Optimally linearizing the alternating direction method of multipliers for convex programming

He, Bingsheng; Ma, Feng; Yuan, Xiaoming

doi:10.1007/s10589-019-00152-3

Cited by 53 publications

(35 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Besides Lemma 2.5, we can derive another estimation for the term (By k+1 − By k ) (λ k − λ k+1 ), which is the result in [20,Lemma 4.4].…”

Section: Optimality Conditions For Problem (11)mentioning

confidence: 91%

“…Li et al [22] proved the global convergence. He et al [20] proposed a linearized version of ADMM with an indefinite proximal term. They considered the case that matrix S = 0 and α = 1 in (1.4), and generated the proximal matrix T as T = τ rI − βB B with r > β B B , τ ∈ (0.75, 1).…”

mentioning

confidence: 99%

“…Inspired by the variable metric semi-proximal ADMM [13,14] and the indefinite proximal ADMM [20], it is worth considering ADMM with a sequence of indefinite proximal matrices. We call the resulting ADMM a variable metric indefinite proximal ADMM (VMIP-ADMM).…”

mentioning

confidence: 99%

“…We call the resulting ADMM a variable metric indefinite proximal ADMM (VMIP-ADMM). Throughout our discussion, we always choose the stepsize α in (1.4c) be 1 as that in [20], which is good enough for such methods in practice and simple for the convergence analysis.…”

mentioning

confidence: 99%

“…• Let S and T k ≡ T be positive semidefinite matrices, VMIP-ADMM turns to be the semi-proximal ADMM (1.4); • Let {T k } be a positive semidefinite sequence, that is, T k 0 for all k. VMIP-ADMM becomes the variable semi-proximal ADMM; • Let S = 0, T k ≡ T be an indefinite matrix, VMIP-ADMM covers the indefiniteproximal ADMM proposed in [20].…”

mentioning

confidence: 99%

See 4 more Smart Citations

Alternating direction method of multipliers with variable metric indefinite proximal terms for convex optimization

Gu¹,

Yamashita²

2020

Numerical Algebra, Control &Amp; Optimization

View full text Add to dashboard Cite

This paper studies a proximal alternating direction method of multipliers (ADMM) with variable metric indefinite proximal terms for linearly constrained convex optimization problems. The proximal ADMM plays an important role in many application areas, since the subproblems of the method are easy to solve. Recently, it is reported that the proximal ADMM with a certain fixed indefinite proximal term is faster than that with a positive semidefinite term, and still has the global convergence property. On the other hand, Gu and Yamashita studied a variable metric semidefinite proximal AD-MM whose proximal term is generated by the BFGS update. They reported that a slightly indefinite matrix also makes the algorithm work well in their numerical experiments. Motivated by this fact, we consider a variable metric indefinite proximal ADMM, and give sufficient conditions on the proximal terms for the global convergence. Moreover, based on the BFGS update, we propose a new indefinite proximal term which can satisfy the conditions for the global convergence. Experiments on several datasets demonstrated that our proposed variable metric indefinite proximal ADMM outperforms most of the comparison proximal ADMMs.

show abstract

“…Besides Lemma 2.5, we can derive another estimation for the term (By k+1 − By k ) (λ k − λ k+1 ), which is the result in [20,Lemma 4.4].…”

Section: Optimality Conditions For Problem (11)mentioning

confidence: 91%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%