First order optimality conditions for nonsmooth quadratic problems in Banach spaces

A new design called nonsmooth quadratic problem (NQP) for nondifferentiable quadratic optimization by duality mappings and selfadjoint operators in Banach spaces is introduced in this paper.We first extend Polyak's Theorem in Banach spaces.Next, we use a generalized version of the celebrated S-Lemma in Banach spaces.Then, we investigate the first order optimality conditions (KKT) of the problem by using the Clarke and the lower global subdifferentials.