A Family of Multi-Parameterized Proximal Point Algorithms

Abstract

In this paper, a multi-parameterized proximal point algorithm combining with a relaxation step is developed for solving convex minimization problem subject to linear constraints. We show its global convergence and sublinear convergence rate from the prospective of variational inequality. Preliminary numerical experiments on testing a sparse minimization problem from signal processing indicate that the proposed algorithm performs better than some well-established methods.

A New Insight on Augmented Lagrangian Method and Its Extensions

“…we can get the following tight result whose proof is similar to that of [1,14] and thus is omitted here for the sake of conciseness.…”

“…Obviously, the solution set of VI(θ, J , M), denoted by M * , is nonempty by the assumption on solution set of the problem (1). A straightforward conjecture is that convergence of P-ALM can be showed if its generated sequence is characterized by a similar inequality to (3) with an extra term converging to zero.…”

“…A benchmark method for solving the problem (1) is the so-called Augmented Lagrangian Method (ALM, [7,13]) which reads…”

“…where r > 0 denotes the penalty parameter and L(x, λ) = θ(x) − λ, Ax − b denotes the Lagrangian function associated with (1). Ignoring some constants, it is easy to check that the above subproblem amounts to x k+1 = arg min…”

“…In this note, we will propose and analyze a new Penalty ALM (abbreviated by P-ALM) for solving (1). For conciseness, we first present this P-ALM as the following:…”