Necessary conditions of optimality for abnormal problems with equality and inequality constraints
Aram Vladimirovich Arutyunov, Dmitry Yu. Karamzin, Фернандо Лобо Перейра · 2009
The aim of this paper is to derive second-order necessary conditions for an abnormal local minimizer of problem (P) which improve the ones presented earlier, [1, 2]. We consider the following mathematical programming problem: f(x) → min, F1(x) = 0, F2(x) ≤ 0. (P). Here, X is a linear space, and f: X → R1, F1: X → Rk1, and F2: X → Rk2 are given mappings (where k1 and k2 are fixed) assumed to be smooth in a sense to be specified. If x0 is abnormal, i.e. im ∂F1∂x (x0) 6 = Rk1, then it is not difficult to find a simple example revealing that the usual second-order necessary conditions do not hold in general. Moreover, in [2], meaningful second order necessary conditions for problem (P) were obtained without a priori normality assumptions of the point x0. In order to formulate these results from [2], let us introduce the Lagrange function L: X ×R1 ×Rk1 ×Rk2 → R1 of problem (P) L(x, λ) = λ0f(x) + 〈λ1, F1(x)〉+ 〈λ2, F2(x)〉, λ = (λ0, λ1, λ2), λ0 ∈ R1, λ1 ∈ Rk1, λ2 ∈ Rk2. Let x0 be a local minimizer for problem (P), and the mappings Fi and f be twice continuously differ-entiable. Denote by Λ(x0) the set of all Lagrange multipliers λ = (λ0, λ1, λ2) satisfying the Lagrange multipliers rule at the point x0: ∂L ∂x