(p,r)-Invex Sets and Functions

Abstract

Notions of invexity of a function and of a set are generalized. The notion of an invex function with respect to η can be further extended with the aid of p-invex sets. Slight generalization of the notion of p-invex sets with respect to η leads to a new class of functions. A family of real functions called, in general, p r -preinvex functions with respect to η (without differentiability) or p r -invex functions with respect to η (in the differentiable case) is introduced. Some (geometric) properties of these classes of functions are derived. Sufficient optimality conditions are obtained for a nonlinear programming problem involving p r -invex functions with respect to η.  2001 Academic Press Key Words: p r -invex set with respect to η; r-invex set with respect to η; p r -pre-invex function with respect to η p r -invex function with respect to η.

Criterion for Generalized Weakly Fuzzy Invex Monotonocities

“…A step forward Yang et al [4] found that there were some errors in [2] and modified the results of RuizGarzon et al [2] and they also proposed the notion of strong pseudo invex monotonicity and quasi-invex monotonicity. In addition, Antczak [5,6] and Suneja et al [7] deliberated the properties and execution of preinvex functions and their generalizations for the nonlinear programming problems. Aghezzaf and Hachimi [8] presented the differentiable type I functions and derived the appropriate duality results for a Mond-Weir type dual.…”

Generalized hybrid $B-(b,\rho,\theta,\tilde{p},\tilde{r})-$invexities and efficiency conditions for multiobjective fractional programming

“…Verma [28] introduced a higher order exponential type generalization -B − (ρ, η, θ,p,r)-invexities -to exponential type first order B − (p,r)−invexities by Antczak [1], and applied to explore parametric sufficient efficiency conditions to semiinfinite minimax fractional programming problems, while Verma [27] introduced and investigated second order (Φ, Ψ, ρ, η, θ)−invexities to the context of parametric sufficient optimality conditions in semiinfinite discrete minimax fractional programming problems. The contribution of Antczak [1][2][3] on first order B−(p, r)−invexities is enormous to the context of nonlinear mathematical programming problems, which have been applied to a class of global parametric sufficient optimality conditions based on first order B −(p, r)−invexities for semiinfinite discrete minimax fractional programming problems. This was followed by Zalmai [37,38] who generalized B − (p, r)−invexities introduced by Antczak [1][2][3], and applied to a class of global parametric sufficient optimality criteria using various assumptions for semiinfinite discrete minimax fractional programming problems.…”

“…The contribution of Antczak [1][2][3] on first order B−(p, r)−invexities is enormous to the context of nonlinear mathematical programming problems, which have been applied to a class of global parametric sufficient optimality conditions based on first order B −(p, r)−invexities for semiinfinite discrete minimax fractional programming problems. This was followed by Zalmai [37,38] who generalized B − (p, r)−invexities introduced by Antczak [1][2][3], and applied to a class of global parametric sufficient optimality criteria using various assumptions for semiinfinite discrete minimax fractional programming problems. 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 fractional programming problems for weakly ε−efficient solutions, while Kim et al [8] have established some ε−optimality conditions for multiobjective fractional optimization problems.…”

On G-invexity-type nonlinear programming problems

“…The term invex (which means invariant convex) was suggested later by Craven [10]. Over the years, many generalizations of this concept have been given in the literature (see, for instance, [1], [2], [3], [5], [6], [7], [8], [9], [12], [13], [14], [15], [16], and others).…”