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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.
Minimax optimization problems are an important class of optimization problems arising from modern machine learning and traditional research areas. While there have been many numerical algorithms for solving smooth convex-concave minimax problems, num
In this work, we present a globalized stochastic semismooth Newton method for solving stochastic optimization problems involving smooth nonconvex and nonsmooth convex terms in the objective function. We assume that only noisy gradient and Hessian inf
Newtons method for polynomial root finding is one of mathematics most well-known algorithms. The method also has its shortcomings: it is undefined at critical points, it could exhibit chaotic behavior and is only guaranteed to converge locally. Based
The Fast Proximal Gradient Method (FPGM) and the Monotone FPGM (MFPGM) for minimization of nonsmooth convex functions are introduced and applied to tomographic image reconstruction. Convergence properties of the sequence of objective function values
Decentralized optimization is a powerful paradigm that finds applications in engineering and learning design. This work studies decentralized composite optimization problems with non-smooth regularization terms. Most existing gradient-based proximal