Remarks on second order generalized derivatives for differentiable functions with Lipschitzian Jacobian
Davide La Torre, Matteo Rocca · 2003
Many definitions of second-order generalized derivatives have been introduced to obtain optimality conditions for optimization problems with C1,1 data. The aim of this note is to show some relations among these definitions. The class of C1,1 functions, that is the class of differentiable functions with Lip-schitzian Jacobian, was first brought to attention by Hiriart-Urruty, Strodiot and Hien Nguyen [8]. The need for investigating such functions, as pointed out in [8] and [9], comes from the fact that several problems of applied mathematics including variational inequalities, semi-infinite programming, penalty functions, augmented La-grangian, proximal point methods, iterated local minimization by decomposition etc. involve differentiable functions with no hope of being twice differentiable. Many second-order generalized derivatives have been introduced to obtain optimality conditions for optimization problems with C1,1 data and for this class of functions numerical methods and minimization algorithms have been proposed too [20, 21]. In this paper we will give some relations among several definitions that one can find in literature and we