  We study a class of monotone inclusions called "self-concordant inclusion"
which covers three fundamental convex optimization formulations as special
cases. We develop a new generalized Newton-type framework to solve this
inclusion. Our framework subsumes three schemes: full-step, damped-step and
path-following methods as specific instances, while allows one to use inexact
computation to form generalized Newton directions. We prove a local quadratic
convergence of both the full-step and damped-step algorithms. Then, we propose
a new two-phase inexact path-following scheme for solving this monotone
inclusion which possesses an
$\mathcal{O}\left(\sqrt{\nu}\log(1/\varepsilon)\right)$-worst-case
iteration-complexity to achieve an $\varepsilon$-solution, where $\nu$ is the
barrier parameter and $\varepsilon$ is a desired accuracy. As byproducts, we
customize our scheme to solve three convex problems: convex-concave
saddle-point, nonsmooth constrained convex program, and nonsmooth convex
program with linear constraints. We also provide three numerical examples to
illustrate our theory and compare with existing methods.
