We study the smooth structure of convex functions by generalizing a powerful concept so-called self-concordance introduced by Nesterov and Nemirovskii in the early 1990s to a broader class of convex functions which we call generalized self-concordant functions. This notion allows us to develop a unified framework for designing Newton-type methods to solve convex optimization problems. The proposed theory provides a mathematical tool to analyze both local and global convergence of Newton-type methods without imposing unverifiable assumptions as long as the underlying functionals fall into our class of generalized self-concordant functions. First, we introduce the class of generalized self-concordant functions which covers the class of standard selfconcordant functions as a special case. Next, we establish several properties and key estimates of this function class which can be used to design numerical methods. Then, we apply this theory to develop several Newton-type methods for solving a class of smooth convex optimization problems involving generalized self-concordant functions. We provide an explicit step-size for a damped-step Newton-type scheme which can guarantee a global convergence without performing any globalization strategy. We also prove a local quadratic convergence of this method and its full-step variant without requiring the Lipschitz continuity of the objective Hessian mapping. Then, we extend our result to develop proximal Newton-type methods for a class of composite convex minimization problems involving generalized self-concordant functions. We also achieve both global and local convergence without additional assumptions. Finally, we verify our theoretical results via several numerical examples, and compare them with existing methods.

Abstract. Many scientific and engineering applications feature nonsmooth convex minimization problems over convex sets. In this paper, we address an important instance of this broad class where we assume that the nonsmooth objective is equipped with a tractable proximity operator and that the convex constraint set affords a self-concordant barrier. We provide a new joint treatment of proximal and self-concordant barrier concepts and illustrate that such problems can be efficiently solved, without the need of lifting the problem dimensions, as in disciplined convex optimization approach. We propose an inexact path-following algorithmic framework and theoretically characterize the worst-case analytical complexity of this framework when the proximal subproblems are solved inexactly. To show the merits of our framework, we apply its instances to both synthetic and real-world applications, where it shows advantages over standard interior point methods. As a by-product, we describe how our framework can obtain points on the Pareto frontier of regularized problems with self-concordant objectives in a tuning free fashion.

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In this paper, we propose a new inexact dual decomposition algorithm for solving separable convex optimization problems. This algorithm is a combination of three techniques: dual Lagrangian decomposition, smoothing and excessive gap. The algorithm has low computational complexity since it consists in only one primal step and two dual steps at each iteration and allows one to solve the subproblem of each component inexactly and in parallel. Moreover, the algorithmic parameters are updated automatically without any tuning strategy as it happens in augmented Lagrangian approaches. We analyse the convergence of the algorithm and estimate its O (1/ε) analytical worst-case complexity for both the primal-dual suboptimality and the primal feasibility violation, where ε is a given accuracy. Extensive numerical tests confirm that our method is numerically more efficient than the classical decomposition methods from the literature.

The self-concordant-like property of a smooth convex function is a new analytical structure that generalizes the self-concordant notion. While a wide variety of important applications feature the selfconcordant-like property, this concept has heretofore remained unexploited in convex optimization. To this end, we develop a variable metric framework of minimizing the sum of a "simple" convex function and a self-concordant-like function. We introduce a new analytic step-size selection procedure and prove that the basic gradient algorithm has improved convergence guarantees as compared to "fast" algorithms that rely on the Lipschitz gradient property. Our numerical tests with real-data sets show that the practice indeed follows the theory.