Semistrict G-Preinvexity and Optimality in Nonlinear Programming

and Applied Analysis 3 Proof. Let x, y ∈ K. From the assumption of f(y+η(x, y)) ≤ f(x), when λ = 0, 1, we can know that f (y + λη (x, y)) ≤ G −1 (λG (f (x)) + (1 − λ)G (f (y))) . (9) Then, there are two cases to be considered. (i) Iff(x) ̸ = f(y), then by the semistrictG-preinvexity of f, we have the following:

On the other hand, Avriel et al. [12] introduced a class of -convex functions which is another generalization of convex functions and obtained some relations with other generalization of convex functions.In [13], Antczak introduced the concept of a class of -invex fuctions, which is a generalization of -convex functions and invex functions.Recently, Antczak [14] introduced a class of -preinvex functions, which is a generalization of -invex [13], preinvex functions [8] and derived some optimality results for constrained optimization problems under -preinvexity.Very recently, Luo and Wu introduced a new class of functions called semistrictly -preinvex functions in [15], which include semistrictly preinvex functions [9] as a special case.They investigated the relationships between semistrictly preinvex functions and -preinvex functions and gave a criterion for semistrict -preinvexity.Moreover, they also proposed three open questions (just as they said: "an interesting topic for our future research is to": (1) investigate -preinvex functions and semicontinuity; (2) explore some properties of semistrictly -preinvex functions; (3) research into some applications in optimization problems under semistrictly preinvexity [15]).

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Abstract and Applied Analysis However, as far as we know, there are few papers dealing with the properties and applications of the semistrictly preinvex functions [16].The questions above in [15] have not been solved, and one condition in [15] is not mild in researching of relationships between -preinvexity and semistrict preinvexity.So, in this paper, we further discuss semistrictly -preinvex functions.The rest of the paper is organized as follows.Firstly, we investigate -preinvex functions and semicontinuity and obtain a criterion of -preinvexity under semicontinuity.Then, we give new relationships between preinvexity and semistrictly -preinvexity, which are different from the recent ones in the literature [15].Finally, we get two optimality results under semistrict -preinvexity for nonlinear programming.The obtained results in this paper improve and extend the existing ones in the literature (e.g., [8,9,11,15,16]).

Preliminaries and Definitions
Throughout this paper, let  be a nonempty subset of   .Let  :  →  be a real-valued function and  :  ×  →   a vector-valued function.Let   () be the range of , that is, the image of  under .Now we recall some definitions.
Remark 7. In order to define an analogous class of semistrictly -preincave functions with respect to , the direction of the inequality in Definition 6 should be changed to the opposite one.
In order to prove our main result, we need Condition C as follows.

Relationships with Semistrictly 𝐺-Preinvexity
In [15] Then, there are two cases to be considered.
From Condition C, By the semistrict -preinvexity of  and (12), we have the following: On the other hand, from Condition C, one can obtain the following: According to (13) and the semistrictly -preinvexity of , we have the following: which contradicts (15).This completes the proof.
Proof.For any ,  ∈  with () ̸ = () and  ∈ (0, 1), by assumption, we have the following: Using ( 18) and the -preinvexity of , we have the following: (ii) Let  < ; that is, From Condition C, we have the following: According to (18) and the -preinvexity of , we get the following: (21) and ( 24) imply that  is a semistrictly -preinvex function on .
The following example illustrates that assumption (18) in Theorem 12 is essential.
It is obvious that  is a -preinvex function with respect to , where () =   ,  = .
Lemma 15 (see [16]).Let  be a nonempty invex set with respect to .Suppose that  is a semistrictly  1 -preinvex function with respect to  and  2 is a continuous and strictly increasing function on   ().If () =  2  −1 1 is convex on the image under  1 of the range of , then  is also semistrictly  2 -preinvex function with respect to the same  on .Theorem 16.Let K be an invex set with respect to  and  :  →  be a semistrictly -preinvex function on .If  is concave on   (), then  is a preinvex function with respect to the same  on .
From Theorem 16 and Definitions 2-4, we can obtain the following Corollary easily.Corollary 17.Let K be an invex set with respect to  and  :  →  is a semistrictly -preinvex function on .If  is concave on   () then  be a prequasi-invex function with respect to the same  on .

Semistrictly 𝐺-Preinvexity and Optimality
In order to solve the open question (3) proposed in [15] (see, the part of Introduction), in this section, we consider nonlinear programming problems with constraint and obtain two optimality results under semistrict -preinvexity.
We consider the following nonlinear programming Problem (P) with inequality constraint: min  () (P) () ≤ 0,  ∈  = 1, . . ., , where  :  → ,   :  → ,  ∈ , and  is a nonempty subset of   .We denote the set of all feasible solutions in (P) by the following: Let  ( ̸ =  * ) be an interior point of .By assumption,  is an invex set with respect to .It follows from the definition of invex set  that there exists  ∈  such that for some  ∈ (0, 1),  =  * +  (,  * ) . ( By assumption,  is semistrictly -preincave with respect to  on .Then, we have the following: where  is an interior point of .From the inequality above, we conclude that no interior of  is an optimal solution of (P), that is, any optimal solution  in problem (P), if exists, must be a boundary point of .This completes the proof.Theorem 19.Let  ∈  be local optimal in problem (P).Moreover, we assume that  is semistrictly -preinvex with respect to  at  on  and the constraint functions   ,  ∈ , are preinvex with respect to  at  on .Then,  is a global optimal solution in problem (P).
(36) Suppose to the contrary,  is not a global minimum in (P), there exists an  * ∈  such that  ( * ) <  () .

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Theorem 18. Suppose the set of all feasible solutions  of problem (P) is an invex set with respect to , and  at least contains two points with nonempty interior.Let  be a nonconstant semistrictly -preincave function with respect to  on .Then no interior of  is an optimal solution of (P), or equivalently, any optimal solution  in problem (P), if exists, must be a boundary point of .Proof.If problem (P) has no solution the theorem is trivially true.Let  be an optimal solution in problem (P).By assumption,  is a nonconstant on .Then, there exists a feasible point * ∈  such that  ( * ) >  () .