Calculation of directional derivatives in max-min problems (Q1974719)

Max-min problems with bounded variables reduce to the maximization of the function \[ \varphi(x)= \min_y \{f(x,y)\mid y\in F(x)\} \] where \(F\) is a multivalued mapping. It is known that \(\varphi\) is nondifferentiable in the general case, but, at the same time, there exist classes of problems with directionally differentiable functions \(\varphi\) to which one can apply the methods of steepest descent. In this paper the authors have obtained formulas for calculating the directional derivatives of the function \(\varphi\) in a problem with an arbitrary, smooth goal function \(f\) and with functional constraints that are linear in \(y\) under the \(\Gamma\)-regularity condition.