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 optimiza- tion 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 un- derlying functionals fall into our generalized self-concordant function class. First, we introduce the class of generalized self-concordant functions, which covers standard self-concordant 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 the generalized self- concordant functions. We provide an explicit step-size for the 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. 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 assumption. Finally, we verify our theoretical results via several numerical examples, and compare them with existing methods.
NewVEM: A Newton Vertex Exchange Method for a Class of Constrained Self-Concordant Minimization Problems
Ling Liang Department of Mathematics University of Tennessee, Knoxville 1403 Circle Drive, Knoxville, TN 37916, USA Kim-Chuan Toh Department of Mathematics；Institute of Operations Research；Analytics National University of Singapore 10 Lower Kent Ridge Road, Singapore 119076 Haizhao Yang Department of Mathematics；Department of Computer Science University of Maryland, College Park 4176 Campus Drive, College Park, MD 20742, USA
Polynomial iteration complexity of a path-following smoothing Newton method for symmetric cone programming
Institute of Mathematics, Hebei University of Technology（河北工业大学数学研究所）；State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and the University of Chinese Academy of Sciences（中国科学院数学与系统科学研究院、中国科学院大学数学科学学院）；School of Mathematical Sciences and LPMC, Nankai University（南开大学数学科学学院）
