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 structural 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.
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