Abstract
In this paper, we show that the sub convexlikeness and subconvexlikeness defined by V. Jeyakumar are equivalent in locally convex topological spaces. We also deal with set-valued vector optimization problems and obtained vector saddle-point theorems and vector Lagrangian theorems.
Keywords:
locally convex topological space; subconvexlikeness; sub convexlikeness; vector Lagrangian multiplier theorems; vector saddle-point theorems MSC:
90C26; 90C48
1. 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 [] introduced the definition of X+-convexlike function. Jeyakumar [] introduced the definition of X+-sub convexlike function and defined X+-subconvexlike function in []. There are many research articles discussing subconvexlike optimization problems, e.g., see [,,,,]. In this paper, by using partial order relations and the absorbing property of bounded convex sets in locally convex topological spaces [], we proved that the sub convexlikeness introduced in [] and subconvexlikeness in [] are equivalent in locally convex topological spaces (including normed linear spaces).
Most papers in set-valued optimization studied the problem with inequality and abstract constraints. In this paper, we consider the set-valued optimization problem with not only inequality and abstract, but also equality constraints. The explicit statement of the equality constraint is very convenient in various applications. For example, recently, mathematical programs with equilibrium constraints have received considerable attention 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., [,,], a mathematical program with equilibrium constraints can be reformulated as an optimization problem with an equality constraint. This paper works with a set-valued optimization problem with inequality, equality as well as abstract constraints. By using the separation theorem for convex sets, we extend or modify some results (theorems of alternatives, saddle-points theorems and Lagrangian theorems) in [,,,,,,] to vector optimization problems with weakened convexities.
2. Preliminary
Let X be a real topological vector space; a subset X+ of X is said to be a convex cone if
We denoted by 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 .
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
Alternatively, if there is no confusion, they may just be denoted by
If , we denoted by
Alternatively,
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
of is said to be the dual cone of the cone X+, where .
Suppose that X and Y are two real topological vector spaces. Let f: X→2Y be a set-valued function, where 2Y denotes the power set of Y.
Let D be a nonempty subset of X. We set and
For , we wrote
The following Definitions 1 and 2 can be found in [].
Definition 1 (convex, bounded, and absorbing).
A subset M of X is said to be convex, if and implies ; M is said to be balanced if and implies ; M is said to be absorbing if, for any given neighborhood U of 0, there exists a positive scalar , such that , where .
Definition 2 (locally convex topological space).
A topological vector space X is called a locally convex topological space if any neighborhood of contains a convex, balanced and absorbing open set.
From [], p. 26 Theorem, p. 33 Definition 1, a normed linear space is a locally convex topological space.
3. The Sub Convexlikeness
This section shows that the definitions of sub convexlikeness and subconvexlikeness provided by Jeyakumar [,] are actually one.
A set-valued function f: X → 2Y is said to be Y+-convex on D if ∀x1, x2 ∈ D, ∀α ∈ [0, 1]; one has
The following definition of convexlikeness was introduced by Ky Fan [].
A set-valued function f: X → 2Y is said to be Y+-convexlike on D if ∀x1, x2 ∈ D, ∀α ∈ [0, 1], ∃x3 ∈ D such that
Jeyakumar [] introduced the following subconvexlikeness.
Definition 3
(subconvexlike). Let Y be a topological vector space and be a nonempty set and Y+ be a convex cone in Y. A set-valued map f: is said to be Y+-subconvexlike on D if , such that , , , holds
Lemma 1 is Lemma 2.3 in [].
Lemma 1.
Let Y be a topological vector space and be a nonempty set and Y+ be a convex cone in Y. A set-valued map f: is Y+-sub-convex-like on D if, and only if, , , , , such that
A bounded function in a topological space can be defined as following Definition 4 (e.g., see Yosida []).
Definition 4 (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 neighborhood U of 0, there exists a positive scalar such that , where . A set-valued map f: is said to bounded map if f (Y) is a bounded subset of Y.
Jeyakumar [] introduced the following sub convexlikeness.
Definition 5 (sub convexlike).
Let Y be a topological vector space and be a nonempty set. A set-valued map f: is said to be Y+-sub convexlike on D if bounded set-valued map u: , , , , , such that
Lemma 2.
Let Y be a locally convex topological space and be a nonempty set Y. A set-valued map f: is Y+-sub-convex-like on D if, and only if, is Y+-convex.
Theorem 1.
Let Y be a locally convex topological space, a nonempty set, and Y+ a convex cone in Y. A set-valued map f: is Y+-sub convexlike on D if, and only if, is Y+-convex.
Proof.
The necessity.
Suppose that f is Y+-sub convexlike.
, such that
Let
Then, . Therefore, neighborhood U of 0, such that is a neighborhood of and
By Definition 2, we may assume that U is convex, balanced and absorbing.
From the assumption of sub convexlikeness, i.e., bounded set-valued map u:, , , , such that
Therefore,
Since U is convex, balanced and absorbing, we may take to be small enough, such that
Therefore,
Then,
Hence, is a Y+-convex set.
The sufficiency.
If is Y+-convex, then, by Lemma 1, f is Y+-subconvexlike. It is clear that Y+-subconvexlikeness implies Y+-sub convexlikeness. □
From Lemma 2 and Theorem 1, we obtained Theorem 2.
Theorem 2.
Let Y be a locally convex topological space, a nonempty set and Y+ a convex cone in Y. A set-valued map f: is Y+-subconvexlike on D if, and only if, f is Y+-sub convexlike on D.
4. Vector Saddle-Point Theorems
This section presents vector saddle-point theorems for set-valued optimization problems.
A set-valued map f: is said to be affine on D if ; therefore,
We introduced the notion of sub affinelike functions, as follows.
Definition 6 (sub affinelike).
A set-valued map f: is said to be Y+-sub affinelike on D if , ; therefore,
Theorem 3.
Let X, Y, Z and W be real topological vector spaces, and are pointed convex cones of Y, Z and W, respectively. Assume that the functions satisfy:
- (a)
- f and g are sub convexlike maps on D, i.e., , , such that
- (b)
- h is a sub affinelike map on D, i.e., such that
- (c)
- ;
(i) and (ii) denote the system:
- (i)
- (ii)
- such that
If (i) has no solutions, then (ii) has solutions.
Moreover, if (ii) has a solution
with , then (i) has no solutions.
Proof.
, such that
By the assumption (b), , such that
Since neighborhood U of 0 in W for which is a neighborhood of .
By Definition 2, we may take to be small enough, such that
Then,
Therefore,
So, is a convex set.
Similarly, and are also convex. Therefore, the set
is convex.
From assumption (c), . We also have since (i) has no solution. Therefore, according to the separation theorem of convex sets of topological vector space, nonzero vector , such that
for , .
Since are convex cones, and B is a linear space, we obtained
Let ; therefore,
Therefore, . Hence, . Similarly, ,
Thus,
Therefore,
which means that (ii) has solutions.
On the other hand, suppose that (ii) has a solution with , i.e.,
We are going to prove that (i) has no solution.
Otherwise, if (i) has a solution , then . Hence, one would have
which is a contradiction. The proof is completed. □
We considered the following optimization problem with set-valued maps:
where f: , , are set-valued maps, Zi+ is a closed convex cone in Zi and D is a nonempty subset of X.
(VP) Y+-min f (x)
Definition 7 (weakly efficient solution).
A point is said to be a weakly efficient solution of (VP) if there exists no satisfying , where
Let
In the sequel, denotes the set of all continuous linear mappings T from W to Y; denotes the set of all non-negative and continuous linear mappings S from Z to Y, where non-negative mapping S means that , write
Definition 8 (vector saddle-point).
is said to be a vector saddle-point of if
where
Theorem 4.
is a vector saddle-point of if and only if , such that
- (i)
- ;
- (ii)
- ;
- (iii)
- .
Proof.
The sufficiency. Suppose that the conditions (i)–(iii) are satisfied. Note that , implies
and the condition (i) states that
So, and imply
Hence,
On the other hand, since , from , we conclude that
Hence,
Consequently,
Therefore, is a vector saddle-point of .
The necessity. Assume that is a vector saddle-point of . From Definition 8, one has
So, , i.e.,
such that
and
Taking we obtained
We aim to show that .
Otherwise, since , if , we would have ,
Because is a closed convex set, by the separate theorem ,
i.e.,
Let ; we obtained , which means that . Meanwhile, yields . Given and let
Then, and
This is a contradiction. Therefore,
At this point, we aim to prove that .
Otherwise, if , then , such that . Similar to the above , such that , . Given and let
Then, and . We proved that , so . Therefore,
Again, a contradiction.
Therefore, . Similarly, one has . From (Lemma 2), we obtain
Hence,
Similarly, we obtained
If , since and is a pointed cone, we have Because is a closed convex set, by the separation theorem , such that
So, since . Taking and defining by
Then,
A contradiction. Therefore,. Thus,
At this point, we aim to prove
Otherwise, if , then , such that . So, . Given and defining , by
Then, . This contradiction implies that we must have
We conclude that
and
We proved that, if is a vector saddle-point of , then the conditions (i)–(iii) hold. □
Theorem 5.
If is a vector saddle-point of , and if , then is a weak efficient solution of (VP).
Proof.
Assume that is a vector saddle-point of ; from Theorem 2, we have
So, (the feasible solution of (VP). , such that , i.e.,
Thus,
Since , one has
Therefore, is a weakly efficient solution of (VP). □
5. Vector Lagrangian Theorems
Definition 9 (vector Lagrangian map).
The vector Lagrangian map of (VP) is defined by the set-valued map
Given , we considered the minimization problem induced by (VP):
Definition 10 (slater constrained qualification (SC)).
Let . We consider that (VP) satisfies the Slater Constrained Qualification at if the following conditions hold:
- (1)
- , s.t. ;
- (2)
- for all j.
According the following Theorem 6, (VPST) can also be considered as a dual problem of (VP).
Theorem 6.
Let . Assume that satisfies 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, is a weakly efficient solution of (VP) if, and only if, , such that is a weakly efficient solution of (VPST).
Proof.
Assume , such that is a weakly efficient solution of (VPST). Then, there exist , such that
If , then , such that , i.e.,
This means that
which is a contradiction.
Therefore,
Hence, is a weakly efficient solution of (VP).
Conversely, suppose that is a weakly efficient solution of (VP). So, , such that there is not any for which That is to say, there is not any , such that
By Theorem 3, , such that
Since and , taking in (1), we obtained
However, and imply that , for which
Hence, , which means
Since implies and implies , such that , we have
Because the Slater Constraint Qualification is satisfied, similar to the proof of Theorem 4, we have . So, we may take , such that
Define the operator and by
It is easy to see that
Therefore
Since , we have . Hence,
Therefore,
And then
i.e.,
Taking , and and applying Theorem 4 to the functions , we have
as well as
since .
Consequently, is a weakly efficient solution of (VPST).
We complete the proof. □
Definition 11 (NNAMCQ).
Let . We say that (VP) satisfies the No Nonzero Abnormal Multiplier Constraint Qualification (NNAMCQ) at , if there is no nonzero vector satisfying the system
where is some neighborhood of .
Similar to the proof of Theorem 6, we obtained Theorem 7.
Theorem 7.
Let Assume that satisfies the generalized convexity condition (a), the generalized affineness condition (b), as well as the inner point condition (c). If is a weakly efficient solution of (VP), then ∃ vector Lagrangian multiplier , such that is a weakly efficient solution of (VPST). Inversely, if (NNAMCQ) holds at , and if ∃ vector Lagrangian multiplier , such that is a weakly efficient solution of (VPST), then is a weakly efficient solution of (VP).
6. Conclusions
Jeyakumar [] introduced the following definition of sub convexlike functions for single-valued functions.
Let Y be a topological vector space and be a nonempty set. A set-valued map f: is said to be Y+-sub convexlike on D if bounded set-valued map u: , , , , , such that
where the partial order is induced by a convex cone of Y.
Jeyakumar [] introduced the following subconvexlikeness.
A set-valued map f: is said to be Y+-subconvexlike on D if , such that , , , holds
In this paper, we proved that the above two generalized convexities are equivalent in locally convex topological spaces. Since Banach spaces are locally convex topological spaces (n-dimensional Euclidean spaces are Banach spaces), we proved that the two definitions of generalized convexities are the same. Then, we solved set-valued vector optimization problems and obtained vector saddle-point theorems and some vector Lagrangian theorems. Our optimization problems have inequality, equality as well as an abstract constraint. Our inequality constraints are generalized convex maps and the generalized convexities are defined by partial order relations.
A set-valued map f: is said to be affine on D if ; there holds
We defined the following sub affinelike maps in order to weaken the condition of the “equality constraints” for optimization problems.
A set-valued map f: is said to be Y+-sub affinelike on D if ; there holds
Then, we considered the following optimization problem with set-valued maps:
where f: and are sub convexlike and are sub affinelike.
(VP) Y+-min f (x)
For a single-valued situation, the above optimization problem (VP) may be written as follows.
Y+-min f (x)
We obtained vector saddle-point theorems and vector Lagrangian theorems for the set-valued optimization problem (VP). Our Theorem 3 is a generalization of theorems of alternatives in [,] and a modification of theorems of alternatives in [,,,,]. Our saddle-points theorems (Theorems 4 and 5) are generalizations of the saddle-point theorem in [,] and modifications of saddle-point theorems in [,]. Our Lagrangian theorems (Theorems 6 and 7) are generalizations of Lagrangian theorems in [] and modifications of those in [,]. We can also extended the results in [] according to our methods used in this paper.
Funding
This research received no external funding.
Data Availability Statement
All relevant data are within the manuscript.
Conflicts of Interest
The author declares no conflict of interest.
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