An envelope-like effect of infinitely many inequality constraints on second-order necessary conditions for minimization problems (Q1113078)

Let X, V and W be Banach spaces. The author considers the following mathematical programming problem: \[ (P)\quad \min imize\quad f(x)\quad subject\quad to\quad g(x)\in K\quad and\quad h(x)=0, \] where f: \(X\to {\mathbb{R}}\), g: \(X\to V\) and h: \(X\to W\) are of class \(C^ 2\) and K is a closed convex cone in V with nonempty interior. He presents new second-order necessary conditions for optimality in problem (P) which are stated in both the primal and the dual form. The latter involves a new term besides the second derivative of the Lagrange function and therefore improves earlier results for (P) given by \textit{A. Ben-Tal} and \textit{J. Zowe} [Math. Program. Study 18, 39-76 (1982; Zbl 0494.49020)]. Finally, the author applies his result to the problem of minimization of a sup-type function.