Nondegenerate Second-order Necessary Conditions of Optimality for Nonlinear Optimization Problems
Aram Vladimirovich Arutyunov, Фернандо Лобо Перейра · 2006
(P ) Minimize f(x) subject to F1(x) ≤ 0, F2(x) = 0, x ∈ C, where X is a vector space, C ⊆ X is a given closed set, f : X → R, F1 : X → R1 and F2 : X → R2 are given smooth mappings, and k1 and k2 are also given positive integers. In order to state our main result, let x0 be our reference point and denote the cone of feasible descent directions for (P ) at x0 by K(x0), and the normal cone to C at x0 in the sense of Mordukhovich (see [6]) by NC(x0). Consider Λ(x0) as the set of Lagrange multipliers λ = (λ0, λ1, λ2), i.e., λ0 ≥ 0, λ1 ∈ IR1 , with λ1 ≥ 0 and 〈λ1, F1(x0)〉 = 0, and λ2 ∈ IR2 , satisfies 0 ∈ ∂L ∂x (x0, λ) + NC(x0). Our conditions proved in [2], where detailed definitions of all objects used here can be found, are briefly stated as follows: