On the characterization of D-preinvex functions

Convexity and some generalizations of convexity play a crucial role in mathematical economics, engineering, management science, and optimization theory. Therefore, it is important to consider wider classes of generalized convex functions and also to seek practical criteria for convexity or generalized convexity (see Refs. [1‐9] and the references therein). A significant generalization of convex functions is the introduction of preinvex functions, which is due to Ben and Mond [4]. Yang and Li [1] presented some properties of preinvex functions; in [2] they introduced two new classes of generalized convex functions called semistrictly preinvex functions and strictly preinvex functions. They established relationships between preinvex functions and semistrictly preinvex functions under a certain set of conditions. Very recently, Peng and Zhu [7] introduced the vector cases of strict preinvexity and semistrict preinvexity and established some relations between them. In this paper, we show that the same results or even a generalized version of their results can be obtained under weaker assumptions. Moreover, we give a new characterization of D-preinvexity and D-semistrict preinvexity under mild conditions. The outline of the paper is as follows. In Section 2, we give some preliminaries. The main results of the paper are presented in Section 3.

Throughout this paper, we will use the following assumptions. Let X be a real topological vector space and Y be a real locally convex vector space, let K ⊂ X be a nonempty subset. Let D ⊂ Y be a nonempty pointed closed convex cone and Y ∗ be the dual space of Y. The dual cone D ∗ of cone D is defined by

From the bipolar theorem, we have the following

Lemma 2.1 For all q ∈ D ∗, 〈 q , d 〉 ≥ 0 if and only if d ∈ D.

Now we will describe some definitions of generalized convexity.

Definition 2.1 ([3])

Let the set K ⊆ X and the vector function η : X × X → X be given. K is said to be an invex set with respect to the function η iff

Definition 2.2 ([7])

Let K ⊆ X be an invex set with respect to η : X × X → X. The vector-valued function f : K → Y is said to be D-preinvex on K iff, ∀ x , y ∈ K, ∀ λ ∈ ( 0 , 1 ),

Definition 2.3 ([7])

Let K ⊆ X be an invex set with respect to η : X × X → X. The vector-valued function f : K → Y is said to be D-semistrictly preinvex on K iff, ∀ x , y ∈ K such that f ( x ) ≠ f ( y ) and ∀ λ ∈ ( 0 , 1 ),

Definition 2.4 ([7])

Let K ⊆ X be an invex set with respect to η : X × X → X. The function f : K → Y is said to be D-strictly preinvex on K iff, ∀ x , y ∈ K, x ≠ y, ∀ λ ∈ ( 0 , 1 ),

In [10], Jeyakumar et al. introduced the ∗-lower semicontinuity for a vector-valued function as follows.

Definition 2.5 ([10])

The vector-valued function f : K → Y is ∗-lower semicontinuous if for every q ∈ D ∗, q ( f ) ( ⋅ ) = 〈 q , f ( ⋅ ) 〉 is lower semicontinuous on K.

In [7], Peng et al. introduced the ∗-upper semicontinuity for a vector-valued function as follows.

Definition 2.6 ([7])

The vector-valued function f : K → Y is ∗-upper semicontinuous if for every q ∈ D ∗, q ( f ) ( ⋅ ) = 〈 q , f ( ⋅ ) 〉 is upper semicontinuous on K.

In order to prove our main result, we need Condition C introduced by Mohan and Neogy [11] as follows.

Condition C Let η : X × X → X. We say that the function η satisfies Condition C iff, ∀ x , y ∈ K, ∀ λ ∈ [ 0 , 1 ],

In this section, we assume always that:

The following result was proved in Ref. [7]; see Theorem 2.2 in Ref. [7].

Theorem 3.1 Let K be a nonempty open invex set in X with respect to η : X × X → X. If f : K → Y is ∗-upper semicontinuous and satisfies f ( y + η ( x , y ) ) ∈ f ( x ) − D, ∀ x , y ∈ K, then f is a D-preinvex function for the same η on K if and only if there exists an α ∈ ( 0 , 1 ) such that

Now we improve the above theorem as follows.

Theorem 3.2 Let f : K → Y be ∗-upper semicontinuous and satisfy f ( y + η ( x , y ) ) ∈ f ( x ) − D, ∀ x , y ∈ K, then f is a D-preinvex function for the same η on K if and only if there exists an α ∈ ( 0 , 1 ) such that

Proof The necessity follows directly from the definition of D-preinvexity for the vector-valued function f. We only need to prove the sufficiency. By Lemma 2.1 in Ref. [7], the set A = { α ∈ [ 0 , 1 ] ∣ f ( y + α η ( x , y ) ) ∈ α f ( x ) + ( 1 − α ) f ( y ) − D , ∀ x , y ∈ K } is dense in the interval [ 0 , 1 ]. Then ∀ α ¯ ∈ ( 0 , 1 ), ∃ { α n } ⊆ ( 0 , 1 ) ∩ A such that α n < α ¯ for each n and α n → α ¯, as n → ∞. Give x , y ∈ K, denote

Define, for each n,

Thus,

Since 0 < α n < α ¯ < 1, we have

which in turn implies that y n ∈ K, by (3.2) and K is invex with respect to η. From Condition C, we have

As α n ∈ A, we have

By the ∗-upper semicontinuity of f on K, for every q ∈ D ∗, q ( f ) ( ⋅ ) is upper semicontinuous, it follows that for any ϵ > 0, there exists an N > 0 such that the following holds:

Hence,

Since ϵ > 0 may be arbitrarily small, then for all q ∈ D ∗, we have

Since q is linear, by Lemma 2.1, we have

Hence, f is a D-preinvex function for the same η on K, this completes the proof. □

Now, we state another result in Ref. [7]; see Theorem 3.3 in Ref. [7].

Theorem 3.3 Let f be a D-preinvex function on K. If there exists an α ∈ ( 0 , 1 ) such that, for each pair x , y ∈ K, x ≠ y,

then f is a D-strictly preinvex function on K.

The above theorem can be improved as follows.

Theorem 3.4 Let f be a D-preinvex function with respect to η : X × X → X on K. For each pair x , y ∈ K, x ≠ y, if there exists an α ∈ ( 0 , 1 ) such that

then f is a strictly D-preinvex function on K.

Proof By contradiction, suppose that there exist x , y ∈ K, x ≠ y, λ ∈ ( 0 , 1 ) such that

Denote

Since f is D-preinvex, we have

We note that the pair x, z and the pair z, y are both distinct under condition (3.5). There exist β 1 , β 2 ∈ ( 0 , 1 ) such that

Denote

From Condition C,

Let μ 1 = λ + ( 1 − λ ) β 1, μ 2 = λ β 2, μ = λ − μ 2 μ 1 − μ 2. It is easy to verify that μ 1 , μ 2 , μ ∈ ( 0 , 1 ). Again from Condition C,

Since f is D-preinvex, we have

Thus, from (3.7)-(3.10), we have

Where

which contradicts (3.6). This completes the proof. □

Remark 3.2 In Theorem 3.3, a uniform α ∈ ( 0 , 1 ) is needed, while in Theorem 3.4 this condition has been weakened to great extent.

By Theorem 2.3 in Ref. [7] and Theorem 3.4 above, we have the following corollary.

Corollary 3.1 Let f be a ∗-lower semicontinuous function on K and satisfy f ( y + η ( x , y ) ) ∈ f ( x ) − D, ∀ x , y ∈ K. For each pair x , y ∈ K, x ≠ y, if there exists an α ∈ ( 0 , 1 ) such that

then f is a strictly D-preinvex function on K.

Corollary 3.2 Let f be a ∗-upper semicontinuous function on K and satisfy f ( y + η ( x , y ) ) ∈ f ( x ) − D, ∀ x , y ∈ K. If there exists an α ∈ ( 0 , 1 ) such that, for each pair x , y ∈ K, x ≠ y,

then f is a strictly D-preinvex function on K.

Theorem 3.5 Let f be a ∗-lower semicontinuous function on K and satisfy f ( y + η ( x , y ) ) ∈ f ( x ) − D, ∀ x , y ∈ K. If there exists an α ∈ ( 0 , 1 ) such that, for each pair x , y ∈ K, f ( x ) ≠ f ( y ) implies

then f is both a D-preinvex function on K and a D-semistrictly preinvex function on K.

Proof First, we prove that f is a D-preinvex function on K. By Theorem 2.3 in Ref. [7], we need to show that for each x , y ∈ K, there exists λ ∈ ( 0 , 1 ) such that

Assume, by contradiction, that there exists x , y ∈ K such that

If f ( x ) ≠ f ( y ), condition (3.11) implies

which contradicts (3.12). Thus, we have f ( x ) = f ( y ), and then (3.12) implies

Since D is a closed convex pointed cone, by the strong separation theorem for a convex set, it follows that there exists k ∗ ∈ D ∗ ∖ { 0 } such that

Let λ = α − 1 α − 2 ∈ ( 0 , 1 ), and let z = y + λ η ( x , y ), then the above inequality reduces to

and

Define

From Condition C, we obtain

Since 1 2 − α ∈ ( 0 , 1 ), it follows from (3.14) that

Conditions (3.11) and (3.13) give

It follows that

which together with (3.15) yields

From (3.17) and condition (3.11), we get

It follows that

which together with (3.17) yields

which contradicts (3.18), hence f is a D-preinvex function on K. Next, the D-semistrict preinvexity of f on K follows from Theorem 3.9 of Ref. [7]. □

In this paper, we firstly obtain a property of D-preinvex functions. We then get a sufficient condition of the strictly D-preinvex functions in terms of intermediate-point D-preinvex functions. We finally obtain a sufficient condition of D-preinvex functions and D-semistrictly preinvex functions. Our results improve and extend the existing ones in the literature.