Asymptotic Convergence Analysis of Some Inexact Proximal Point Algorithms for Minimization

In this paper, we prove that the inexact proximal point algorithm (PPA) in the form of Bonnans-Gilbert-Lemarechal-Sagastizabal's "general algorithmic pattern" (GAP-1) converges linearly under mild conditions. We also propose another variant (GAP-2) of inexact PPA that shares the same convergence property as GAP-1 but makes more sense numerically. Based on this essential result, we further prove the linear convergence for the outer iteration of the bundle method without requiring the differentiability of the objective function or the uniqueness of the solution. We also prove the linear convergence for the outer iteration of Correa-Lemarechal's "implementable form" of PPA and derive its rate.