Generalized ( ? , ? ) - ? -invexity and generalized ( ? , ? ) - ? -invariant-monotonicity

Abstract In this paper several kinds of ( ρ , θ ) - η -invariant-monotone functions and generalized ( ρ , θ ) - η -invariant-monotone functions are considered. For a given Frechet differentiable function f , ∇ f (gradient of f ), ( ρ , θ ) - η -invariant-monotonicity of ∇ f is related to ( ρ , θ ) - η -invexity of f . It is also shown that ( ρ , θ ) - η -invariant-monotonicity and generalized ( ρ , θ ) - η -invariant-monotonicity are proper generalizations of invariant-monotonicity and generalized invariant-monotonicity.