On the eta -(1,2) approximated optimization problems

Let X be a nonempty subset of Rn, x 0 be an interior point of X, f : X → R be a differentiable function at x 0 , g : X → Rm be a twice differentiable function at x 0 and η : X × X → Rn be a function. In this paper, we attach to the optimization problem (P)    min f(x) x ∈ X g(x) 5 0, the (1, 2)-η- approximated optimization problem (AP)    min F(x) := f(x 0 ) + ∇f(x 0 ), η(x, x0 ) x ∈ X G(x) := g(x 0 ) + [∇g(x 0 )](η(x, x0 ))+ + 1 2 [∇2g(x 0 )](η(x, x0 )), η(x, x0 ) 5 0, and we will study the relations between the optimal solutions of Problem (P), the optimal solutions of Problem (AP), the saddle points of Problem (P) and saddle points of Problem (AP).