  The paper proposes and justifies a new algorithm of the proximal Newton type
to solve a broad class of nonsmooth composite convex optimization problems
without strong convexity assumptions. Based on advanced notions and techniques
of variational analysis, we establish implementable results on the global
convergence of the proposed algorithm as well as its local convergence with
superlinear and quadratic rates. For certain structured problems, the obtained
local convergence conditions do not require the local Lipschitz continuity of
the corresponding Hessian mappings that is a crucial assumption used in the
literature to ensure a superlinear convergence of other algorithms of the
proximal Newton type. The conducted numerical experiments of solving the $l_1$
regularized logistic regression model illustrate the possibility of applying
the proposed algorithm to deal with practically important problems.
