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Rescaled proximal methods for linearly constrained convex problems

Abstract
We present an inexact interior point proximal method to solve linearly constrained convex problems. In fact, we derive a primal-dual algorithm to solve the KKT conditions of the optimization problem using a modified version of the rescaled proximal method. We also present a pure primal method. The proposed proximal method has as distinctive feature the possibility of allowing inexact inner steps even for Linear Programming. This is achieved by using an error criterion that bounds the subgradient of the regularized function, instead of using -subgradients of the original objective function. Quadratic convergence for LP is also proved using a more stringent error criterion.