Abstract
In this paper, we introduce the concepts of KT-G-invexity and WD$-G-invexity for the considered differentiable optimization problem with inequality constraints. Using KT-G-invexity notion, we prove new necessary and sufficient optimality conditions for a new class of such nonconvex differentiable optimization problems. Further, the so-called G-Wolfe dual problem is defined for the considered extremum problem with inequality constraints. Under WD-G-invexity assumption, the necessary and sufficient conditions for weak duality between the primal optimization problem and its G-Wolfe dual problem are also established.
Highlights
In the paper, we consider the following constrained optimization problem: Further, we denote an index set of active inequality constraints at point x ∈ X as follows:J (x) = {j ∈ J : gj (x) = 0} .subject to minimize f (x) gj(x) ≦ 0, j ∈ J = {1, ...m}, (P)x ∈ X, where f : X → R and gj : X → R, j ∈ J, are differentiable functions defined on a nonempty open set X ⊂ Rn.For the purpose of simplifying our presentation, we will introduce some notation which will be used frequently throughout this paper
In this paper, following Martin [14] and Antczak [4], we introduce the definitions of KT -G-invexity and W D-G-invexity notions for the considered differentiable optimization problem (P) with inequality constraints
The so-called KT -G-invexity and W D-G-invexity notions defined for the considered differentiable optimization problem (P) with inequality constraints are generalizations the Ginvexity notions introduced by Antczak [4] and the concepts of KT -invexity and W D-invexity introduced by Martin [14], respectively
Summary
We consider the following constrained optimization problem: Further, we denote an index set of active inequality constraints at point x ∈ X as follows:. In [4], Antczak generalized Hanson’s definition of a (differentiable) invex function and he introduced the concept of G-invexity for differentiable constrained optimization problems He formulated and proved new necessary optimality conditions of G-F. In this paper, following Martin [14] and Antczak [4], we introduce the definitions of KT -G-invexity and W D-G-invexity notions for the considered differentiable optimization problem (P) with inequality constraints For such an extremum problem (P), we define the socalled G-Karush-Kuhn-Tucker point (G-KKT point) and we prove that every G-Karush-KuhnTucker point of problem (P) is its global minimizer if and only if problem (P) is KT -G-invex. W D-G-invexity in proving the necessary and sufficient optimality conditions and the necessary and sufficient conditions for weak duality for a new class of nonconvex differentiable optimization problems
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