On the Convergence Properties of a Majorized Alternating Direction Method of Multipliers for Linearly Constrained Convex Optimization Problems with Coupled Objective Functions

Cui, Ying; Li, Xudong; Sun, Defeng; Toh, Kim-Chuan

doi:10.1007/s10957-016-0877-2

Cited by 51 publications

(60 citation statements)

References 17 publications

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“…Proof. The inclusion (7) follows from the last statement in Lemma 3.1. Let us now show that (8) holds.…”

Section: Let Us First Introduce the Elements Required By The Setting mentioning

confidence: 91%

“…In section 3.2, we will show that a generalized ADMM can be seen as an instance of the HPE framework specifying, in particular, how this triple (z k ,z k , η k ) can be obtained. Third, if M is positive definite and σ = η 0 = 0, then (8) implies that η k = 0 and z k =z k for every k, and hence that M (z k−1 − z k ) ∈ T (z k ) in view of (7). Therefore, the HPE error conditions (7)-(8) can be viewed as a relaxation of an iteration of the exact proximal point method.…”

Section: A Hpe-type Frameworkmentioning

confidence: 99%

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Iteration-complexity analysis of a generalized alternating direction method of multipliers

2018

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This paper analyzes the iteration-complexity of a generalized alternating direction method of multipliers (G-ADMM) for solving linearly constrained convex problems. This ADMM variant, which was first proposed by Bertsekas and Eckstein, introduces a relaxation parameter α ∈ (0, 2) into the second ADMM subproblem. Our approach is to show that the G-ADMM is an instance of a hybrid proximal extragradient framework with some special properties, and, as a by product, we obtain ergodic iteration-complexity for the G-ADMM with α ∈ (0, 2], improving and complementing related results in the literature. Additionally, we also present pointwise iteration-complexity for the G-ADMM.2000 Mathematics Subject Classification: 47H05, 49M27, 90C25, 90C60, 65K10.

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“…Proof. The inclusion (7) follows from the last statement in Lemma 3.1. Let us now show that (8) holds.…”

Section: Let Us First Introduce the Elements Required By The Setting mentioning

confidence: 91%

Section: A Hpe-type Frameworkmentioning

confidence: 99%

Iteration-complexity analysis of a generalized alternating direction method of multipliers

2018

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“…Hence, (a) follows from the above inequality, the fact that 0 ∈ T (z * ) and r k ∈ T (z k ) (see (10)), and the monotonicity of T . (b) Using (a), (8) and Assumption 2.1, we find…”

Section: A Proof Of Theorems 22 and 23mentioning

confidence: 99%

Pointwise and Ergodic Convergence Rates of a Variable Metric Proximal Alternating Direction Method of Multipliers

Gonçalves

Alves

Melo

2018

J Optim Theory Appl

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In this paper, we obtain global O(1/ √ k) pointwise and O(1/k) ergodic convergence rates for a variable metric proximal alternating direction method of multipliers (VM-PADMM) for solving linearly constrained convex optimization problems. The VM-PADMM can be seen as a class of ADMM variants, allowing the use of degenerate metrics (defined by noninvertible linear operators). We first propose and study nonasymptotic convergence rates of a variable metric hybrid proximal extragradient (VM-HPE) framework for solving monotone inclusions. Then, the abovementioned convergence rates for the VM-PADMM are obtained essentially by showing that it falls within the latter framework. To the best of our knowledge, this is the first time that global pointwise (resp. pointwise and ergodic) convergence rates are obtained for the VM-PADMM (resp. VM-HPE framework).2000 Mathematics Subject Classification: 90C25, 90C60, 49M27, 47H05, 47J22, 65K10.Key words: alternating direction method of multipliers, variable metric, pointwise and ergodic convergence rates, hybrid proximal extragradient method, convex program.• by letting H k = βI, β > 0, and θ = 1, the resulting method becomes similar to Algorithm 7 in [2], where a composite structure of f is considered and ergodic convergence rates were obtained under the additional conditions that B = I in (1) and the dual solution set of (1) be bounded.Contributions of the paper. We obtain an O(1/k) global convergence rate for an ergodic sequence associated to the VM-PADMM (2)-(4) with θ ∈ (0, ( √ 5 + 1)/2), which provides, for given tolerances ρ, ε > 0, triples (x, y,γ), (r x , r y , r γ ) and scalars ε x , ε y ≥ 0 such that r x ∈ ∂ εx f (x) − A * γ , r y ∈ ∂ εy g(y) − B * γ , r γ = Ax + By − b, max r x * x , r y * y , r γ * γ ≤ ρ, ε x +ε y ≤ ε,

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“…With (34) it is easy to verify that Assumptions 2 and 3 can be satisfied. In the simulation setup, we let the number of agents N = 10 with connections |E | = 15, the number of training samples b i = 10 (∀i ∈ V), input and output dimension n = 10, ν = 1.…”

Section: A Convergence Experimentsmentioning

confidence: 99%

Mobility-Aware Content Preference Learning in Decentralized Caching Networks

Xiao

Skoglund

2020

IEEE Trans. Cogn. Commun. Netw.

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Due to the drastic increase of mobile traffic, wireless caching is proposed to serve repeated requests for content download. To determine the caching scheme for decentralized caching networks, the content preference learning problem based on mobility prediction is studied. We first formulate preference prediction as a decentralized regularized multi-task learning (DRMTL) problem without considering the mobility of mobile terminals (MTs). The problem is solved by a hybrid Jacobian and Gauss-Seidel proximal multi-block alternating direction method (ADMM) based algorithm, which is proven to conditionally converge to the optimal solution with a rate O(1/k). Then we use the tool of Markov renewal process to predict the moving path and sojourn time for MTs, and integrate the mobility pattern with the DRMTL model by reweighting the training samples and introducing a transfer penalty in the objective. We solve the problem and prove that the developed algorithm has the same convergence property but with different conditions. Through simulation we show the convergence analysis on proposed algorithms. Our real trace driven experiments illustrate that the mobilityaware DRMTL model can provide a more accurate prediction on geography preference than DRMTL model. Besides, the hit ratio achieved by most popular proactive caching (MPC) policy with preference predicted by mobility-aware DRMTL outperforms the MPC with preference from DRMTL and random caching (RC) schemes.

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On the Convergence Properties of a Majorized Alternating Direction Method of Multipliers for Linearly Constrained Convex Optimization Problems with Coupled Objective Functions

Cited by 51 publications

References 17 publications

Iteration-complexity analysis of a generalized alternating direction method of multipliers

Iteration-complexity analysis of a generalized alternating direction method of multipliers

Pointwise and Ergodic Convergence Rates of a Variable Metric Proximal Alternating Direction Method of Multipliers

Mobility-Aware Content Preference Learning in Decentralized Caching Networks

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