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We investigate the proximal point algorithm (PPA) and its inexact extensionsunder an error bound condition, which guarantees a global linear convergence ifthe proximal regularization parameter is larger than the error bound conditionparameter. We propose an adaptive generalized proximal point algorithm (AGPPA),which adaptively updates the proximal regularization parameters based on someimplementable criteria. We show that AGPPA achieves linear convergence withoutany knowledge of the error bound condition parameter, and the rate only differsfrom the optimal one by a logarithm term. We apply AGPPA on convex minimizationproblem and analyze the iteration complexity bound of the resulting algorithm.Our framework and the complexity results apply to arbitrary linearly convergentinner solver and allows a hybrid with any locally fast convergent method. Weillustrate the performance of AGPPA by applying it to solve large-scale linearprogramming (LP) problem. The resulting complexity bound has a weakerdependence on the Hoffman constant and scales with the dimension better thanlinearized ADMM. In numerical experiments, our algorithm demonstrates improvedperformance in obtaining solution of medium accuracy on large-scale LP problem.

An adaptive proximal point algorithm framework and application to large-scale optimization