A necessary condition for the extremal problems under Gâteaux differentiability
The following so called smooth-convex optimization problem is considered: \(f_ 0(x,u)\to \inf\) subject to \(f_ i(x,u)\leq 0\), \(i=1,...,n\), \(F(x,u)=0\), \(u\in U\), where \(f_ i\) are functions on \(X\times U\), F: \(X\times U\to Y\) is an operator, U is a given subset of Y (X, Y are Banach spaces). The necessary local minimum condition is obtained under the following assumptions: \(f_ 0,f_ 1,...,f_ n\) are Fréchet-differentiable with respect to x and convex with respect to u, the operator F is Gâteaux- differentiable with respect to x and convex with respect to u. These assumptions are weaker than those assumed by \textit{A. D. Ioffe} and \textit{V. M. Tikhomirov} [Theory of extremal problems, (Russian) (1974; Zbl 0292.90042)].
scientific article; zbMATH DE number 4073831