New class of G-Wolfe-type symmetric duality model and duality relations under Gf$G_{f}$-bonvexity over arbitrary cones

*Correspondence: lakshminarayanmishra04@gmail.com 4Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology (VIT) University, Vellore, India Full list of author information is available at the end of the article Abstract This paper is devoted to theoretical aspects in nonlinear optimization, in particular, duality relations for some mathematical programming problems. In this paper, we introduce a new generalized class of second-order multiobjective symmetric G-Wolfe-type model over arbitrary cones and establish duality results under Gf -bonvexity/Gf -pseudobonvexity assumptions. We construct nontrivial numerical examples which are Gf -bonvex/Gf -pseudobonvex but neither η-bonvex/η-pseudobonvex nor η-invex/η-pseudoinvex.


Introduction
It is an undeniable fact that all of us are optimizers as we all make decisions for the sole purpose of maximizing our quality of life, productivity in time, and our welfare in some way or another. Since this is an ongoing struggle for creating the best possible among many inferior designs and is always the core requirement of human life, this fact yields the development of a massive number of techniques in this area, starting from the early ages of civilization until now. The efforts and lives behind this aim dedicated by many brilliant philosophers, mathematicians, scientists, and engineers have brought a high level of civilization we enjoy today. The decision process is relatively easier when there is a single criterion or object in mind. The process gets complicated when we have to make decisions in the presence of more than one criteria to judge the decisions. In such circumstances a single decision that optimizes all the criteria simultaneously may not exist. For handling such type of situations, we use multiobjective programming, also known as multiattribute optimization, which is the process of simultaneously optimizing two or more conflicting objectives subject to certain constraints. Multiobjective optimization problems can be found in various fields such as product and process design, finance, aircraft design, the oil and gas industry, automobile design, and other where optimal decisions need to be taken in the presence of trade-offs between two or more conflicting objectives.
This paper is organized as follows. In Sect. 2, we give some preliminaries and definitions used in this paper and also a nontrivial example for such type functions. In Sect. 3, we formulate second-order multiobjective symmetric G-Wolfe-type dual programs over arbitrary cones. We prove weak, strong, and converse duality theorems by using G f -bonvexity/G f -pseudobonvexity assumptions over arbitrary cones. Finally, we construct nontrivial numerical examples that are G f -bonvex/G f -pseudobonvex but neither η-bonvex/η-pseudobonvex nor η-invex/η-pseudoinvex functions.

Preliminaries and definitions
Let f = (f 1 , f 2 , f 3 , . . . , f k ) : X → R k be a vector-valued differentiable function defined on a nonempty open set X ⊆ R n , and let I f i (X), i = 1, . . . , k, be the range of f i , that is, the image of X under f i . Let G f = (G f 1 , G f 2 , . . . , G f k ) : R → R k be a differentiable function such that every component G f i : I f i (X) → R is strictly increasing on the range of I f i , i = 1, 2, 3, . . . , k.

Definition 2.1
The positive polar cone S * of a cone S ⊆ R s is defined by Consider the following vector minimization problem: Definition 2.2ȳ ∈ S 0 is an efficient solution of (MP) if there exists no other y ∈ S 0 such that f r (y) < f r (ȳ) for some r = 1, 2, 3, . . . , k and f i (y) ≤ f i (ȳ) for all i = 1, 2, 3, . . . , k.

Definition 2.3
If there exists a function η : S × S → R n such that for all y ∈ S, then f is called invex at v ∈ S with respect to η.

Definition 2.4
If there exist G f i : I f i (S) → R and η : S × S → R n such that for all y ∈ S, then f i is called G f i -pseudoinvex at u ∈ S with respect to η.

Definition 2.5
If there exist G f i : I f i (S) → R and η : S × S → R n such that for all y ∈ S and p ∈ R n ,

Definition 2.6
If there exist functions G f and η : S × S → R n such that for all y ∈ S and p ∈ R n , where f 1 (y) = y 10 , f 2 (y) = arcsin y, f 3 (y) = arctan y, f 4 (y) = arccot y, and let Let η : [-1, 1] × [-1, 1] → R be given as To show that f is G f -bonvex at v = 0 with respect to η, we have to claim that Putting the values of f i , G f i , i = 1, 2, 3, 4, into the last expression, after simplifying at the Fig. 2).
Therefore f 3 is not η-bonvex at v = 0 with respect to p. Hence f is not η-bonvex at v = 0 with respect to p. Next, where f 1 (y) = ( e 2y -1 e y ), f 2 (y) = y 3 , and To show that f is G f -pseudobonvex at v = 0 with respect to η, we have to claim that, for i = 1, 2, Let Substituting the values of η and f 1 at the point v = 0, we get φ 1 ≥ 0 for all y ∈ [-2, 2] and p.

Second-order multiobjective G-Wolfe-type symmetric dual program
Consider the following pair of second-order multiobjective G-Wolfe-type dual programs over arbitrary cones.
Let Y 0 and Z 0 be the sets of feasible solutions of (GWP) and (GWD), respectively.
Then the following inequalities cannot hold together: and R r (y, z, λ, p) < S r (v, w, λ, q) for at least one r ∈ K.
Proof Proof follows the lines of Theorem 3.2.

Concluding remarks
In this paper, we have formulated a second-order symmetric G-Wolfe-type dual problem for a nonlinear multiobjective optimization problem with cone constraints. A number of duality relations are further established under G f -bonvexity/G f -pseudobonvexity assumptions on the function f . We have discussed various numerical examples to show the existence of G f -bonvex/G f -pseudobonvex functions. The question arises whether the duality results developed in this paper hold for G-Wolfe-or mixed-type higher-order multiobjective optimization problems. This may be the future direction for the researchers working in this area.