Generalized sufficiency criteria in continuous-time programming with application to a class of variational-type inequalities

Control problems under $$(p,r)-\rho -(\eta ,\theta )$$ ( p , r ) - ρ - ( η , θ ) -invexity

“…We construct an example of a ( p, r ) − ρ − (η, θ )-invex function which generalizes ρ − (η, θ )-invex function [21] and ( p, r )-invex function [1].…”

“…We introduce the concept of ( p, r )−ρ −(η, θ )-invexity which is the generalization of ρ − (η, θ )-invexity and ( p, r )-invexity, introduced by Zalmai [21] and Antczak [1], respectively. Definition 2.1 ([11]) Let f : R n → R be a differentiable function and p, r be arbitrary real numbers, ρ ∈ R. The function f is said to be ( p, r ) − ρ − (η, θ )-invex with respect to η, θ : R n × R n → R n at u, if any one of the following conditions holds…”

Second and higher order duality in Banach space under -invexity

“…The Mond-Weir higher Order dual (MWHD) of the primal (HP) is given by 12) ⟨y * , g(u)⟩ + ⟨y * , k(u, p)⟩ ≥ 0,…”

“…Hanson [8] introduced the concept of invexity and had shown that Kuhn-Tucker conditions are the sufficient conditions for optimality. Many researchers [9][10][11] have discussed various properties, extensions, and applications of ρ−(η, θ )-invexity (for example see [12]). Behera et al [13] introduced generalized ρ − (η, θ )-invex functions between Banach spaces and obtained Kuhn-Tucker sufficient optimality conditions.…”

Generalized (a, ?)-(?, ?)-Convexity in Multi-objective Optimization

“…Recently, the concept of d-(η, θ)-invexity which generalized the notion of η-invexity, has been introduced by Zalmai [7]. While, Nahak and Mohapatra [8] introduced the notation of d-ρ-(η, θ)-invexity with the directional derivative of f in the direction of η(x, u), which is further generalization of the notion of d-(η, θ)-invexity.…”