Nondifferentiable mathematical programming involving "Equation missing" -invexity

Abstract

In this paper, we define new vector generalized convexity, namely nondifferentiable vector (G f , β f )-invexity, for a given locally Lipschitz vector function f . Basing on this new nondifferentiable vector generalized invexity, we have managed to deal with nondifferentiable nonlinear programming problems under some assumptions. Firstly, we present G-Karush-Kuhn-Tucker necessary optimality conditions for nonsmooth mathematical programming problems. With the new vector generalized invexity assumption, we also obtain G-Karush-Kuhn-Tucker sufficient optimality conditions for the same programming problems. Moreover, we establish duality results for this kind of multiobjective programming problems. In the end, a suitable example illustrates that the new optimality results are more useful for some class of optimization problems than the optimality conditions with invex functions. MSC: 90C26

Generalized Minimax Programming with Nondifferentiable (G,β)-Invexity

“…Lemma 2 (see [18]). Let be a real-valued Lipschitz continuous function defined on and denote the image of under by ( ); let : ( ) → R be a differentiable function such that ( ) is continuous on ( ) and ( ) ≥ 0 for each ∈ ( ).…”

“…Next, we derive -sufficient optimality conditions for problem (P) under the assumption of ( , )-invexity proposed in [18].…”

“…Note that the function ∘ may not be differentiable even if the function is differentiable. Yuan et al [18] introduced the ( , )-invexity concept for the locally Lipschtiz function . This ( , )-invexity extended Antczak'sinvexity concept to the nonsmooth case.…”

“…This ( , )-invexity extended Antczak'sinvexity concept to the nonsmooth case. In this paper, we deal with nondifferentiable generalized minimax programming problem (P) with the vector ( , )-invexity proposed in [18]. Here, the generalized minimax programming problem (P) is presented as follows:…”

Mixed Saddle Point and Its Equivalence with an Efficient Solution under Generalized (V, <i>p</i>)-Invexity

“…In [14] (Reddy and Mukherjee), some problems consisting of nonsmooth composite multiobjective programs have been treated with (V, ρ)-invexity type conditions and also vector saddle point theorems were obtained for composite programs. Yuan, Liu and Lai [15] defined new vector generalized convexity.…”