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
Highlights
We focus on the following convex minimization problem with linear equality constraints min f (x) s.t
The work in [22] proposed a linearized augmented Lagrangian method (ALM) aiming at linearizing the x-subproblem such that its closed-form solution can be derived
Remark 2: Note that 1/s in step 5 plays a role of penalty parameter in ALM, while r can be treated as the Algorithm 1 RM-proximal point algorithm (PPA) for Solving Problem (1)
Summary
We focus on the following convex minimization problem with linear equality constraints min f (x) s.t. In many real applications, [2], [4], [12], the coefficient matrix A is not an identity matrix (or does not satisfy AAT = Im), which makes it difficult even infeasible for solving this subproblem of ALM To overcome such difficulty, the work in [22] proposed a linearized ALM aiming at linearizing the x-subproblem such that its closed-form solution can be derived. [5], [8], [12], [24] turn out to be special cases of our multi-parameterized proximal matrix (see Remark 2 for details) In this sense, our proposed algorithm can be viewed as a general customized PPA for solving problem (1). For any x ∈ Rn, the symbol x means the standard Euclidean norm of x
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