} - x^{2} \in i n t X_{+},

A, B \subseteq X,

we denoted by

A ≺_{X_{+}} B, if x ≺_{X_{+}} y for \forall x \in A, \forall y \in B, A ≺ ≺_{X_{+}} B, if x ≺ ≺_{X_{+}} y for \forall x \in A, \forall y \in B .

Or,

A ≺ B, if x ≺ y for \forall x \in A, \forall y \in B, A ≺ ≺ B, if x ≺ ≺ y for \forall x \in A, \forall y \in B .

A linear functional on X is a continuous linear function from X to R (1-dimensinal Euclidean space). The set X ^* of all linear functionals on X is the dual space of X. The subset

X_{+}^{*} = {ξ \in X^{*} : 〈x, ξ〉 \geq 0, \forall x \in X_{+}}

X^{*}

is said to be the dual cone of the cone X₊ , where

〈x, ξ〉 = ξ (x)

Suppose that X and Y are two real topological vector spaces. Let f: X→2^Y be a set-valued function, where 2^Y denotes the power set of Y.

Let D be a nonempty subset of X. Setting

f (D) = \cup_{x \in D} f (x)

, and

〈f (x), η〉 = {〈y, η〉 : y \in f (x)}, 〈f (D), η〉 = \cup_{x \in D} 〈f (x), η〉 .

For

x \in D, η \in Y^{*}

, we write

〈f (x), η〉 \geq 0, if 〈y, η〉 \geq 0, \forall y \in f (x), 〈f (D), η〉 \geq 0, if 〈f (x), η〉 \geq 0, \forall x \in D .

The following Definitions 1.1 and 1.2 can be found in [15].

Definition 1.1 (convex, bounded, and absorbing) A subset M of X is said to be convex, if

x_{1}, x_{2} \in M

and

0 < α < 1

implies

α x_{1} + (1 - α) x_{2} \in M

; M is said to be balanced if

x \in M

and

| α | \leq 1

implies

α x \in M

; M is said to be absorbing if for any given neighbourhood U of 0, there exists a positive scalar

β

such that

β^{- 1} M \subseteq U

, where

β^{- 1} M = {x \in X; x = β^{- 1} v; v \in M}

Definition 1.2 (locally convex topological space) A topological vector space X is called a locally convex topological space if any neighborhood of

0_{X}

contains a convex, balanced, and absorbing open set.

From [15, pp.26 Theorem, pp.33 Definition 1], a normed linear space is a locally convex topological space.

2. The Sub Convexlikeness

This section shows that the definitions of sub convexlikeness and subconvexlikeness given by Jeyakumar [1, 2] are actually one.

A set-valued function f: X→ 2^Y is said to be Y₊ -convex on D if ∀x¹, x²∈D, ∀α∈[0, 1], one has

α f (x^{1}) + (1 - α) f (x^{2}) ≺_{Y_{+}} f (α x^{1} + (1 - α) x^{2}) .

The following definition of convexlikeness was introduced by Ky Fan [8].

A set-valued function f: X→2^Y is said to be Y₊ -convexlike on D if ∀x¹, x²∈D, ∀α∈[0, 1], ∃x³∈D such that

α f (x^{1}) + (1 - α) f (x^{2}) ≺_{Y_{+}} f (x^{3})

Jeyakumar [2] introduced the following subconvexlikeness.

Definition 2.1 (subconvexlike) Let Y be a topological vector space and

D \subseteq X

be a nonempty set and Y₊ be a convex cone in Y. A set-valued map f :

D \to 2^{Y}

is said to be Y₊-subconvexlike on D if

\exists θ \in int Y_{+}

such that

\forall x_{1}, x_{2} \in D

\forall ε > 0

\forall α \in [0, 1]

\exists x_{3} \in D

here holds

ε θ + α f (x_{1}) + (1 - α) f (x_{2}) ≺_{Y_{+}} f (x_{3}) .

Lemma 2.1 is [16, Lemma 2.3].

Lemma 2.1 Let Y be a topological vector space and

D \subseteq X

be a nonempty set and Y₊ be a convex cone in Y. A set-valued map f :

D \to 2^{Y}

is Y₊-subconvexlike on D if and only if

\forall θ \in int Y_{+}

\forall x_{1}, x_{2} \in D

\forall α \in [0, 1]

\exists x_{3} \in D

such that

θ + α f (x_{1}) + (1 - α) f (x_{2}) ≺_{Y_{+}} f (x_{3}) .

A bounded function in a topological space can be defined as following Definition 2.2 (e.g,, see Yosida [15]).

Definition 2.2 (bounded set-valued map) A subset M of a real topological vector space Y is said to be a bounded subset if for any given neighbourhood U of 0, there exists a positive scalar

β

such that

β^{- 1} M \subseteq U

, where

β^{- 1} M = {y \in Y; y = β^{- 1} v; v \in M}

. A set-valued map f :

D \to Y

is said to bounded map if f (Y) is a bounded subset of Y.

Jeyakumar [1] introduced the following sub convexlikeness.

Definition 2.3 (sub convexlike) Let Y be a topological vector space and

D \subseteq X

be a nonempty set. A set-valued map f :

D \to 2^{Y}

is said to be Y₊-sub convexlike on D if

\exists

bounded set-valued map u:

D \to Y

\forall x_{1}, x_{2} \in D

\forall ε > 0

\forall α \in [0, 1]

\exists x_{3} \in D

such that

ε u + α f (x_{1}) + (1 - α) f (x_{2}) ≺_{Y_{+}} f (x_{3}) .

The following Lemma 2.1 is from Li and Wang [16, Lemma 2.3].

Lemma 2.2 Let Y be a locally convex topological space and

D \subseteq X

be a nonempty set Y. A set-valued map f :

D \to 2^{Y}

is Y₊-subconvexlike on D if and only if

f (D) + int Y_{+}

is Y₊-convex.

Theorem 2.1 Let Y be a locally convex topological space,

D \subseteq X

a nonempty set, and Y₊ a convex cone in Y. A set-valued map f :

D \to 2^{Y}

is Y₊-sub convexlike on D if and only if

f (D) + int Y_{+}

is Y₊-convex.

Proof. The necessity.

Suppose that f is Y₊-sub convexlike..

\forall z_{1} = y_{1} + y_{0}^{1}, z_{2} = y_{2} + y_{0}^{2} \in f (D) + int Y_{+},

\exists x_{1}, x_{2} \in D

such that

y_{1} \in f (x_{1}),

y_{2} \in f (x_{2}) .

Let

y_{0}^{} = α y_{0}^{1} + (1 - α) y_{0}^{2},

then

y_{0}^{} \in int Y_{+}^{}

. Therefore,

\exists

neighbourhood U of 0 such that

y_{+}^{0} + U

is a neighbourhood of

y_{+}^{0}

and

y_{+}^{0} + U \subseteq int Y_{+} .

By Definition 1.2, we may assume that U is convex, balanced, and absorbing.

From the assumption of sub convexlikeness, i.e.,

\exists

bounded set-valued map u:

x_{1}, x_{2} \in D

ε > 0

α \in [0, 1]

\exists

x_{3} \in D

such that

ε u + α f (x_{1}) + (1 - α) f (x_{2}) \subseteq f (x_{3}) + Y_{+} .

Therefore

\begin{array}{l} α z_{1} + (1 - α) z_{2} \\ = α y_{1} + (1 - α) y_{2} + α y_{+}^{1} + (1 - α) y_{+}^{2} \\ \subseteq f (x_{3}) - ε u + Y_{+} + y_{+}^{0} \end{array}

Since U is convex, balanced, and absorbing, we may take

ε > 0

small enough such that

- ε u \subseteq U .

Therefore

- ε u + y_{+}^{0} \subseteq y_{+}^{0} + U \subseteq int Y_{+} .

And then

\begin{array}{l} α z_{1} + (1 - α) z_{2} \\ = α y_{1} + (1 - α) y_{2} + α y_{+}^{1} + (1 - α) y_{+}^{2} \\ \subseteq f (x_{3}) + int Y_{+} \\ \subseteq f (D) + int Y {}_{+}. \end{array}

Hence,

f (D) + int Y_{+}

is a Y₊-convex set.

The sufficiency.

f (D) + int Y_{+}

is Y₊-convex, then, by Lemma 2.1, f is Y₊-subconvexlike. And, it is clear that Y₊-subconvexlikeness implies Y₊-sub convexlikeness.

From Lemma 2.2 and Theorem 2.1 one has Theorem 2.2.

Theorem 2.2 Let Y be a locally convex topological space and

D \subseteq X

a nonempty set, and Y₊ a convex cone in Y. A set-valued map f :

D \to 2^{Y}

is Y₊-subconvexlike on D if and only if f is Y₊-sub convexlike on D.

3. Vector Saddle-Point Theorems

This section works on vector saddle-point theorems for set-valued optimization problems.

A set-valued map f:

D \to 2^{Y}

is said to be affine on D if

\forall x_{1}, x_{2} \in D, \forall β \in R

, there holds

β f (x_{1}) + (1 - β) f (x_{2}) = f (β x_{1} + (1 - β) x_{2})

We introduce the notion of sub affinelike functions as follows.

Definition 3.1 (sub affinelike) A set-valued map f :

D \to 2^{Y}

is said to be Y₊-sub affinelike on D if

\forall x_{1}, x_{2} \in D, \forall α \in (0, 1),

\exists

v \in int Y_{+}

\exists x_{3} \in D

there holds

v + α f (x_{1}) + (1 - α) f (x_{2}) = f (x_{3}) .

Theorem 3.1 Let X, Y, Z and W be real topological vector spaces,

D \subseteq X .

Y_{+}, Z_{+}

W_{+}

pointed convex cones of Y, Z and W, respectively. Assume that functions

f : D \to Y, g : D \to Z, h : D \to W

satisfy that

(a): f and g are sub convexlike maps on D, i.e., $\forall u_{1} \in int Y_{+}, \forall u_{2} \in int Z_{+},$ $\forall α \in (0, 1), \forall x^{1}, x^{2} \in D$ , $\exists x', x'' \in D$ such that

$\begin{array}{l} u_{1} + α f (x^{1}) + (1 - α) f (x^{2}) ≺ f (x'), \\ u_{2} + α g (x^{1}) + (1 - α) g (x^{2}) ≺ g (x''); \end{array}$

(b): h is a sub affinelike map on D, i.e., $\forall α \in (0, 1), \forall x^{1}, x^{2} \in D,$ $\exists x''' \in D,$ $v \in int W_{+}$ such that

$v + α h (x^{1}) + (1 - α) h (x^{2}) = h (x''');$

(c): $int h (D) \neq \emptyset$ ;

and (i) and (ii) denote the systems

(i): $\exists x \in D, s . t ., f (x) ≺ ≺ 0, g (x) ≺ 0, h (x) = 0;$

(ii): $\exists (ξ, η, ζ) \in (Y_{+}^{*} \times Z_{+}^{*} \times W^{*}) \ {(0_{Y}, 0_{Z}, 0_{W})}$ such that

$ξ (f (x)) + η (g (x)) + ς (h (x)) \geq 0, \forall x \in D .$

If (i) has no solution then (ii) has solutions.

Moreover if (ii) has a solution

(ξ, η, ς)

with

ξ \neq 0_{Y^{*}}

then (i) has no solutions.

Proof.

\forall w_{1}, w_{2} \in \cup_{t > 0} t h (D) + int W_{+},

\forall α \in (0, 1),

\exists x_{1}, x_{2} \in D,

\exists b_{1}, b_{2} \in int W_{+},

\exists t_{1}, t_{2} > 0

such that

\begin{array}{l} α w_{1} + (1 - α) w_{2} \\ = α t_{1} h (x_{1}) + (1 - α) t_{2} h (x_{2}) + α b_{1} + (1 - α) b_{2} \\ = (α t_{1} + (1 - α) t_{2}) [\frac{α t_{1}}{α t_{1} + (1 - α) t_{2}} h (x_{1}) + \frac{(1 - α) t_{2}}{α t_{1} + (1 - α) t_{2}} h (x_{2})] + α b_{1} + (1 - α) b_{2} . \end{array}

By the assumption (b),

\exists x_{3} \in D,

\exists v \in int W_{+},

\forall ε > 0

such that

\frac{α t_{1}}{α t_{1} + (1 - α) t_{2}} h (x_{1}) + \frac{(1 - α) t_{2}}{α t_{1} + (1 - α) t_{2}} h (x_{2}) = h (x_{3}) - ε v .

Since

v \in int Z_{+},

\exists

neighbourhood U of 0 in W for which

V = α b_{1} + (1 - α) b_{2} + U

is a neighbourhood of

α b_{1} + (1 - α) b_{2}

By Definition 1.2, we may take

ε > 0

small enough such that

- ε (α t_{1} + (1 - α) t_{2}) v \subseteq U .

Then,

α b_{1} + (1 - α) b_{2} - ε (α t_{1} + (1 - α) t_{2}) v \subseteq V \subseteq int W_{+} .

Therefore,

\begin{array}{l} α w_{1} + (1 - α) w_{2} \\ \subseteq α t_{1} h (x_{1}) + (1 - α) t_{2} h (x_{2}) + α b_{1} + (1 - α) b_{2} \\ = (α t_{1} + (1 - α) t_{2}) [\frac{α t_{1}}{α t_{1} + (1 - α) t_{2}} h (x_{1}) + \frac{(1 - α) t_{2}}{α t_{1} + (1 - α) t_{2}} h (x_{2})] + α b_{1} + (1 - α) b_{2} \\ = (α t_{1} + (1 - α) t_{2}) h (x_{3}) + α b_{1} + (1 - α) b_{2} - ε (α t_{1} + (1 - α) t_{2}) v \\ \subseteq \cup_{t > 0} t h (D) + int W_{+} . \end{array}

So,

\cup_{t > 0} t h (D) + int Y_{+},

is a convex set.

Similarly,

\cup_{t > 0} t f (D) + int Y_{+},

and

\cup_{t > 0} t g (D) + int Z_{+}

are also convex. Therefore, the set

C = (\cup_{t > 0} t f (D) + int Y_{+}) \times (\cup_{t > 0} t g (D) + int Z_{+}) \times (\cup_{t > 0} t h (D) + int W_{+})

is convex.

From the assumption (c),

int C \neq \emptyset

. We also have

(0_{Y}, 0_{Z}, 0_{W}) \notin B

since (i) has no solution. Therefore, according to the separation theorem of convex sets of topological vector space,

\exists

nonzero vector

(ξ, η, ς) \in Y^{*} \times Z^{*} \times W^{*}

such that

ξ (t_{1} f (x) + y^{0}) + η (g t_{2} (x) + z^{0}) + ς (t_{3} h (x) + w^{0}) \geq 0,

for

\forall t_{1}, t_{2}, t_{3} > 0

\forall x \in D,

\forall y^{0} \in int Y_{+}, \forall z^{0} \in int Z_{+}, \forall w^{0} \in B

Since

int Y_{+}, int Z_{+}

are convex cones, and B is a linear space, one gets

ξ (t_{1} f (x) + λ_{1} y^{0}) + η (t_{2} g (x) + λ_{2} z^{0}) + ς (t_{3} h (x) + λ_{3} w^{0}) \geq 0

\forall x \in D, \forall y^{0} \in int Y_{+}, \forall z^{0} \in int Z {}_{+}, \forall w^{0} \in B, \forall λ_{i} > 0, (i = 1, 2, 3), \forall t_{i} > 0, (i = 1, 2, 3)

Let

λ_{i} \to 0 (i = 2, 3), t_{i} \to 0 (i = 1, 2, 3)

one has

ξ (y^{0}) \geq 0, \forall y^{0} \in int Y_{+} .

Therefore

ξ (y) \geq 0, \forall y \in Y_{+}

. Hence

ξ \in Y_{+}^{*}

. Similarly,

η \in int Z_{+}

ς \in int W_{+}

Thus

(ξ, η, ς) \in Y_{+}^{*} \times Z_{+}^{*} \times W^{*} .

Therefore,

ξ (f (x)) + η (g (x)) + ς (h (x)) \geq 0, x \in D .

Which means that (ii) has solutions.

On the other hand, suppose that (ii) has a solution

(ξ, η, ς)

with

ξ \neq 0_{Y_{+}^{*}}

, i.e.,

ξ (f (x)) + η (g (x)) + ς (h (x)) \geq 0, x \in D .

We are going to prove that (i) has no solution.

Otherwise, if (i) has a solution

\tilde{x} \in D

, then

f (\tilde{x}) ≺ ≺ 0, g (\tilde{x}) \leq 0, h (\tilde{x}) = 0

. Hence, one would have

ξ (f (\tilde{x})) + η (g (\tilde{x})) + ς (h (\tilde{x})) < 0 .

Which is a contradiction. The proof is completed.

We consider the following optimization problem with set-valued maps:

\begin{array}{l} (VP) & Y_{+} - \min f (x) \\ s . t . g_{i} (x) \cap (- Z_{i +}) \neq 0, i = 1, 2, \cdot \cdot \cdot, m, \\ 0 \in h_{j} (x), j = 1, 2, \cdot \cdot \cdot, n, \\ x \in D, \end{array}

where f:

X \to 2^{Y}

g_{i} : X \to 2^{Z_{i}}

h_{j} : X \to 2^{W_{j}}

are set-valued maps, Z_i₊ is a closed convex cone in Z_i and D is a nonempty subset of X.

Definition 3.2 (weakly efficient solution)A point

\bar{x} \in F

is said to be a weakly efficient solution of (VP) if there exists no

x \in D

satisfying

f (\bar{x}) ≻ ≻ f (x)

, where

F : = {x \in D : g (x) ≺ 0, h (x) = 0} .

Let

P \min [A, Y_{+}] = {y \in A : (y - A) \cap int Y_{+} = \emptyset}, P \max [A, Y_{+}] = {y \in A : (A - y) \cap int Y_{+} = \emptyset} .

In the sequel,

B (W, Y)

denotes the set of all continuous linear mappings T from W to Y;

B^{+} (Z, Y)

denotes the set of all non-negative and continuous linear mappings S from Z to Y, where non-negative mapping S means that

S (z) \in Y_{+}, \forall z \in Z

Let

L (\bar{x}, \bar{S}, \bar{T}) = f (\bar{x}) + \bar{S} (g (\bar{x})) + \bar{T} (h (\bar{x}))

Definition 3.3 (vector saddle-point)

(\bar{x}, \bar{S}, \bar{T}) \in X \times B^{+} (Z, Y) \times B (W, Y)

is said to be a vector saddle-point of

L (\bar{x}, \bar{S}, \bar{T})

L (\bar{x}, \bar{S}, \bar{T}) \in P \min [L (X, \bar{S}, \bar{T}), Y_{+}] \cap P \max [L (\bar{x}, B^{+} (Z, Y), B (W, Y)), Y_{+}] .

Where

\begin{array}{l} P \max [L (\bar{x}, B^{+} (Z, Y), B (W, Y)), Y_{+}] \\ = {μ : μ = P \max [L (\bar{x}, S, T), Y_{+}], (S, T) \in B^{+} (Z, Y) \times B (W, Y)} . \end{array}

Theorem 3.2

(\bar{x}, \bar{S}, \bar{T}) \in X \times B^{+} (Z, Y) \times B (W, Y)

is a vector saddle-point of

L (\bar{x}, \bar{S}, \bar{T})

, if and only if

\exists \bar{y} \in f (\bar{x}), \bar{z} \in g (\bar{x})

, such that

(i): $\bar{y} \in P \min [L (X, \bar{S}, \bar{T}), Y_{+}]$ ,
(ii): $g (\bar{x}) \subset - Z_{+}, h (\bar{x}) = {0}$ ,
(iii): $(f (\bar{x}) - \bar{y} - \bar{S} (\bar{z})) \cap int Y_{+} = \emptyset$ .

Proof. The sufficiency. Suppose that the conditions (i)-(iii) are satisfied. Note that

- g (\bar{x}) \subseteq Z_{+}

h (\bar{x}) = {0}

imply

- S (g (\bar{x})) \subseteq Y_{+}, T (h (\bar{x})) = {0}, \forall (S, T) \in B^{+} (Z, Y) \times B (W, Y),

(3.1)

and the condition (i) states that

{\bar{y} - [f (X) + \bar{S} (g (X)) + \bar{T} (h (X))]} \cap int Y_{+} = \emptyset,

Y_{+} + int Y_{+} \subseteq Y_{+}

and

- S (\bar{z}) \in Y_{+}

together imply

{\bar{y} + \bar{S} (\bar{z}) + \bar{T} (\bar{w}) - [f (X) + \bar{S} (g (X)) + \bar{T} (h (X))]} \cap int Y_{+} = \emptyset .

Hence

\bar{y} + \bar{S} (\bar{z}) + \bar{T} (\bar{w}) \in P \min [L (X, \bar{S}, \bar{T}), Y_{+}] .

On the other hand, since

(f (\bar{x}) - [\bar{y} + S (\bar{z})]) \cap int Y_{+} = \emptyset

, from (3.1), and from

int Y_{+} + Y_{+} \subseteq int Y_{+}

we conclude that

{\cup_{(S, T) \in B^{+} (Z, Y) \times B (W, Y)} [f (\bar{x}) + S (g (\bar{x})) + T (h (\bar{x}))] - [\bar{y} + \bar{S} (\bar{z}) + \bar{T} (\bar{w})]} \cap int Y_{+} = \emptyset .

Hence

\bar{y} + \bar{S} (\bar{z}) + \bar{T} (\bar{w}) \in P \max [L (\bar{x}, B^{+} (Z, Y), B (W, Y)), Y_{+}]

Consequently,

L (\bar{x}, \bar{S}, \bar{T}) \cap P \min [L (X, \bar{S}, \bar{T}), Y_{+}] \cap P \max [L (\bar{x}, B^{+} (Z, Y), B (W, Y)), Y_{+}] \neq \emptyset .

Therefore

(\bar{x}, \bar{S}, \bar{T}) \in X \times B^{+} (Z, Y) \times B (W, Y)

is a vector saddle-point of

L (\bar{x}, \bar{S}, \bar{T})

The necessity. Assume that

(\bar{x}, \bar{S}, \bar{T}) \in X \times B^{+} (Z, Y) \times B (W, Y)

is a vector saddle-point of

L (\bar{x}, \bar{S}, \bar{T})

. From Definition 3.3 one has

L (\bar{x}, \bar{S}, \bar{T}) \cap P \min [L (X, \bar{S}, \bar{T}), Y_{+}] \cap P \max [L (\bar{x}, B^{+} (Z, Y) \times B (W, Y)), Y_{+}] \neq \emptyset .

So,

\exists \bar{y} \in f (\bar{x}), \bar{z} \in g (\bar{x}), \bar{w} \in h (\bar{x})

, i.e.,

\bar{y} + S (\bar{z}) + T (\bar{w}) \in L (\bar{x}, \bar{S}, \bar{T}) = f (\bar{x}) + S (g (\bar{x})) + T (h (\bar{x})),

such that

\begin{array}{l} \{f (\bar{x}) + S (g (\bar{x})) + T (h (\bar{x})) - [\bar{y} + \bar{S} (\bar{z}) + \bar{T} (\bar{w})] \} \cap int Y_{+} = \emptyset, \\ \forall (S, T) \in B^{+} (Z, Y) \times B (W, Y), \end{array}

(3.2)

and

(\bar{y} + \bar{S} (\bar{z}) + \bar{T} (\bar{w}) - [f (X) + \bar{S} (g (X)) + \bar{T} (h (X))]) \cap int Y_{+} = \emptyset .

(3.3)

Taking

T = \bar{T}

in (3.2) we get

S (z) - \bar{S} (\bar{z}) \notin int Y_{+}, \forall z \in g (\bar{x}), \forall S \in B^{+} (Z, Y) .

(3.4)

Aim to show that

- \bar{z} \in Z_{+}

Otherwise, since

0 \in - Z_{+}

, if

- \bar{z} \notin Z_{+}

, we would have

- \bar{z} \neq 0

Because

Z_{+}

is a closed convex set, by the separate theorem

\exists η \in Z^{*} \ {0}

η (t z_{+}) > η (- \bar{z}), \forall z \in Z_{+}, \forall t > 0 .

(3.5)

i.e.,

η (z_{+}) > \frac{1}{t} η (- \bar{z}), \forall z \in Z_{+}, \forall t > 0 .

Let

t \to \infty

we obtain

η (z_{+}) \geq 0, \forall z \in Z_{+}

. Which means that

η \in Z_{+}^{*} \ {0}

. Meanwhile,

0 \in Z_{+}

and (3.5) yield that

η (\bar{z}) > 0

. Given

\tilde{z} \in int Z_{+}

and let

S (z) = \frac{η (z)}{η (\bar{z})} \tilde{z} + \bar{S} (z) .

Then

\bar{S} \in B^{+} (Z, Y)

and

S (\bar{z}) - \bar{S} (\bar{z}) = \tilde{z} \in int Y_{+} .

Contradicting to (3.4). Therefore

- \bar{z} \in Z_{+} .

Now, aim to prove that

- g (\bar{x}) \subseteq Z_{+}

Otherwise, if

- g (\bar{x}) ⊄ Z_{+}

, then

\exists z_{0} \in g (\bar{x})

such that

0 \neq - z_{0} \notin Z_{+}

. Similar to the above

\exists η_{0} \in Z^{*} \ {0}

such that

η_{0} \in Z_{+}^{*} \ {0}

η_{0} (z_{0}) > 0

. Given

\tilde{z} \in int Z_{+}

and let

S_{0} (z) = \frac{η_{0} (z)}{η_{0} (z_{0})} \tilde{z} .

Then

S_{0} \in B^{+} (Z, Y)

and

S_{0} (z_{0}) = \tilde{z} \in int Y_{+}

. And we have proved that

- \bar{z} \in Z_{+}

, so

- \bar{S} (\bar{z}) \in Y_{+}

. Therefore

S_{0} (z_{0}) - \bar{S} (\bar{z}) \in int Y_{+} + Y_{+} \subseteq int Y_{+} .

Again, contradicting to (3.4).

Therefore

- g (\bar{x}) \subseteq Z_{+}

. Similarly, one has

- h (\bar{x}) \subseteq W_{+}

. From (3.2) we get

[T (h (\bar{x})) - \bar{T} (\bar{w})] \cap int Y_{+} = \emptyset .

Hence

T (\bar{w}) - \bar{T} (\bar{w}) \notin int Y_{+}, \forall T \in B (W, Y) .

(3.6)

Similarly, from (3.2) again we have

T (w) - \bar{T} (\bar{w}) \notin int Y_{+}, \forall w \in h (\bar{x}), \forall T \in B (W, Y) .

(3.7)

\bar{w} \neq 0

, since

- h (\bar{x}) \subseteq W_{+}

and

W_{+}

is a pointed cone, we have

\bar{w} \notin W_{+} .

Because

Y_{+}

is a closed convex set, by the separation theorem

\exists ς \in W^{*}

, such that

ς (w) < ς (\bar{w}), \forall w \in W_{+} .

(3.8)

ς (\bar{w}) \neq 0

since

0 \in W_{+}

. Taking

y^{0} \in int Y_{+}

and define

T^{0} \in B^{+} (W, Y)

T^{0} (w) = \frac{ς (w)}{ς (\bar{w})} y^{0} + \bar{T} (w) .

Then

T^{0} (\bar{w}) - \bar{T} (\bar{w}) = y^{0} \in int Y_{+},

Contradicting to (3.6). Therefore

\bar{w} = 0

.Thus

0 \in h (\bar{x}) .

Now, we’d like to prove

h (\bar{x}) = {0} .

Otherwise, if

w^{0} \in h (\bar{x}) : w^{0} \neq 0

, similar to (3.8)

\exists ς^{0} \in W^{*}

, such that

ς^{0} (w) < ς^{0} (w^{0}), \forall w \in W_{+}

. So

ς^{0} (w^{0}) \neq 0

. Given

y_{0} \in int Y_{+}

and define

T_{0} \in B (W, Y)

, by

T_{0} (w) = \frac{ς^{0} (w)}{ς^{0} (w^{0})} y_{0} .

Then

T_{0} (w^{0}) = y_{0} \in int Y_{+}

, i.e.,

\bar{w} = O

)

T^{0} (w^{0}) - \bar{T} (\bar{w}) \in int Y_{+}

. Contradicting to (3.7). Therefore we must have

h (\bar{x}) = {0} .

(3.9)

Combining (3.2), (3.3), (3.9), and we conclude that

\bar{y} \in P \min [L (X, \bar{S}, \bar{T}), Y_{+}],

(3.10)

and

(f (\bar{x}) - \bar{y} - \bar{S} (\bar{z})) \cap int Y_{+} = \emptyset .

We have proved that, if

(\bar{x}, \bar{S}, \bar{T}) \in X \times B^{+} (Z, Y) \times B (W, Y)

is a vector saddle-point of

L (\bar{x}, \bar{S}, \bar{T})

, then the conditions (i)-(iii) hold.

Theorem 3.3 If

(\bar{x}, \bar{S}, \bar{T}) \in X \times B^{+} (Z, Y) \times B (W, Y)

is a vector saddle-point of

L (\bar{x}, \bar{S}, \bar{T})

, and if

0 \in \bar{S} (g (\bar{x}))

, then

\bar{x}

is a weak efficient solution of (VP).

Proof. Assume that

(\bar{x}, \bar{S}, \bar{T}) \in D \times B^{+} (Z, Y) \times B (W, Y)

is a vector saddle-point of

L (\bar{x}, \bar{S}, \bar{T})

, from Theorem 3.2 we have

- S (g (\bar{x})) \subseteq Y_{+}, h (\bar{x}) = {0} .

(3.11)

\bar{x} \in D

(the feasible solution of (VP). And

\exists \bar{y} \in f (\bar{x})

such that

\bar{y} \in

P \min [L (X, \bar{S}, \bar{T}), Y_{+}]

, i.e.

(\bar{y} - [f (X) + \bar{S} (g (X)) + \bar{T} (h (X)]) \cap int Y_{+} = \emptyset .

Thus

(\bar{y} - [f (D) + \bar{S} (g (\bar{x})) + \bar{T} (h (\bar{x})]) \cap int Y_{+} = \emptyset .

Since

0 \in \bar{S} (g (\bar{x}))

, by (3.11) , one has

(\bar{y} - f (D)) \cap int Y_{+} = \emptyset .

Therefore,

\bar{x}

is a weakly efficient solution of (VP).

4. Vector Lagrangian Theorems

Definition 4.1 (vector Lagrangian map) The vector Lagrangian map

L : X \times B^{+} (Z, Y) \times B (W, Y) \to 2^{Y}

of (VP) is defined by the set-valued map

L (x, S, T) = f (x) + S (g (x)) + T (h (x)) .

Given

(S, T) \in B^{+} (Z, Y) \times B (W, Y)

, we consider the minimization problem induced by (VP):

(VPST) \begin{array}{l} Y_{+} - \min L (x, S, T), \\ s . t ., x \in D . \end{array}

Definition 4.2 (slater constrained qualification (SC)) Let

\bar{x} \in F

. We say that (VP) satisfies the Slater Constrained Qualification at

\bar{x}

if the following conditions hold:

(1): $\exists x \in D$ , s.t. $h_{j} (x) = 0, g_{i} (x) ≺ ≺ 0$ ;
(2): $0 \in int h_{j} (D)$ for all j.

According the following Theorem 4.1, (VPST) can also be considered as a dual problem of (VP).

Theorem 4.1 Let

\bar{x} \in D

. Assume that

f (x) - f (\bar{x}), g (x), h (x)

satisfy the generalized convexity condition (a), the generalized affineness condition (b), as well as the inner point condition (c), and (VP) satisfies the Slater Constrained Qualification (SC). Then,

\bar{x} \in D

is a weakly efficient solution of (VP) if and only if

\exists (S, T) \in B^{+} (Z, Y) \times B (W, Y)

such that

\bar{x} \in D

is a weakly efficient solution of (VPST).

Proof. Assume

\exists (S, T) \in B^{+} (Z, Y) \times B (W, Y)

such that

\bar{x} \in D

is a weakly efficient solution of (VPST). Then there exist

\bar{y} \in f (\bar{x}), \bar{z} \in g (\bar{x}), \bar{w} \in h (\bar{x})

, such that

(\bar{y} + S (\bar{z}) + T (\bar{w}) - [f (D) + S (g (D)) + T (h (D))]) \cap int Y_{+} = \emptyset,

(\bar{y} - f (D)) \cap int Y_{+} \neq \emptyset

, then

\exists y \in f (D)

such that

\bar{y} - y \in int Y_{+}

, i.e.,

(\bar{y} + S (\bar{z}) + T (\bar{w}) - [y + S (\bar{z}) + T (\bar{w})]) \in int Y_{+} .

Which means that

(\bar{y} + S (\bar{z}) + T (\bar{w}) - [f (D) + S (g (D)) + T (h (D))]) \cap int Y_{+} \neq \emptyset .

Which is a contradiction.

Therefore

(\bar{y} - f (D)) \cap int Y_{+} = \emptyset .

Hence,

\bar{x} \in D

is a weakly efficient solution of (VP).

Conversely, suppose that

\bar{x} \in D

is a weakly efficient solution of (VP). So

\exists \bar{y} \in f (\bar{x})

such that there is not any

x \in D

for which

f (x) - \bar{y} \in - int Y_{+} .

That is to say, there is not any

x \in X

such that

f (x) - \bar{y} \in - int Y_{+}, g (x) \in - Z_{+}, 0_{W} \in h (x) .

By Theorem 3.1,

\exists (ξ, η, ς) \in Y_{+}^{*} \times Z_{+}^{*} \times W^{*} \ {(0_{Y^{*}}, 0_{Z^{*}}, 0_{W^{*}})}

such that

ξ (f (x) - \bar{y}) + η (g (x)) + ς (h (x)) \geq 0, \forall x \in D .

(4.1)

Since

\bar{y} \in f (\bar{x})

and

0_{W} \in h (\bar{x})

, take

x = \bar{x}

in (1) we obtain

η (g (\bar{x})) \geq 0 .

But

\bar{x} \in D

and

η \in Z_{+}^{*}

imply that

\exists \bar{z} \in g (\bar{x}) \cap (- Z_{+})

for which

η (\bar{z}) \leq 0 .

Hence

η (\bar{z}) = 0

, which means

0 \in η (g (\bar{x})) .

(4.2)

Since

x \in D

implies

0_{W} \in h (x)

, and

g (x) \cap (- Z_{+}) \neq \emptyset

implies

\exists z \in g (x) \cap (- Z_{+})

such that

η (z) \leq 0

, we have

ξ (f (x) - \bar{y}) \geq 0, \forall x \in D .

Because the Slater Constraint Qualification is satisfied, similar to the proof of Theorem 3.2, we have

ξ \neq 0_{Y^{*}}

. So we may take

y_{0} \in int Y_{+}

such that

ξ (y_{0}) = 1.

Define the operator

S : Z \to Y

and

T : W \to Y

S (z) = η (z) y_{0}, T (w) = ς (w) y_{0} .

(4.3)

It is easy to see that

\begin{array}{l} S \in B^{+} (Z, Y), S (Z_{+}) = η (Z_{+}) y_{0} \subseteq Y_{+}, \\ T \in B (W, Y) . \end{array}

And (4.2) implies

S (g (\bar{x})) = η (g (\bar{x})) y_{0} \in 0 \cdot Y_{+} = 0_{Y} .

(4.4)

Since

\bar{x} \in D

, we have

0_{W} \in h (\bar{x})

. Hence

0_{Y} \in T (h (\bar{x})) .

(4.5)

Therefore, by (4.4) and (4.5) one gets

\bar{y} \in f (\bar{x}) \subseteq f (\bar{x}) + S (g (\bar{x})) + T (h (\bar{x})) .

From (4.1) and (4.3)

\begin{array}{l} ξ [f (x) + S (g (x)) + T (h (x))] \\ = ξ (f (x)) + η ((g (x)) ξ (y_{0}) + ς (h (x)) ξ (y_{0}) \\ = ξ (f (x)) + η (g (x)) + ς (h (x)) \\ \geq ξ (\bar{y}), \forall x \in D . \end{array}

i.e.,

ξ [f (x) - \bar{y}) + S (g (x)) + T (h (x))] \geq 0, \forall x \in D .

(4.6)

Taking

F (x) = f (x) + S (g (x)) + T (h (x))

G (x) = {0_{Z}}

and

H (x) = {0_{W}}

, applying Theorem 3.2 to the functions

F (x) - \bar{y}, G (x), H (x)

, then (4.6) deduces that

(\bar{y} - [f (D) + S (g (D)) + T (h (D))] \cap int Y_{+} = \emptyset,

and

\bar{y} \in F (\bar{x}) = f (\bar{x}) + S (g (\bar{x})) + T (h (\bar{x})),

since

0_{Y} \in S (g (\bar{x})), 0_{Y} \in T (h (\bar{x}))

Consequently,

\bar{x} \in D

is a weakly efficient solution of (VPST).

We complete the proof.

Definition 4.3 (NNAMCQ) Let

\bar{x} \in F

. We say that (VP) satisfies the No Nonzero Abnormal Multiplier Constraint Qualification (NNAMCQ) at

\bar{x}

if there is no nonzero vector

(η, ς) \in Π_{i = 1}^{m} Z_{i}^{*} \times Π_{j = 1}^{n} W {}_{j}^{*}

satisfying the system

\begin{array}{l} \min_{x \in D \cap U (\bar{x})} [\sum_{i = 1}^{m} η_{i} g_{i} (x) + \sum_{j = 1}^{n} ς_{j} h_{j} (x)] = 0 \\ \sum_{i = 1}^{m} η_{i} g_{i} (\bar{x}) = 0, \end{array}

where

U (\bar{x})

is some neighborhood of

\bar{x}

Similar to the proof of Theorem 4.1, one has Theorem 4.2.

Theorem 4.2 Let

\bar{x} \in D .

Assume that

f (x) - f (\bar{x}), g (x), h (x)

satisfy the generalized convexity condition (a), the generalized affineness condition (b), as well as the inner point condition (c). If

\bar{x}

is a weakly efficient solution of (VP), then ∃vector Lagrangian multiplier

(S, T) \in B^{+} (Z, Y) \times B (W, Y)

such that

\bar{x} \in D

is a weakly efficient solution of (VPST). Inversely, if (NNAMCQ) holds at

\bar{x} \in D

, and if ∃vector Lagrangian multiplier

(S, T) \in B^{+} (Z, Y) \times B (W, Y)

such that

\bar{x}

is a weakly efficient solution of (VPST), then

\bar{x}

is a weakly efficient solution of (VP).

5. Conclusions.

Jeyakumar [1] introduced the following definition of sub convexlike functions for single-valued functions.

Let Y be a topological vector space and

D \subseteq X

be a nonempty set. A set-valued map f :

D \to 2^{Y}

is said to be Y₊-sub convexlike on D if

\exists

bounded set-valued map u:

D \to Y

\forall x_{1}, x_{2} \in D

\forall ε > 0

\forall α \in [0, 1]

\exists x_{3} \in D

such that

ε u + α f (x_{1}) + (1 - α) f (x_{2}) ≺_{Y_{+}} f (x_{3}),

where the partial order is induced by a convex cone

Y_{+}

of Y.

Jeyakumar [2] introduced the following subconvexlikeness.

A set-valued map f :

D \to 2^{Y}

is said to be Y₊-subconvexlike on D if

\exists θ \in int Y_{+}

such that

\forall x_{1}, x_{2} \in D

\forall ε > 0

\forall α \in [0, 1]

\exists x_{3} \in D

here holds

ε θ + α f (x_{1}) + (1 - α) f (x_{2}) ≺_{Y_{+}} f (x_{3}) .

In this paper, we prove that the above two generalized convexities are equivalent.

A set-valued map f:

D \to 2^{Y}

is said to be affine on D if

\forall x_{1}, x_{2} \in D, \forall β \in R

, there holds

β f (x_{1}) + (1 - β) f (x_{2}) = f (β x_{1} + (1 - β) x_{2})

We define the following sub affinelike maps, in order to weaken the condition of the “equality constraints” for optimization problems.

A set-valued map f :

D \to 2^{Y}

is said to be Y₊-sub affinelike on D if

\forall x_{1}, x_{2} \in D, \forall α \in (0, 1),

\forall v \in int W_{+},

\exists x_{3} \in D

there holds

v + α f (x_{1}) + (1 - α) f (x_{2}) = f (x_{3}) .

And then, we consider the following optimization problem with set-valued maps:

\begin{array}{l} (VP) & Y_{+} - \min f (x) \\ s . t . g_{i} (x) \cap (- Z_{i +}) \neq \emptyset, i = 1, 2, \cdot \cdot \cdot, m, \\ 0 \in h_{j} (x), j = 1, 2, \cdot \cdot \cdot, n, \\ x \in D, \end{array}

where f:

X \to 2^{Y}

and

g_{i} : X \to 2^{Z_{i}}

are sub convexlike, and

h_{j} : X \to 2^{W_{j}}

are sub affinelike.

For a single-valued situation, above optimization problem (VP) may be written as follows.

\begin{array}{l} Y_{+} - \min f (x) \\ s . t . g_{i} (x) ≺ 0, i = 1, 2, \cdot \cdot \cdot, m, \\ h_{j} (x) = 0, j = 1, 2, \cdot \cdot \cdot, n, \\ x \in D . \end{array}

We obtain some vector saddle-point theorems and some vector Lagrangian theorems for the set-valued optimization problem (VP). Our Theorem 3.1 is a generalization of theorems of alternatives in [1, 2], a modification of theorems of alternatives in [5, 10, 11, 17, 18]. Our saddle-points theorems (Theorems 3.2 and 3.3) are generalizations of the saddle-point theorem in [16, 20], and modifications of saddle-point theorems in [4]. Our Lagrangian theorems (Theorems 4.1 and 4.2) are generalizations of Lagrangian theorems in [16] and modifications of those in [19].

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Submitted:

14 June 2023

Posted:

15 June 2023

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Abstract

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Subject: Computer Science and Mathematics - Applied Mathematics

0. Introduction

1. Preliminary

Let X be a real topological vector space, a subset X₊ of X is said to be a convex cone if

α x^{1} + β x^{2} \in X_{+}, \forall x^{1}, x^{2} \in X_{+}, \forall α, β \geq 0 .

We denote by

0_{X}

the zero element in the topological space X and simply by 0 if there is no confusion.

A convex cone X₊ of X is called a pointed cone if

X_{+} \cap (- X_{+}) = {0}

A real topological vector space X with a pointed cone is said to be an ordered topological liner space. We denote intX₊ the topological interior of X_{+ .} The partial order on X is defined by

x^{1} ≺_{X_{+}} x^{2}, if x^{1} - x^{2} \in X_{+}, x^{1} ≺ ≺_{X_{+}} x^{2}, if x^{1} - x^{2} \in i n t X_{+},

Or, if there is no confusion, they may just be denoted by

x^{1} ≺ x^{2}, if x^{1} - x^{2} \in X_{+}, x^{1} ≺ ≺ x^{2}, if x^{1} - x^{2} \in i n t X_{+},

A, B \subseteq X,

we denoted by

A ≺_{X_{+}} B, if x ≺_{X_{+}} y for \forall x \in A, \forall y \in B, A ≺ ≺_{X_{+}} B, if x ≺ ≺_{X_{+}} y for \forall x \in A, \forall y \in B .

Or,

A ≺ B, if x ≺ y for \forall x \in A, \forall y \in B, A ≺ ≺ B, if x ≺ ≺ y for \forall x \in A, \forall y \in B .

A linear functional on X is a continuous linear function from X to R (1-dimensinal Euclidean space). The set X ^* of all linear functionals on X is the dual space of X. The subset

X_{+}^{*} = {ξ \in X^{*} : 〈x, ξ〉 \geq 0, \forall x \in X_{+}}

X^{*}

is said to be the dual cone of the cone X₊ , where

〈x, ξ〉 = ξ (x)

Suppose that X and Y are two real topological vector spaces. Let f: X→2^Y be a set-valued function, where 2^Y denotes the power set of Y.

Let D be a nonempty subset of X. Setting

f (D) = \cup_{x \in D} f (x)

, and

〈f (x), η〉 = {〈y, η〉 : y \in f (x)}, 〈f (D), η〉 = \cup_{x \in D} 〈f (x), η〉 .

For

x \in D, η \in Y^{*}

, we write

〈f (x), η〉 \geq 0, if 〈y, η〉 \geq 0, \forall y \in f (x), 〈f (D), η〉 \geq 0, if 〈f (x), η〉 \geq 0, \forall x \in D .

The following Definitions 1.1 and 1.2 can be found in [15].

Definition 1.1 (convex, bounded, and absorbing) A subset M of X is said to be convex, if

x_{1}, x_{2} \in M

and

0 < α < 1

implies

α x_{1} + (1 - α) x_{2} \in M

; M is said to be balanced if

x \in M

and

| α | \leq 1

implies

α x \in M

; M is said to be absorbing if for any given neighbourhood U of 0, there exists a positive scalar

β

such that

β^{- 1} M \subseteq U

, where

β^{- 1} M = {x \in X; x = β^{- 1} v; v \in M}

Definition 1.2 (locally convex topological space) A topological vector space X is called a locally convex topological space if any neighborhood of

0_{X}

contains a convex, balanced, and absorbing open set.

From [15, pp.26 Theorem, pp.33 Definition 1], a normed linear space is a locally convex topological space.

2. The Sub Convexlikeness

This section shows that the definitions of sub convexlikeness and subconvexlikeness given by Jeyakumar [1, 2] are actually one.

A set-valued function f: X→ 2^Y is said to be Y₊ -convex on D if ∀x¹, x²∈D, ∀α∈[0, 1], one has

α f (x^{1}) + (1 - α) f (x^{2}) ≺_{Y_{+}} f (α x^{1} + (1 - α) x^{2}) .

The following definition of convexlikeness was introduced by Ky Fan [8].

A set-valued function f: X→2^Y is said to be Y₊ -convexlike on D if ∀x¹, x²∈D, ∀α∈[0, 1], ∃x³∈D such that

α f (x^{1}) + (1 - α) f (x^{2}) ≺_{Y_{+}} f (x^{3})

Jeyakumar [2] introduced the following subconvexlikeness.

Definition 2.1 (subconvexlike) Let Y be a topological vector space and

D \subseteq X

be a nonempty set and Y₊ be a convex cone in Y. A set-valued map f :

D \to 2^{Y}

is said to be Y₊-subconvexlike on D if

\exists θ \in int Y_{+}

such that

\forall x_{1}, x_{2} \in D

\forall ε > 0

\forall α \in [0, 1]

\exists x_{3} \in D

here holds

ε θ + α f (x_{1}) + (1 - α) f (x_{2}) ≺_{Y_{+}} f (x_{3}) .

Lemma 2.1 is [16, Lemma 2.3].

Lemma 2.1 Let Y be a topological vector space and

D \subseteq X

be a nonempty set and Y₊ be a convex cone in Y. A set-valued map f :

D \to 2^{Y}

is Y₊-subconvexlike on D if and only if

\forall θ \in int Y_{+}

\forall x_{1}, x_{2} \in D

\forall α \in [0, 1]

\exists x_{3} \in D

such that

θ + α f (x_{1}) + (1 - α) f (x_{2}) ≺_{Y_{+}} f (x_{3}) .

A bounded function in a topological space can be defined as following Definition 2.2 (e.g,, see Yosida [15]).

Definition 2.2 (bounded set-valued map) A subset M of a real topological vector space Y is said to be a bounded subset if for any given neighbourhood U of 0, there exists a positive scalar

β

such that

β^{- 1} M \subseteq U

, where

β^{- 1} M = {y \in Y; y = β^{- 1} v; v \in M}

. A set-valued map f :

D \to Y

is said to bounded map if f (Y) is a bounded subset of Y.

Jeyakumar [1] introduced the following sub convexlikeness.

Definition 2.3 (sub convexlike) Let Y be a topological vector space and

D \subseteq X

be a nonempty set. A set-valued map f :

D \to 2^{Y}

is said to be Y₊-sub convexlike on D if

\exists

bounded set-valued map u:

D \to Y

\forall x_{1}, x_{2} \in D

\forall ε > 0

\forall α \in [0, 1]

\exists x_{3} \in D

such that

ε u + α f (x_{1}) + (1 - α) f (x_{2}) ≺_{Y_{+}} f (x_{3}) .

The following Lemma 2.1 is from Li and Wang [16, Lemma 2.3].

Lemma 2.2 Let Y be a locally convex topological space and

D \subseteq X

be a nonempty set Y. A set-valued map f :

D \to 2^{Y}

is Y₊-subconvexlike on D if and only if

f (D) + int Y_{+}

is Y₊-convex.

Theorem 2.1 Let Y be a locally convex topological space,

D \subseteq X

a nonempty set, and Y₊ a convex cone in Y. A set-valued map f :

D \to 2^{Y}

is Y₊-sub convexlike on D if and only if

f (D) + int Y_{+}

is Y₊-convex.

Proof. The necessity.

Suppose that f is Y₊-sub convexlike..

\forall z_{1} = y_{1} + y_{0}^{1}, z_{2} = y_{2} + y_{0}^{2} \in f (D) + int Y_{+},

\exists x_{1}, x_{2} \in D

such that

y_{1} \in f (x_{1}),

y_{2} \in f (x_{2}) .

Let

y_{0}^{} = α y_{0}^{1} + (1 - α) y_{0}^{2},

then

y_{0}^{} \in int Y_{+}^{}

. Therefore,

\exists

neighbourhood U of 0 such that

y_{+}^{0} + U

is a neighbourhood of

y_{+}^{0}

and

y_{+}^{0} + U \subseteq int Y_{+} .

By Definition 1.2, we may assume that U is convex, balanced, and absorbing.

From the assumption of sub convexlikeness, i.e.,

\exists

bounded set-valued map u:

x_{1}, x_{2} \in D

ε > 0

α \in [0, 1]

\exists

x_{3} \in D

such that

ε u + α f (x_{1}) + (1 - α) f (x_{2}) \subseteq f (x_{3}) + Y_{+} .

Therefore

\begin{array}{l} α z_{1} + (1 - α) z_{2} \\ = α y_{1} + (1 - α) y_{2} + α y_{+}^{1} + (1 - α) y_{+}^{2} \\ \subseteq f (x_{3}) - ε u + Y_{+} + y_{+}^{0} \end{array}

Since U is convex, balanced, and absorbing, we may take

ε > 0

small enough such that

- ε u \subseteq U .

Therefore

- ε u + y_{+}^{0} \subseteq y_{+}^{0} + U \subseteq int Y_{+} .

And then

\begin{array}{l} α z_{1} + (1 - α) z_{2} \\ = α y_{1} + (1 - α) y_{2} + α y_{+}^{1} + (1 - α) y_{+}^{2} \\ \subseteq f (x_{3}) + int Y_{+} \\ \subseteq f (D) + int Y {}_{+}. \end{array}

Hence,

f (D) + int Y_{+}

is a Y₊-convex set.

The sufficiency.

f (D) + int Y_{+}

is Y₊-convex, then, by Lemma 2.1, f is Y₊-subconvexlike. And, it is clear that Y₊-subconvexlikeness implies Y₊-sub convexlikeness.

From Lemma 2.2 and Theorem 2.1 one has Theorem 2.2.

Theorem 2.2 Let Y be a locally convex topological space and

D \subseteq X

a nonempty set, and Y₊ a convex cone in Y. A set-valued map f :

D \to 2^{Y}

is Y₊-subconvexlike on D if and only if f is Y₊-sub convexlike on D.

3. Vector Saddle-Point Theorems

This section works on vector saddle-point theorems for set-valued optimization problems.

A set-valued map f:

D \to 2^{Y}

is said to be affine on D if

\forall x_{1}, x_{2} \in D, \forall β \in R

, there holds

β f (x_{1}) + (1 - β) f (x_{2}) = f (β x_{1} + (1 - β) x_{2})

We introduce the notion of sub affinelike functions as follows.

Definition 3.1 (sub affinelike) A set-valued map f :

D \to 2^{Y}

is said to be Y₊-sub affinelike on D if

\forall x_{1}, x_{2} \in D, \forall α \in (0, 1),

\exists

v \in int Y_{+}

\exists x_{3} \in D

there holds

v + α f (x_{1}) + (1 - α) f (x_{2}) = f (x_{3}) .

Theorem 3.1 Let X, Y, Z and W be real topological vector spaces,

D \subseteq X .

Y_{+}, Z_{+}

W_{+}

pointed convex cones of Y, Z and W, respectively. Assume that functions

f : D \to Y, g : D \to Z, h : D \to W

satisfy that

(a): f and g are sub convexlike maps on D, i.e., $\forall u_{1} \in int Y_{+}, \forall u_{2} \in int Z_{+},$ $\forall α \in (0, 1), \forall x^{1}, x^{2} \in D$ , $\exists x', x'' \in D$ such that

$\begin{array}{l} u_{1} + α f (x^{1}) + (1 - α) f (x^{2}) ≺ f (x'), \\ u_{2} + α g (x^{1}) + (1 - α) g (x^{2}) ≺ g (x''); \end{array}$

(b): h is a sub affinelike map on D, i.e., $\forall α \in (0, 1), \forall x^{1}, x^{2} \in D,$ $\exists x''' \in D,$ $v \in int W_{+}$ such that

$v + α h (x^{1}) + (1 - α) h (x^{2}) = h (x''');$

(c): $int h (D) \neq \emptyset$ ;

and (i) and (ii) denote the systems

(i): $\exists x \in D, s . t ., f (x) ≺ ≺ 0, g (x) ≺ 0, h (x) = 0;$

(ii): $\exists (ξ, η, ζ) \in (Y_{+}^{*} \times Z_{+}^{*} \times W^{*}) \ {(0_{Y}, 0_{Z}, 0_{W})}$ such that

$ξ (f (x)) + η (g (x)) + ς (h (x)) \geq 0, \forall x \in D .$

If (i) has no solution then (ii) has solutions.

Moreover if (ii) has a solution

(ξ, η, ς)

with

ξ \neq 0_{Y^{*}}

then (i) has no solutions.

Proof.

\forall w_{1}, w_{2} \in \cup_{t > 0} t h (D) + int W_{+},

\forall α \in (0, 1),

\exists x_{1}, x_{2} \in D,

\exists b_{1}, b_{2} \in int W_{+},

\exists t_{1}, t_{2} > 0

such that

\begin{array}{l} α w_{1} + (1 - α) w_{2} \\ = α t_{1} h (x_{1}) + (1 - α) t_{2} h (x_{2}) + α b_{1} + (1 - α) b_{2} \\ = (α t_{1} + (1 - α) t_{2}) [\frac{α t_{1}}{α t_{1} + (1 - α) t_{2}} h (x_{1}) + \frac{(1 - α) t_{2}}{α t_{1} + (1 - α) t_{2}} h (x_{2})] + α b_{1} + (1 - α) b_{2} . \end{array}

By the assumption (b),

\exists x_{3} \in D,

\exists v \in int W_{+},

\forall ε > 0

such that

\frac{α t_{1}}{α t_{1} + (1 - α) t_{2}} h (x_{1}) + \frac{(1 - α) t_{2}}{α t_{1} + (1 - α) t_{2}} h (x_{2}) = h (x_{3}) - ε v .

Since

v \in int Z_{+},

\exists

neighbourhood U of 0 in W for which

V = α b_{1} + (1 - α) b_{2} + U

is a neighbourhood of

α b_{1} + (1 - α) b_{2}

By Definition 1.2, we may take

ε > 0

small enough such that

- ε (α t_{1} + (1 - α) t_{2}) v \subseteq U .

Then,

α b_{1} + (1 - α) b_{2} - ε (α t_{1} + (1 - α) t_{2}) v \subseteq V \subseteq int W_{+} .

Therefore,

\begin{array}{l} α w_{1} + (1 - α) w_{2} \\ \subseteq α t_{1} h (x_{1}) + (1 - α) t_{2} h (x_{2}) + α b_{1} + (1 - α) b_{2} \\ = (α t_{1} + (1 - α) t_{2}) [\frac{α t_{1}}{α t_{1} + (1 - α) t_{2}} h (x_{1}) + \frac{(1 - α) t_{2}}{α t_{1} + (1 - α) t_{2}} h (x_{2})] + α b_{1} + (1 - α) b_{2} \\ = (α t_{1} + (1 - α) t_{2}) h (x_{3}) + α b_{1} + (1 - α) b_{2} - ε (α t_{1} + (1 - α) t_{2}) v \\ \subseteq \cup_{t > 0} t h (D) + int W_{+} . \end{array}

So,

\cup_{t > 0} t h (D) + int Y_{+},

is a convex set.

Similarly,

\cup_{t > 0} t f (D) + int Y_{+},

and

\cup_{t > 0} t g (D) + int Z_{+}

are also convex. Therefore, the set

C = (\cup_{t > 0} t f (D) + int Y_{+}) \times (\cup_{t > 0} t g (D) + int Z_{+}) \times (\cup_{t > 0} t h (D) + int W_{+})

is convex.

From the assumption (c),

int C \neq \emptyset

. We also have

(0_{Y}, 0_{Z}, 0_{W}) \notin B

since (i) has no solution. Therefore, according to the separation theorem of convex sets of topological vector space,

\exists

nonzero vector

(ξ, η, ς) \in Y^{*} \times Z^{*} \times W^{*}

such that

ξ (t_{1} f (x) + y^{0}) + η (g t_{2} (x) + z^{0}) + ς (t_{3} h (x) + w^{0}) \geq 0,

for

\forall t_{1}, t_{2}, t_{3} > 0

\forall x \in D,

\forall y^{0} \in int Y_{+}, \forall z^{0} \in int Z_{+}, \forall w^{0} \in B

Since

int Y_{+}, int Z_{+}

are convex cones, and B is a linear space, one gets

ξ (t_{1} f (x) + λ_{1} y^{0}) + η (t_{2} g (x) + λ_{2} z^{0}) + ς (t_{3} h (x) + λ_{3} w^{0}) \geq 0

\forall x \in D, \forall y^{0} \in int Y_{+}, \forall z^{0} \in int Z {}_{+}, \forall w^{0} \in B, \forall λ_{i} > 0, (i = 1, 2, 3), \forall t_{i} > 0, (i = 1, 2, 3)

Let

λ_{i} \to 0 (i = 2, 3), t_{i} \to 0 (i = 1, 2, 3)

one has

ξ (y^{0}) \geq 0, \forall y^{0} \in int Y_{+} .

Therefore

ξ (y) \geq 0, \forall y \in Y_{+}

. Hence

ξ \in Y_{+}^{*}

. Similarly,

η \in int Z_{+}

ς \in int W_{+}

Thus

(ξ, η, ς) \in Y_{+}^{*} \times Z_{+}^{*} \times W^{*} .

Therefore,

ξ (f (x)) + η (g (x)) + ς (h (x)) \geq 0, x \in D .

Which means that (ii) has solutions.

On the other hand, suppose that (ii) has a solution

(ξ, η, ς)

with

ξ \neq 0_{Y_{+}^{*}}

, i.e.,

ξ (f (x)) + η (g (x)) + ς (h (x)) \geq 0, x \in D .

We are going to prove that (i) has no solution.

Otherwise, if (i) has a solution

\tilde{x} \in D

, then

f (\tilde{x}) ≺ ≺ 0, g (\tilde{x}) \leq 0, h (\tilde{x}) = 0

. Hence, one would have

ξ (f (\tilde{x})) + η (g (\tilde{x})) + ς (h (\tilde{x})) < 0 .

Which is a contradiction. The proof is completed.

We consider the following optimization problem with set-valued maps:

\begin{array}{l} (VP) & Y_{+} - \min f (x) \\ s . t . g_{i} (x) \cap (- Z_{i +}) \neq 0, i = 1, 2, \cdot \cdot \cdot, m, \\ 0 \in h_{j} (x), j = 1, 2, \cdot \cdot \cdot, n, \\ x \in D, \end{array}

where f:

X \to 2^{Y}

g_{i} : X \to 2^{Z_{i}}

h_{j} : X \to 2^{W_{j}}

are set-valued maps, Z_i₊ is a closed convex cone in Z_i and D is a nonempty subset of X.

Definition 3.2 (weakly efficient solution)A point

\bar{x} \in F

is said to be a weakly efficient solution of (VP) if there exists no

x \in D

satisfying

f (\bar{x}) ≻ ≻ f (x)

, where

F : = {x \in D : g (x) ≺ 0, h (x) = 0} .

Let

P \min [A, Y_{+}] = {y \in A : (y - A) \cap int Y_{+} = \emptyset}, P \max [A, Y_{+}] = {y \in A : (A - y) \cap int Y_{+} = \emptyset} .

In the sequel,

B (W, Y)

denotes the set of all continuous linear mappings T from W to Y;

B^{+} (Z, Y)

denotes the set of all non-negative and continuous linear mappings S from Z to Y, where non-negative mapping S means that

S (z) \in Y_{+}, \forall z \in Z

Let

L (\bar{x}, \bar{S}, \bar{T}) = f (\bar{x}) + \bar{S} (g (\bar{x})) + \bar{T} (h (\bar{x}))

Definition 3.3 (vector saddle-point)

(\bar{x}, \bar{S}, \bar{T}) \in X \times B^{+} (Z, Y) \times B (W, Y)

is said to be a vector saddle-point of

L (\bar{x}, \bar{S}, \bar{T})

L (\bar{x}, \bar{S}, \bar{T}) \in P \min [L (X, \bar{S}, \bar{T}), Y_{+}] \cap P \max [L (\bar{x}, B^{+} (Z, Y), B (W, Y)), Y_{+}] .

Where

\begin{array}{l} P \max [L (\bar{x}, B^{+} (Z, Y), B (W, Y)), Y_{+}] \\ = {μ : μ = P \max [L (\bar{x}, S, T), Y_{+}], (S, T) \in B^{+} (Z, Y) \times B (W, Y)} . \end{array}

Theorem 3.2

(\bar{x}, \bar{S}, \bar{T}) \in X \times B^{+} (Z, Y) \times B (W, Y)

is a vector saddle-point of

L (\bar{x}, \bar{S}, \bar{T})

, if and only if

\exists \bar{y} \in f (\bar{x}), \bar{z} \in g (\bar{x})

, such that

(i): $\bar{y} \in P \min [L (X, \bar{S}, \bar{T}), Y_{+}]$ ,
(ii): $g (\bar{x}) \subset - Z_{+}, h (\bar{x}) = {0}$ ,
(iii): $(f (\bar{x}) - \bar{y} - \bar{S} (\bar{z})) \cap int Y_{+} = \emptyset$ .

Proof. The sufficiency. Suppose that the conditions (i)-(iii) are satisfied. Note that

- g (\bar{x}) \subseteq Z_{+}

h (\bar{x}) = {0}

imply

- S (g (\bar{x})) \subseteq Y_{+}, T (h (\bar{x})) = {0}, \forall (S, T) \in B^{+} (Z, Y) \times B (W, Y),

(3.1)

and the condition (i) states that

{\bar{y} - [f (X) + \bar{S} (g (X)) + \bar{T} (h (X))]} \cap int Y_{+} = \emptyset,

Y_{+} + int Y_{+} \subseteq Y_{+}

and

- S (\bar{z}) \in Y_{+}

together imply

{\bar{y} + \bar{S} (\bar{z}) + \bar{T} (\bar{w}) - [f (X) + \bar{S} (g (X)) + \bar{T} (h (X))]} \cap int Y_{+} = \emptyset .

Hence

\bar{y} + \bar{S} (\bar{z}) + \bar{T} (\bar{w}) \in P \min [L (X, \bar{S}, \bar{T}), Y_{+}] .

On the other hand, since

(f (\bar{x}) - [\bar{y} + S (\bar{z})]) \cap int Y_{+} = \emptyset

, from (3.1), and from

int Y_{+} + Y_{+} \subseteq int Y_{+}

we conclude that

{\cup_{(S, T) \in B^{+} (Z, Y) \times B (W, Y)} [f (\bar{x}) + S (g (\bar{x})) + T (h (\bar{x}))] - [\bar{y} + \bar{S} (\bar{z}) + \bar{T} (\bar{w})]} \cap int Y_{+} = \emptyset .

Hence

\bar{y} + \bar{S} (\bar{z}) + \bar{T} (\bar{w}) \in P \max [L (\bar{x}, B^{+} (Z, Y), B (W, Y)), Y_{+}]

Consequently,

L (\bar{x}, \bar{S}, \bar{T}) \cap P \min [L (X, \bar{S}, \bar{T}), Y_{+}] \cap P \max [L (\bar{x}, B^{+} (Z, Y), B (W, Y)), Y_{+}] \neq \emptyset .

Therefore

(\bar{x}, \bar{S}, \bar{T}) \in X \times B^{+} (Z, Y) \times B (W, Y)

is a vector saddle-point of

L (\bar{x}, \bar{S}, \bar{T})

The necessity. Assume that

(\bar{x}, \bar{S}, \bar{T}) \in X \times B^{+} (Z, Y) \times B (W, Y)

is a vector saddle-point of

L (\bar{x}, \bar{S}, \bar{T})

. From Definition 3.3 one has

L (\bar{x}, \bar{S}, \bar{T}) \cap P \min [L (X, \bar{S}, \bar{T}), Y_{+}] \cap P \max [L (\bar{x}, B^{+} (Z, Y) \times B (W, Y)), Y_{+}] \neq \emptyset .

So,

\exists \bar{y} \in f (\bar{x}), \bar{z} \in g (\bar{x}), \bar{w} \in h (\bar{x})

, i.e.,

\bar{y} + S (\bar{z}) + T (\bar{w}) \in L (\bar{x}, \bar{S}, \bar{T}) = f (\bar{x}) + S (g (\bar{x})) + T (h (\bar{x})),

such that

\begin{array}{l} \{f (\bar{x}) + S (g (\bar{x})) + T (h (\bar{x})) - [\bar{y} + \bar{S} (\bar{z}) + \bar{T} (\bar{w})] \} \cap int Y_{+} = \emptyset, \\ \forall (S, T) \in B^{+} (Z, Y) \times B (W, Y), \end{array}

(3.2)

and

(\bar{y} + \bar{S} (\bar{z}) + \bar{T} (\bar{w}) - [f (X) + \bar{S} (g (X)) + \bar{T} (h (X))]) \cap int Y_{+} = \emptyset .

(3.3)

Taking

T = \bar{T}

in (3.2) we get

S (z) - \bar{S} (\bar{z}) \notin int Y_{+}, \forall z \in g (\bar{x}), \forall S \in B^{+} (Z, Y) .

(3.4)

Aim to show that

- \bar{z} \in Z_{+}

Otherwise, since

0 \in - Z_{+}

, if

- \bar{z} \notin Z_{+}

, we would have

- \bar{z} \neq 0

Because

Z_{+}

is a closed convex set, by the separate theorem

\exists η \in Z^{*} \ {0}

η (t z_{+}) > η (- \bar{z}), \forall z \in Z_{+}, \forall t > 0 .

(3.5)

i.e.,

η (z_{+}) > \frac{1}{t} η (- \bar{z}), \forall z \in Z_{+}, \forall t > 0 .

Let

t \to \infty

we obtain

η (z_{+}) \geq 0, \forall z \in Z_{+}

. Which means that

η \in Z_{+}^{*} \ {0}

. Meanwhile,

0 \in Z_{+}

and (3.5) yield that

η (\bar{z}) > 0

. Given

\tilde{z} \in int Z_{+}

and let

S (z) = \frac{η (z)}{η (\bar{z})} \tilde{z} + \bar{S} (z) .

Then

\bar{S} \in B^{+} (Z, Y)

and

S (\bar{z}) - \bar{S} (\bar{z}) = \tilde{z} \in int Y_{+} .

Contradicting to (3.4). Therefore

- \bar{z} \in Z_{+} .

Now, aim to prove that

- g (\bar{x}) \subseteq Z_{+}

Otherwise, if

- g (\bar{x}) ⊄ Z_{+}

, then

\exists z_{0} \in g (\bar{x})

such that

0 \neq - z_{0} \notin Z_{+}

. Similar to the above

\exists η_{0} \in Z^{*} \ {0}

such that

η_{0} \in Z_{+}^{*} \ {0}

η_{0} (z_{0}) > 0

. Given

\tilde{z} \in int Z_{+}

and let

S_{0} (z) = \frac{η_{0} (z)}{η_{0} (z_{0})} \tilde{z} .

Then

S_{0} \in B^{+} (Z, Y)

and

S_{0} (z_{0}) = \tilde{z} \in int Y_{+}

. And we have proved that

- \bar{z} \in Z_{+}

, so

- \bar{S} (\bar{z}) \in Y_{+}

. Therefore

S_{0} (z_{0}) - \bar{S} (\bar{z}) \in int Y_{+} + Y_{+} \subseteq int Y_{+} .

Again, contradicting to (3.4).

Therefore

- g (\bar{x}) \subseteq Z_{+}

. Similarly, one has

- h (\bar{x}) \subseteq W_{+}

. From (3.2) we get

[T (h (\bar{x})) - \bar{T} (\bar{w})] \cap int Y_{+} = \emptyset .

Hence

T (\bar{w}) - \bar{T} (\bar{w}) \notin int Y_{+}, \forall T \in B (W, Y) .

(3.6)

Similarly, from (3.2) again we have

T (w) - \bar{T} (\bar{w}) \notin int Y_{+}, \forall w \in h (\bar{x}), \forall T \in B (W, Y) .

(3.7)

\bar{w} \neq 0

, since

- h (\bar{x}) \subseteq W_{+}

and

W_{+}

is a pointed cone, we have

\bar{w} \notin W_{+} .

Because

Y_{+}

is a closed convex set, by the separation theorem

\exists ς \in W^{*}

, such that

ς (w) < ς (\bar{w}), \forall w \in W_{+} .

(3.8)

ς (\bar{w}) \neq 0

since

0 \in W_{+}

. Taking

y^{0} \in int Y_{+}

and define

T^{0} \in B^{+} (W, Y)

T^{0} (w) = \frac{ς (w)}{ς (\bar{w})} y^{0} + \bar{T} (w) .

Then

T^{0} (\bar{w}) - \bar{T} (\bar{w}) = y^{0} \in int Y_{+},

Contradicting to (3.6). Therefore

\bar{w} = 0

.Thus

0 \in h (\bar{x}) .

Now, we’d like to prove

h (\bar{x}) = {0} .

Otherwise, if

w^{0} \in h (\bar{x}) : w^{0} \neq 0

, similar to (3.8)

\exists ς^{0} \in W^{*}

, such that

ς^{0} (w) < ς^{0} (w^{0}), \forall w \in W_{+}

. So

ς^{0} (w^{0}) \neq 0

. Given

y_{0} \in int Y_{+}

and define

T_{0} \in B (W, Y)

, by

T_{0} (w) = \frac{ς^{0} (w)}{ς^{0} (w^{0})} y_{0} .

Then

T_{0} (w^{0}) = y_{0} \in int Y_{+}

, i.e.,

\bar{w} = O

)

T^{0} (w^{0}) - \bar{T} (\bar{w}) \in int Y_{+}

. Contradicting to (3.7). Therefore we must have

h (\bar{x}) = {0} .

(3.9)

Combining (3.2), (3.3), (3.9), and we conclude that

\bar{y} \in P \min [L (X, \bar{S}, \bar{T}), Y_{+}],

(3.10)

and

(f (\bar{x}) - \bar{y} - \bar{S} (\bar{z})) \cap int Y_{+} = \emptyset .

We have proved that, if

(\bar{x}, \bar{S}, \bar{T}) \in X \times B^{+} (Z, Y) \times B (W, Y)

is a vector saddle-point of

L (\bar{x}, \bar{S}, \bar{T})

, then the conditions (i)-(iii) hold.

Theorem 3.3 If

(\bar{x}, \bar{S}, \bar{T}) \in X \times B^{+} (Z, Y) \times B (W, Y)

is a vector saddle-point of

L (\bar{x}, \bar{S}, \bar{T})

, and if

0 \in \bar{S} (g (\bar{x}))

, then

\bar{x}

is a weak efficient solution of (VP).

Proof. Assume that

(\bar{x}, \bar{S}, \bar{T}) \in D \times B^{+} (Z, Y) \times B (W, Y)

is a vector saddle-point of

L (\bar{x}, \bar{S}, \bar{T})

, from Theorem 3.2 we have

- S (g (\bar{x})) \subseteq Y_{+}, h (\bar{x}) = {0} .

(3.11)

\bar{x} \in D

(the feasible solution of (VP). And

\exists \bar{y} \in f (\bar{x})

such that

\bar{y} \in

P \min [L (X, \bar{S}, \bar{T}), Y_{+}]

, i.e.

(\bar{y} - [f (X) + \bar{S} (g (X)) + \bar{T} (h (X)]) \cap int Y_{+} = \emptyset .

Thus

(\bar{y} - [f (D) + \bar{S} (g (\bar{x})) + \bar{T} (h (\bar{x})]) \cap int Y_{+} = \emptyset .

Since

0 \in \bar{S} (g (\bar{x}))

, by (3.11) , one has

(\bar{y} - f (D)) \cap int Y_{+} = \emptyset .

Therefore,

\bar{x}

is a weakly efficient solution of (VP).

4. Vector Lagrangian Theorems

Definition 4.1 (vector Lagrangian map) The vector Lagrangian map

L : X \times B^{+} (Z, Y) \times B (W, Y) \to 2^{Y}

of (VP) is defined by the set-valued map

L (x, S, T) = f (x) + S (g (x)) + T (h (x)) .

Given

(S, T) \in B^{+} (Z, Y) \times B (W, Y)

, we consider the minimization problem induced by (VP):

(VPST) \begin{array}{l} Y_{+} - \min L (x, S, T), \\ s . t ., x \in D . \end{array}

Definition 4.2 (slater constrained qualification (SC)) Let

\bar{x} \in F

. We say that (VP) satisfies the Slater Constrained Qualification at

\bar{x}

if the following conditions hold:

(1): $\exists x \in D$ , s.t. $h_{j} (x) = 0, g_{i} (x) ≺ ≺ 0$ ;
(2): $0 \in int h_{j} (D)$ for all j.

According the following Theorem 4.1, (VPST) can also be considered as a dual problem of (VP).

Theorem 4.1 Let

\bar{x} \in D

. Assume that

f (x) - f (\bar{x}), g (x), h (x)

\bar{x} \in D

is a weakly efficient solution of (VP) if and only if

\exists (S, T) \in B^{+} (Z, Y) \times B (W, Y)

such that

\bar{x} \in D

is a weakly efficient solution of (VPST).

Proof. Assume

\exists (S, T) \in B^{+} (Z, Y) \times B (W, Y)

such that

\bar{x} \in D

is a weakly efficient solution of (VPST). Then there exist

\bar{y} \in f (\bar{x}), \bar{z} \in g (\bar{x}), \bar{w} \in h (\bar{x})

, such that

(\bar{y} + S (\bar{z}) + T (\bar{w}) - [f (D) + S (g (D)) + T (h (D))]) \cap int Y_{+} = \emptyset,

(\bar{y} - f (D)) \cap int Y_{+} \neq \emptyset

, then

\exists y \in f (D)

such that

\bar{y} - y \in int Y_{+}

, i.e.,

(\bar{y} + S (\bar{z}) + T (\bar{w}) - [y + S (\bar{z}) + T (\bar{w})]) \in int Y_{+} .

Which means that

(\bar{y} + S (\bar{z}) + T (\bar{w}) - [f (D) + S (g (D)) + T (h (D))]) \cap int Y_{+} \neq \emptyset .

Which is a contradiction.

Therefore

(\bar{y} - f (D)) \cap int Y_{+} = \emptyset .

Hence,

\bar{x} \in D

is a weakly efficient solution of (VP).

Conversely, suppose that

\bar{x} \in D

is a weakly efficient solution of (VP). So

\exists \bar{y} \in f (\bar{x})

such that there is not any

x \in D

for which

f (x) - \bar{y} \in - int Y_{+} .

That is to say, there is not any

x \in X

such that

f (x) - \bar{y} \in - int Y_{+}, g (x) \in - Z_{+}, 0_{W} \in h (x) .

By Theorem 3.1,

\exists (ξ, η, ς) \in Y_{+}^{*} \times Z_{+}^{*} \times W^{*} \ {(0_{Y^{*}}, 0_{Z^{*}}, 0_{W^{*}})}

such that

ξ (f (x) - \bar{y}) + η (g (x)) + ς (h (x)) \geq 0, \forall x \in D .

(4.1)

Since

\bar{y} \in f (\bar{x})

and

0_{W} \in h (\bar{x})

, take

x = \bar{x}

in (1) we obtain

η (g (\bar{x})) \geq 0 .

But

\bar{x} \in D

and

η \in Z_{+}^{*}

imply that

\exists \bar{z} \in g (\bar{x}) \cap (- Z_{+})

for which

η (\bar{z}) \leq 0 .

Hence

η (\bar{z}) = 0

, which means

0 \in η (g (\bar{x})) .

(4.2)

Since

x \in D

implies

0_{W} \in h (x)

, and

g (x) \cap (- Z_{+}) \neq \emptyset

implies

\exists z \in g (x) \cap (- Z_{+})

such that

η (z) \leq 0

, we have

ξ (f (x) - \bar{y}) \geq 0, \forall x \in D .

Because the Slater Constraint Qualification is satisfied, similar to the proof of Theorem 3.2, we have

ξ \neq 0_{Y^{*}}

. So we may take

y_{0} \in int Y_{+}

such that

ξ (y_{0}) = 1.

Define the operator

S : Z \to Y

and

T : W \to Y

S (z) = η (z) y_{0}, T (w) = ς (w) y_{0} .

(4.3)

It is easy to see that

\begin{array}{l} S \in B^{+} (Z, Y), S (Z_{+}) = η (Z_{+}) y_{0} \subseteq Y_{+}, \\ T \in B (W, Y) . \end{array}

And (4.2) implies

S (g (\bar{x})) = η (g (\bar{x})) y_{0} \in 0 \cdot Y_{+} = 0_{Y} .

(4.4)

Since

\bar{x} \in D

, we have

0_{W} \in h (\bar{x})

. Hence

0_{Y} \in T (h (\bar{x})) .

(4.5)

Therefore, by (4.4) and (4.5) one gets

\bar{y} \in f (\bar{x}) \subseteq f (\bar{x}) + S (g (\bar{x})) + T (h (\bar{x})) .

From (4.1) and (4.3)

\begin{array}{l} ξ [f (x) + S (g (x)) + T (h (x))] \\ = ξ (f (x)) + η ((g (x)) ξ (y_{0}) + ς (h (x)) ξ (y_{0}) \\ = ξ (f (x)) + η (g (x)) + ς (h (x)) \\ \geq ξ (\bar{y}), \forall x \in D . \end{array}

i.e.,

ξ [f (x) - \bar{y}) + S (g (x)) + T (h (x))] \geq 0, \forall x \in D .

(4.6)

Taking

F (x) = f (x) + S (g (x)) + T (h (x))

G (x) = {0_{Z}}

and

H (x) = {0_{W}}

, applying Theorem 3.2 to the functions

F (x) - \bar{y}, G (x), H (x)

, then (4.6) deduces that

(\bar{y} - [f (D) + S (g (D)) + T (h (D))] \cap int Y_{+} = \emptyset,

and

\bar{y} \in F (\bar{x}) = f (\bar{x}) + S (g (\bar{x})) + T (h (\bar{x})),

since

0_{Y} \in S (g (\bar{x})), 0_{Y} \in T (h (\bar{x}))

Consequently,

\bar{x} \in D

is a weakly efficient solution of (VPST).

We complete the proof.

Definition 4.3 (NNAMCQ) Let

\bar{x} \in F

. We say that (VP) satisfies the No Nonzero Abnormal Multiplier Constraint Qualification (NNAMCQ) at

\bar{x}

if there is no nonzero vector

(η, ς) \in Π_{i = 1}^{m} Z_{i}^{*} \times Π_{j = 1}^{n} W {}_{j}^{*}

satisfying the system

\begin{array}{l} \min_{x \in D \cap U (\bar{x})} [\sum_{i = 1}^{m} η_{i} g_{i} (x) + \sum_{j = 1}^{n} ς_{j} h_{j} (x)] = 0 \\ \sum_{i = 1}^{m} η_{i} g_{i} (\bar{x}) = 0, \end{array}

where

U (\bar{x})

is some neighborhood of

\bar{x}

Similar to the proof of Theorem 4.1, one has Theorem 4.2.

Theorem 4.2 Let

\bar{x} \in D .

Assume that

f (x) - f (\bar{x}), g (x), h (x)

satisfy the generalized convexity condition (a), the generalized affineness condition (b), as well as the inner point condition (c). If

\bar{x}

is a weakly efficient solution of (VP), then ∃vector Lagrangian multiplier

(S, T) \in B^{+} (Z, Y) \times B (W, Y)

such that

\bar{x} \in D

is a weakly efficient solution of (VPST). Inversely, if (NNAMCQ) holds at

\bar{x} \in D

, and if ∃vector Lagrangian multiplier

(S, T) \in B^{+} (Z, Y) \times B (W, Y)

such that

\bar{x}

is a weakly efficient solution of (VPST), then

\bar{x}

is a weakly efficient solution of (VP).

5. Conclusions.

Jeyakumar [1] introduced the following definition of sub convexlike functions for single-valued functions.

Let Y be a topological vector space and

D \subseteq X

be a nonempty set. A set-valued map f :

D \to 2^{Y}

is said to be Y₊-sub convexlike on D if

\exists

bounded set-valued map u:

D \to Y

\forall x_{1}, x_{2} \in D

\forall ε > 0

\forall α \in [0, 1]

\exists x_{3} \in D

such that

ε u + α f (x_{1}) + (1 - α) f (x_{2}) ≺_{Y_{+}} f (x_{3}),

where the partial order is induced by a convex cone

Y_{+}

of Y.

Jeyakumar [2] introduced the following subconvexlikeness.

A set-valued map f :

D \to 2^{Y}

is said to be Y₊-subconvexlike on D if

\exists θ \in int Y_{+}

such that

\forall x_{1}, x_{2} \in D

\forall ε > 0

\forall α \in [0, 1]

\exists x_{3} \in D

here holds

ε θ + α f (x_{1}) + (1 - α) f (x_{2}) ≺_{Y_{+}} f (x_{3}) .

In this paper, we prove that the above two generalized convexities are equivalent.

A set-valued map f:

D \to 2^{Y}

is said to be affine on D if

\forall x_{1}, x_{2} \in D, \forall β \in R

, there holds

β f (x_{1}) + (1 - β) f (x_{2}) = f (β x_{1} + (1 - β) x_{2})

We define the following sub affinelike maps, in order to weaken the condition of the “equality constraints” for optimization problems.

A set-valued map f :

D \to 2^{Y}

is said to be Y₊-sub affinelike on D if

\forall x_{1}, x_{2} \in D, \forall α \in (0, 1),

\forall v \in int W_{+},

\exists x_{3} \in D

there holds

v + α f (x_{1}) + (1 - α) f (x_{2}) = f (x_{3}) .

And then, we consider the following optimization problem with set-valued maps:

\begin{array}{l} (VP) & Y_{+} - \min f (x) \\ s . t . g_{i} (x) \cap (- Z_{i +}) \neq \emptyset, i = 1, 2, \cdot \cdot \cdot, m, \\ 0 \in h_{j} (x), j = 1, 2, \cdot \cdot \cdot, n, \\ x \in D, \end{array}

where f:

X \to 2^{Y}

and

g_{i} : X \to 2^{Z_{i}}

are sub convexlike, and

h_{j} : X \to 2^{W_{j}}

are sub affinelike.

For a single-valued situation, above optimization problem (VP) may be written as follows.

\begin{array}{l} Y_{+} - \min f (x) \\ s . t . g_{i} (x) ≺ 0, i = 1, 2, \cdot \cdot \cdot, m, \\ h_{j} (x) = 0, j = 1, 2, \cdot \cdot \cdot, n, \\ x \in D . \end{array}

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On the Sub Convexlike Optimization Problems

Abstract

0. Introduction

1. Preliminary

2. The Sub Convexlikeness

3. Vector Saddle-Point Theorems

4. Vector Lagrangian Theorems

5. Conclusions.

References

Abstract

0. Introduction

1. Preliminary

2. The Sub Convexlikeness

3. Vector Saddle-Point Theorems

4. Vector Lagrangian Theorems

5. Conclusions.

References

MDPI Initiatives

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