Optimality Conditions for Set-Valued Optimization Problems

A BSTRACT . In this paper, we first prove that the generalized subconvexlikeness introduced by Yang, Yang and Chen [1] and the presubconvelikeness introduced by Zeng [2] are equivalent. We discuss set-valued nonconvex optimization problems and obtain some optimality conditions


Introduction
Set-valued optimization is a vibrant and expanding branch of mathematics that deals with optimization problems where the objectives and/or the constraints are set-valued maps.Corley [3] pointed out that the dual problem of a multiobjective optimization involves the optimization of a setvalued map, while Klein and Thompson [4] gave some examples in Economics where it is necessary to use set-valued maps instead of single-valued maps.There are many recent developments about set-values optimization problems, e.g., [5][6][7][8][9].
Convex and generalized convex optimization is a rich branch of mathematics.Many interesting and useful definitions of generalized convexities were introduced.Borwein [10] proposed the definition of cone convexity, Fan [11] introduced the definition of convexlikeness.Yang, Yang and Chen [1] defined the generalized subconvexlike functions, while Zeng [2] introduced a presubconvexlikeness.
In this paper, we first prove that the generalized subconvexlikeness introduced by Yang, Yang, and Chen [1] and the presubconvexlikeness introduced by Zeng [2] are equivalent, in locally convex topological spaces.And then, we deal with set-valued optimization problems and obtain some optimality conditions.
A subset Y + of a real linear topological space Y is a cone if yY  +  for all yY +  and 0   .We denote by 0 Y the zero element in the linear topological space Y and simply by 0 if there is no confusion.
A convex cone is one for which  .A pointed cone is one for which ( ) {0} YY ++ −=.Let Y be a real linear topological space with pointed convex cone Y + .We denote the partial order induced by Y + as follows: , where intY + denotes the topological interior of a set Y + .Let X, Zi, Wj be real linear topological spaces and Y be an ordered linear topological space with the partial order induced by a pointed convex cone We recall some notions of generalized convexity of set-valued maps.First we recall the notion of cone-convexity of a set-valued map introduced by Borwein XY → is said to be Y + -convex on D if and only if The following notion of generalized convexity is a set-valued map version of Ky Fan convexity [11] (Ky Fan's definition was for vector-valued optimization problems).
The following concept of generalized subconvexlikeness was introduced by Yang, Yang and Chen [1] ( [1] introduced subconvexlikeness for vector-valued optimization).
Definition 1.3 (Generalized subconvexlike) Let Y be a linear topological space and DX  be a nonempty set and Y + be a convex cone in Y.A set-valued map f : DY → is said to be generalized The following Lemma 1.1 is from Chen and Rong [12, Proposition 3.1].

Lemma 1.1 A function f :
A bounded function in a real linear topological space can be fined as following Definition 1.4 (e.g,, see Yosida [13]).
Definition 1.4 (Bounded set-valued map) A subset M of a real linear topological space Y is said to be a bounded subset if for any given neighbourhood U of 0,  positive scalar  such that The following Definition 1.5 was introduced by Zeng [2] for single-valued functions.Definition 1.5 (Presubconvexlike) Let Y be a linear topological space and DX  be a nonempty set and Y + be a convex cone in Y.A set-valued map f : DY → is said to be Y + -presubconvexlike on It is obvious that It is important to note that the concept of convexlike or any weaker concepts are only nontrivial if Y is not the one-dimensional Euclidean space since any real-valued function is R + -convexlike.

The Equivalence of Generalized Subconvexlikeness and Presubconvexlikeness
In this section, we are going to prove that Definition 1.4 (Generalized subconvexlikeness) and (2) M is said to be balanced if yM  and | | 1 (3) M is said to be absorbing if for any given neighbourhood U of 0, there exists a positive scalar , From Definition 2.2,  neighbourhood U of 0 such that U is convex, balanced, and absorbing, and , where uU + is a neighbourhood of u.Therefore, we may take This and the convexity of intY + imply that And so Let Y be a locally convex topological space and DX  be a nonempty set and Proof.The necessity.
Suppose that f is Y + -presubconvexlike on, aim to show that 0 ( ( ) int ) (1 ) , y y y . By Definition 2.2, without loss of generality, we may assume that U is convex, balanced, and absorbing.
From the assumption of and  > 0 such that (1 ) (1 ) (1 ) Since U is convex, balanced, and absorbing, by Definition 2.2, we may take And then The given int uY +  can be consider as a bounded function.
By Propositions 1 and 2 one has Theorem 2.1.
Theorem 2.1 Let Y be a locally convex topological space and DX  be a nonempty set, and

Optimal Conditions
We consider the following optimization problem with set-valued maps: (VP) xD  where f : are set-valued maps, Zi+ is a closed convex cone in Zi and D is a nonempty subset of X.
For a set-valued map f : XY → , we denote by ( ) ( ) We now explain the kind of optimality we consider here.Let F be the feasible set of (VP), i.e.
: { : ( We are looking for a weakly efficient solution of (VP) defined as follows.
Definition 3.1 (Weakly Efficient Solution) A point xF  is said to be a weakly efficient solution of (VP) with a weakly efficient value Consider the set-valued optimization problem (VP).From now on we assume that Y+, Zi+ are pointed convex cones with nonempty interior of intY + , int i Z + , respectively.The following three assumptions will be used in this paper.
Proof.Suppose that System 1 has no solution, then 0 B  .Since (A1) holds, the set B is convex.
By assumption, Since intY+, intZi+ are convex cones, we have

Hence
x is a weakly efficient solution of (VP) with () y f x  .From Theorem 3.2 and 3.3 one has Theorem 3.  XR → are functions and D is a nonempty subset of X.
Applying Theorem 3.1 to the above single-valued optimization problem we have the following Fritz John type necessary optimality condition.
Theorem 4.1 Let x be an optimal solution of (P).Suppose the following generalized convexity assumption holds:  0,( 0,1,2, , ) Then,  nonzero vector ( , , We now study some cases where the generalized convexity holds and consequently the Fritz John condition in the above theorem holds.

Theorem 4.2 Let
x be an optimal solution of (P).Suppose one of the following set of assumptions hold.
(I) All functions gi are nonnegative on the set D and n = 0 (i.e.there is no equality constraints).
(II) All functions f, gi are nonnegative on the set D and n = 1.Then,  non-zero vector Proof.From Theorem 4.1, it suffices to prove that the generalized convexity assumption holds.
First assume that assumption (I) holds.Let 12 , Since g is nonnegative on set D, for small enough (0, ]   one has Case 2:

( ) ( ) f x f x 
. In this case by choosing 32 xx = similarly as in case 1 we can prove (1)   and (2).Hence the generalized convexity assumption holds.Now assume that assumption (II) holds.Let 12 , Let 31 xx = .Then since f, gi are nonnegative, similarly as in (I) one can find a small enough 0 0   and 1 0   such that Otherwise if h( 2x )  0, then one can find 2 0 Let 32 xx = .Then since f, gi are nonnegative, similarly as in (I) one can find a small enough 0 0   and 1 0   such that (3) and ( 4) hold.Hence the generalized convexity assumption holds.
Theorem 4.3 (Kuhn-Tucker Type Necessary Optimality Condition) Let x be an optimal solution of (P).Suppose all assumptions in Theorem 4.2 hold and there is no nonzero vector ( , )   (5) And Zeng [2] introduced the presubconvexlikeness as follows.
Let Y be a topological vector space and DX  be a nonempty set and Y + be a convex cone in Y.A set-valued map f : DY → is said to be Y + -presubconvexlike on D if  bounded set-valued map u: DY → such that 12 , The inclusions ( 5) and ( 6) may be written as In this paper, we proved that the above two generalized convexities are equivalent.
And then, we worked with nonconvex set-valued optimization problems and attained some optimality conditions.Our Fritz John Type Necessary Optimality Condition (Theorem 3.1) and Kuhn-Tucker Type Necessary Optimality Condition (Theorem 4.3) extend the classic results in Clarke [14].Our Proposition 3.1 are modifications of the alternative theorems in [15,16].Our Theorem 3.2 (sufficient optimality condition) extends Theorem 23 in [9].Our Strong Duality Theorem (Theorem 3.3) extends Theorem 7 in Li and Chen [8].
Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publication of this paper.
[10].Definition 1.1 (Convexity) Let X, Y be real linear topological spaces, DX  a nonempty convex set and Y + a convex cone in Y.A set-valued map f : and (A2) holds (which is equivalent to saying that int B ) or dimensional.Therefore by the separation theorem,  nonzero vector

Theorem 3 . 1 [
Fritz John Type Necessary Optimality Condition] Assume that the generalized convexity assumption (A1) is satisfied and either (A2) or (A3) holds.If xF  is a weakly efficient solution of (VP) with xF  is a weakly efficient solution of (VP) with () y f x  , by definition the following system ,( ( ) ) ( int )

3 .Theorem 3 . 3 (x
Strong Duality) Suppose all assumptions in Theorem 3.1 hold and there is no nonzero vector ( , ) be a solution of problem (P).Then the strong duality holds.That is,

 4 .
Applications to Single-Valued Optimization Problems Consider the optimization problem: xD  where f, gi, hj: and Chen[1] defined the following generalized subconvexlike functions.([1] introduced subconvexlikeness for vector-valued optimization).Let Y be a topological vector space and DX  be a nonempty set and Y + be a convex cone in Y.A set-valued map f : DY → is said to be generalized Y + -subconvexlike on D if int uY order induced by the convex cone Y + .
we have