Mathematical Programming Based on Sufficient Optimality Conditions and Higher Order Exponential Type Generalized Invexities

First, a class of comprehensive higher order exponential type generalized B-(b, ρ, η, ω, θ, p̃, r̃, s̃)-invexities is introduced, which encompasses most of the existing generalized invexity concepts in the literature, including the Antczak type first order B-(b, η, p̃, r̃)-invexities as well as the Zalmai type (α, β, γ, η, ρ, θ)-invexities, and then a wide range of parametrically sufficient optimality conditions leading to the solvability for discrete minimax fractional programming problems are established with some other related results. To the best of our knowledge, the obtained results are new and general in nature relating the investigations on generalized higher order exponential type invexities.


Introduction
Recently, Zalmai [41], in a series of publications based on the work of Antczak [1,2,3], generalized the exponential type of invexities and applied to a class of global parametric sufficient optimality criteria using various assumptions for semiinfinite discrete minimax fractional programming problems.Furthermore, Zalmai [41], applying certain suitable partitioning schemes investigated various sets of generalized parametric sufficient optimality results each of which is in fact a family of such results whose members can easily be identified by appropriate choices of certain sets and functions.Antczak [1,2,3] introduced and studied first order exponential type B-(p, r)invexities and applied investigating nonlinear mathematical programming problems, especially in [2] Antczak proved some optimality conditions for a class of generalized fractional programming problems involving B-(p, r)invex functions.This work was followed by developing various duality models relating to fractional programming problems in the literature.Verma [30] introduced the second order (Φ, Ψ, ρ, η, θ)-invexities to the context of parametric sufficient optimality conditions in semiinfinite discrete minimax fractional programming, while Zalmai and Zhang [42] have established a set of necessary efficiency conditions and a fairly large number of global nonparametric sufficient efficiency results under various frameworks for generalized (η, ρ)-invexity for semiinfinite discrete minimax fractional programming problems.There exists an enormous literature on generalized first order as well as second order invexities with applications.Verma [25] also developed a general framework for a class of (ρ, η, θ)-invex functions to examine some parametric sufficient efficiency conditions for multiobjective The general invexity theory has been investigated in several directions.We generalize the notion of the first order Antczak type B-(p, r)-invexiies to the case of the second order B-(b, ρ, η, ω, θ, p, r, s)-invexities.These notions of the second order invexity encompass most of the existing notions in the literature.Let f be a twice continuously differentiable real-valued function defined on X.

Definition 2.2
The function f is said to be second order strictly B − (b, ρ, η, ω, θ, p, r, s)-invex at x * ∈ X if there exist functions η, ω, θ : X × X → R n and b : X × X → [0, ∞), and real numbers r, s and p such that for all x ∈ X and x * )∥ 2 for p = 0, r = 0 and s = 0.

Definition 2.6
The function f is said to be second order B − (b, ρ, η, ω, θ, p, r, s)-quasiinvex at x * ∈ X if there exist functions η, ω, θ : X × X → R n and b : X × X → [0, ∞), and real numbers r, s and p such that for all x ∈ X and z

Definition 2.7
The function f is said to be second order strictly ), and real numbers r, s and p such that for all x ∈ X and The function f is said to be second order prestrictly B − (b, ρ, η, ω, θ, p, r, s)-quasiinvex at x * ∈ X if there exist functions η, ω, θ : X × X → R n and b : X × X → [0, ∞), and real numbers r and p such that for all x ∈ X and x * )∥ 2 ≤ 0 for p = 0, s = 0 and r = 0, equivalently, Next, we present some examples which shall reflect the interrelationship among the basic definitions introduced (and applied) in this paper.

Example 2.1
The function f is said to be second order B − (b, ρ, η, θ, p, r, s)-pseudoinvex with respect to η and b at x * ∈ X if there exist functions η, θ : X × X → R n and b : X × X → [0, ∞), and real numbers r, s and p such that for all ) ) ≥ 0 for p ̸ = 0, r ̸ = 0 and s ̸ = 0.

Example 2.2
The function f is said to be second order B − (b, ρ, η, θ, p, r)-pseudoinvex with respect to η and b at x * ∈ X if there exist functions η, θ : X × X → R n and b : X × X → [0, ∞), and real numbers r and p such that for all x ∈ X and Example 2.3 (Zalmai [41]) The function f is said to be first order B − (b, ρ, η, θ, p, r)-pseudoinvex with respect to η and b at ), and real numbers r and p such that for all x ∈ X and z ∈ R n , We shall use the following auxiliary results which are crucial to the overall development of the main results on hand.
for each i ∈ p, let f i and g i be twice continuously differentiable at x * , for each j ∈ q, let the function z → G j (z, t) be twice continuously differentiable at x * for all t ∈ T j , and for each k ∈ r, let the function z → H k (z, s) be twice continuously differentiable at x * for all s ∈ S k .If x * is an optimal solution of (P), if the second order generalized Abadie constraint qualification holds at x * , and if for any critical direction y, the set cone where ⟨y, where , and ν * \ν * 0 is the complement of the set ν * 0 relative to the set ν * .
The proof applying (iv) is similar to that of (iii), but still we include it as follows: if x ∈ Q, then it follows from (3.1) and (3.2) that and Then, in light of the equivalent form for the strict B − ( b, ρ, η, ω, θ, p, r, s)−pseudo-invexity of B j (., v * ) at x * , we have It follows from (3.3), (3.21) and (3.22) that Thus, we have Since u * i > 0 for each i ∈ {1, • • •, p}, we conclude using Lemma 2.1 that Since x ∈ Q is arbitrary, x * is an optimal solution to (P).
We Remark that when functions f i , g i and H j have first-order derivatives, the established results seem to be specialized to B − (p, r)−invexities frameworks introduced by Antczak [1, 2, 3] and later generalized and investigated by Zalmai [41], Zalmai and Zhang [42] and others.gi(x * ) ≥ 0, g i (x * ) > 0 and H j for j ∈ {1, • • •, m} be differentiable at x * ∈ Q, and let there exist u * ∈ U = {u ∈ R p : u > 0, Σ p i=1 u i = 1} and v * ∈ R m + such that Suppose, in addition, that any one of the following assumptions holds: (i) E i (. ; x * , u * ) ∀ i ∈ {1, • • •, p} are B − (b, ρ, η, θ, p, r)-pseudoinvex with respect to η and b at x * ∈ X if there exist a function η : X × X → R n , a function b : X × X → [0, ∞), and real numbers r and p such that for all x ∈ X and z ∈ R n with b(x, x * ) > 0, and B j (. , v * ) ∀ j ∈ {1, • • •, m} are B − (b, ρ, η, θ, p, r)-quasiinvex with respect to η and b at x * ∈ X if there exist a function η : X × X → R n , a function b : X × X → [0, ∞), and real numbers r and p such that for all x ∈ X, z ∈ R n , and ρ(x, x * ) ≥ 0.