Abstract
The proximal alternating direction method of multipliers (P-ADMM) is an efficient first-order method for solving the separable convex minimization problems. Recently, He et al. have further studied the P-ADMM and relaxed the proximal regularization matrix of its second subproblem to be indefinite. This is especially significant in practical applications since the indefinite proximal matrix can result in a larger step size for the corresponding subproblem and thus can often accelerate the overall convergence speed of the P-ADMM. In this paper, without the assumptions that the feasible set of the studied problem is bounded or the objective function’s component theta_{i}(cdot) of the studied problem is strongly convex, we prove the worst-case mathcal{O}(1/t) convergence rate in an ergodic sense of the P-ADMM with a general Glowinski relaxation factor gammain(0,frac{1+sqrt{5}}{2}), which is a supplement of the previously known results in this area. Furthermore, some numerical results on compressive sensing are reported to illustrate the effectiveness of the P-ADMM with indefinite proximal regularization.
Highlights
Let θi : Rni → (–∞, +∞] (i =, ) be two lower semicontinuous proper functions
We only focus our attention on the proximal alternating direction method of multipliers (P-ADMM)
The global convergence of the P-ADMM with γ = has been proved in [, ] for some concrete models of ( ), and in [ ], Xu and Wu√presented an elegant analysis of the global convergence of the for the general model. He et al [ ] have further studied the P-ADMM and get some substantial advances by relaxing the matrix G in the proximal regularization term of its second subproblem to be indefinite. This is quite preferred in practical applications since the indefinite proximal matrix can result in a larger step size for the subproblem and maybe accelerate the overall convergence speed of the P-ADMM
Summary
This work aims to solve the following two-block separable convex minimization problem: min θ (x ) + θ (x )|A x + A x = b , ( ) As one of the first-order methods, the following Algorithm , that is proximal alternating direction method of multipliers (P-ADMM) [ – ] is quite efficient for solving ( ) or related problems, especially for large scale case. He et al [ ] have further studied the P-ADMM and get some substantial advances by relaxing the matrix G in the proximal regularization term of its second subproblem to be indefinite.
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