  We introduce new global and local inexact oracle concepts for a wide class of
convex functions in composite convex minimization. Such inexact oracles
naturally come from primal-dual framework, barrier smoothing, inexact
computations of gradients and Hessian, and many other situations. We also
provide examples showing that the class of convex functions equipped with the
newly inexact second-order oracles is larger than standard self-concordant as
well as Lipschitz gradient function classes. Further, we investigate several
properties of convex and/or self-concordant functions under the inexact
second-order oracles which are useful for algorithm development. Next, we apply
our theory to develop inexact proximal Newton-type schemes for minimizing
general composite convex minimization problems equipped with such inexact
oracles. Our theoretical results consist of new optimization algorithms,
accompanied with global convergence guarantees to solve a wide class of
composite convex optimization problems. When the first objective term is
additionally self-concordant, we establish different local convergence results
for our method. In particular, we prove that depending on the choice of
accuracy levels of the inexact second-order oracles, we obtain different local
convergence rates ranging from $R$-linear and $R$-superlinear to $R$-quadratic.
In special cases, where convergence bounds are known, our theory recovers the
best known rates. We also apply our settings to derive a new primal-dual method
for composite convex minimization problems. Finally, we present some
representative numerical examples to illustrate the benefit of our new
algorithms.
