Abstract
In this work, several extended approximately invex vector-valued functions of higher order involving a generalized Jacobian are introduced, and some examples are presented to illustrate their existences. The notions of higher-order (weak) quasi-efficiency with respect to a function are proposed for a multi-objective programming. Under the introduced generalization of higher-order approximate invexities assumptions, we prove that the solutions of generalized vector variational-like inequalities in terms of the generalized Jacobian are the generalized quasi-efficient solutions of nonsmooth multi-objective programming problems. Moreover, the equivalent conditions are presented, namely, a vector critical point is a weakly quasi-efficient solution of higher order with respect to a function.
Highlights
Convexity and its generalizations played a critical role in multi-objective programming problems
Approximate convexity and invexity are two significant generalized versions of convexity, which tried to weaken the convexity hypotheses to study the relations between vector variational-like inequalities and multi-objective programming problems
In Section, we study the relations between vector critical points and weakly quasi-efficient solutions of higher order for (NMP) with respect to a vector-valued function
Summary
Convexity and its generalizations played a critical role in multi-objective programming problems. A point x ∈ X is called a quasi-efficient solution of order m for (NMP) with respect to ψ , if there exist a function ψ : X × X → Rn and α ∈ int(Rp+) such that, for any x ∈ X, the following cannot hold: f (x) ≤ f (x ) – α ψ(x, x ) m.
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