Second-Order Necessary Conditions for Set Constrained Nonsmooth Optimization Problems via Second-Order Projective Tangent Cones

Abstract

We present second-order necessary optimality
conditions for constrained minimization problems involving locally
Lipschitz functions. We extend some second-order necessary
conditions of extremum due to J.P Penot
for twice Fréchet differentiable objective functions subject
to a constraint set given just as a set. We describe the
second-order tangent sets and then the higher-order tangent sets to
the null-set of a mapping between two Banach spaces. Also we
characterize the higher-order adjacent sets and the related
higher-order tangent cones in Pavel sense to the null-sets of such a
mapping. The effectiveness of our results is illustrated on some
examples.