Composite convex optimization with global and local inexact oracles

We introduce new global and local inexact oracle concepts for a wide class of convex functions in composite convex minimization. Such inexact oracles naturally arise in many situations, including primal–dual frameworks, barrier smoothing, and inexact evaluations of gradients and Hessians. We also provide examples showing that the class of convex functions equipped with the newly inexact oracles is larger than standard self-concordant and Lipschitz gradient function classes. Further, we investigate several properties of convex and/or self-concordant functions under our inexact oracles which are useful for algorithmic development. Next, we apply our theory to develop inexact proximal Newton-type schemes for minimizing general composite convex optimization 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 linear and superlinear to 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 involving linear operators. Finally, we present some representative numerical examples to illustrate the benefit of the new algorithms.