On the regularity condition for the extremal problem under locally Lipschitz inclusion constraints (Q1074101)

The following vector optimization problem is considered: (P) optimize f(x) subject to \(x\in C\), \(0\in F(x)\), where (i) the optimality is taken with respect to a convex cone M, \(M\neq -M\), in the n-dimensional Euclidean space \({\mathbb{R}}^ n\), neither necessarily closed nor pointed, (ii) f is a locally Lipschitz function from a Hilbert space X into \({\mathbb{R}}^ n\), (iii) C is a closed subset of X, (iv) F is a locally Lipschitz set-valued mapping from X into a Hilbert space Y with non- empty, closed and convex images. The author extends the definition of Lagrange multipliers for (P) given by \textit{J. Zowe} and \textit{S. Kurcyusz} [ibid. 5, 49-62 (1979; Zbl 0401.90104)]. Then regularity conditions are imposed in order to ensure the existence of the Lagrange multipliers (they generalize the regularity condition given by Zowe and Kurcyusz). The main result reads: if (P) is regular at an optimal solution \(\bar x\) and the set valued mapping \((y^*,x)\to \partial_ xC^ F(y^*,x)\) has a closed graph in the \(w^*\)-topologies of \(X^*\) and \(Y^*\) and the strong topology of X, then the Lagrange multipliers of (P) exist. In the above \(\partial_ x\) is the generalized gradient in x and \(C^ F(y^*,x)=\sup \{<y^*,v>:\) \(v\in F(x)\}\) for \(y^*\in Y^*\), \(x\in X\). This result is applied to a class of discrete systems where the regularity assumption automatically holds.