On vector optimization problems and vector variational inequalities using convexificators

A study on vector variational-like inequalities using convexificators and application to its bi-level form

“…Remark 3. Theorem 3.2 extends the characterization of ∂ * * -convexity of a vectorvalued function by the monotonicity of its convexificator in ( [16], Theorem 3.3) to the framework of invexity while relaxing the convexity. This result can be applied to those functions which are ∂ * * -invex, but not ∂ * * -convex as in example 2.1.…”

“…It may be observed that if η(x, y) is replaced by x − y, then the above definitions of η-monotonicity, strict η-monotonicity of ∂ * * f , ∂ * * -invexity, strict ∂ * *invexity reduce respectively to monotonicity, strict monotonicity, convexity, strict convexity in the framework of convexificators as studied in [16]. In this sense, all the latter forms can be considered as the respective particular cases of the former forms.…”

“…In this paper, since we are dealing with convexificators, instead of C(x) or C, we use R n + while defining the variational-like inequalities for the sake of simplicity. It may be noted that, if η(x, y) = x−y and ϕ(y * , x−y) = y * , x−y , then ∂ * * -WSVVLI and ∂ * * -WMVVLI reduce to the weak formulations for, Stampacchia vector variational inequality (Stampacchia ∂ * * -WVVI): Findx ∈ K, such that for any x ∈ K, there existsx * ∈ ∂ * * ϕ(x), such that, x * , x −x n / ∈ −int R n + , Minty vector variational inequality (Minty ∂ * * -WVVI): Findx ∈ K, such that for any x ∈ K, there exists x * ∈ ∂ * * ϕ(x), such that, x * , x −x n / ∈ −int R n + , as studied by Laha and Mishra [16].…”

“…This result can be applied to those functions which are ∂ * * -invex, but not ∂ * * -convex as in example 2.1. In this sense it is applicable to broader class of functions and useful for relaxing the convexity assumption as compared to the result in Laha and Mishra [16].…”

“…They introduced the generalized pseudoinvexity and generalized invexity in the sense of Clarke generalized directional derivative. Laha and Mishra [16] derived necessary and sufficient conditions for a point to be a solution of the vector optimization problem using vector variational inequality as a tool in the framework of convexificators.…”