Tangent cone and contingent cone to the intersection of two closed sets
Let \(X\) be a normed vector space and \(C\subset X\) a nonempty closed set; let \(T_C(x)\) denote the tangent cone to \(C\) at \(x\in C\). For \(X=\mathbb R^n\), \textit{R. T. Rockafellar} [Nonlinear Anal., Theory Methods Appl.\ 3, 145--154 (1979; Zbl 0443.26010)] has shown that, if \(x\in C_1\cap C_2\), then \[ T_{C_1}(x)\cap T_{C_2}(x)\subset T_{C_1\cap C_2}(x), \] provided that \(T_{C_1}(x)\cap \mathrm{int}\,T_{C_2}(x)\neq \emptyset\). This last hypothesis is essential as illustrated by example 1.1 of this paper. The author studies sufficient conditions for the equation above to hold. He also studies the analogous inclusion for the contingent cones of the sets \(C_i\), and gives interesting characterizations of the cones, involving Clarke and Dini derivatives of the constraint functions. Finally, he presents new constraint qualifications for the Karush-Kuhn-Tucker conditions.
Mangasarian-Fromovitz constraint qualification
scientific article; zbMATH DE number 5736444
scientific article