Interior proximal and multiplier methods based on second order homogeneous kernels

We study a class of interior proximal algorithms and nonquadratic multiplier methods for solving convex programs, where the usual proximal quadratic term is replaced by an homogeneous functional of order two, defined in terms of a convex function. We prove, under mild assumptions, several new convergence results in both the primal and dual framework allowing also for approximate minimization. In particular, we introduce a new class of interior proximal methods which is globally convergent and allows for generating C-infinity multiplier methods with bounded Hessians which exhibit strong convergence properties. We also consider linearly constrained convex problems and establish global quadratic convergence rates results for linear programs. We then study in detail a particular realization of these algorithms, leading to a new class of logarithmic-quadratic interior point algorithms which are shown to enjoy several attractive properties for solving constrained convex optimization problems.