Abstract
In the first part of this paper we point out some basic properties of the critical cones used in second-order optimality conditions and give a simple proof of a strong second-order necessary optimality condition by assuming a “modified” first-order Abadie constraint qualification. In the second part we give some insights on second-order constraint qualifications related to second-order local approximations of the feasible set.
Highlights
In the first part of this paper we point out some basic properties of the critical cones used in second-order optimality conditions and give a simple proof of a strong second-order necessary optimality condition by assuming a “modified” firstorder Abadie constraint qualification
In the second part we give some insights on second-order constraint qualifications related to second-order local approximations of the feasible set
The study of second-order optimality conditions for a nonlinear programming problem is a classical subject in mathematical programming theory: second-order optimality conditions have a long history that begins with a modern approach in the Karush’s Master Thesis (1939)
Summary
The study of second-order optimality conditions for a nonlinear programming problem is a classical subject in mathematical programming theory: second-order optimality conditions have a long history that begins with a modern approach in the Karush’s Master Thesis (1939). The generalized Lagrangian function (or Fritz John-Lagrange function) associated with (P) is defined as: L1(x, u0, u, w) = u0 f (x) + uigi(x) + w jh j(x) i=1 j=1 where u0 0, ui 0, ∀i ∈ M (ui, i = 0, 1, ..., m; w j ∈ R, j = 1, ..., p, not all zero) As it is well-known, the two more used first-order necessary optimality conditions for (P) are given by the two following statements. The theorem states the classical second-order necessary conditions for (local) optimality of a point x0 ∈ K. The (KKT) conditions hold at x0 with associated unique multipliers vectors u and w; the following additional second-order necessary conditions hold at x0 : z⊤∇2xL(x0, u, w)z 0, ∀z ∈ Z(x0). This has been remarked by Anitescu (2000), Arutyunov (1991), Baccari (2004)
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