Kybernetika 46 no. 3, 472-487, 2010

On second-order Taylor expansion of critical values

Stephan Bütikofer, Diethard Klatte and Bernd Kummer


Studying a critical value function $\vi$ in parametric nonlinear programming, we recall conditions guaranteeing that $\vi$ is a $C^{1,1}$ function and derive second order Taylor expansion formulas including second-order terms in the form of certain generalized derivatives of $D \vi$. Several specializations and applications are discussed. These results are understood as supplements to the well-developed theory of first- and second-order directional differentiability of the optimal value function in parametric optimization.


Taylor expansion, parametric programs, critical value function, generalized derivatives, envelope theorems, Lipschitz stability, $C^{1,1}$ optimization


49J52, 49K40, 90C31, 65K10