Generalized Self-Concordant Functions: A Recipe for Newton-Type Methods

Understand

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.