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

Adona, Vando A.; Gonçalves, Max L. N.; Melo, Jefferson G.

doi:10.1007/s10898-018-0697-z

Cited by 13 publications

(16 citation statements)

References 38 publications

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“…horizon length T of the computation of each iteration are summarised in Table 1 3 , and the memory requirement of the algorithm is O(T ), as only two bandwidth-2 matrices (defined in Appendix D) and the variables themselves require storage. The ADMM iteration ( 6) is a particular case of 'Generalized ADMM', for which it was demonstrated in [30] that O(1/ ) iterations are required to meet criterion (7) for a given problem. To the best of the authors' knowledge there are no results that demonstrate how the iteration complexity varies with horizon length (i.e.…”

Section: B Variable Updates and Algorithm Complexitymentioning

confidence: 99%

Optimal Power Allocation in Battery/Supercapacitor Electric Vehicles Using Convex Optimization

East

Cannon

2020

IEEE Trans. Veh. Technol.

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This paper presents a framework for optimizing the power allocation between a battery and supercapacitor in an electric vehicle energy storage system. A convex optimal control formulation is proposed that minimizes total energy consumption whilst enforcing hard constraints on power output and total energy stored in the battery and supercapacitor. An alternating direction method of multipliers (ADMM) algorithm is proposed, for which computational and memory requirements scale linearly with the length of the prediction horizon (and can be reduced using parallel processing). The optimal controller is compared with a low-pass filter against an all-battery baseline in numerical simulations, where it is shown to provide significant improvement in battery degradation (inferred through reductions of 71.4% in peak battery power, 21.0% in root-mean-squared battery power, and 13.7% in battery throughput), and a reduction of 5.7% in energy consumption. It is also shown that the ADMM algorithm can solve the optimization problem in a fraction of a second for prediction horizons of more than 15 minutes, and is therefore a promising candidate for online receding-horizon control.

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Section: B Variable Updates and Algorithm Complexitymentioning

confidence: 99%

Optimal Power Allocation in Battery/Supercapacitor Electric Vehicles Using Convex Optimization

East

Cannon

2020

IEEE Trans. Veh. Technol.

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“…We now state an ergodic iteration-complexity bound for the modified HPE framework, whose proof can be found in [21,Theorem 2.3] (see also [9,Theorem 3.4] for a more general result). Theorem 2.5.…”

Section: Modified Hpe Frameworkmentioning

confidence: 99%

“…Subsequently, [6] and [7] analyzed ergodic and pointwise iteration-complexities of a partially proximal ADMM, respectively. We refer the reader to [21,22,23,24,25,26,27,28] where iteration-complexities of other ADMM variants have been considered. The complexity analyses of the present paper are based on showing that the proposed method falls within the setting of a hybrid proximal extragradient framework whose iteration-complexity bounds were established in [9].…”

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confidence: 99%

A Partially Inexact Proximal Alternating Direction Method of Multipliers and Its Iteration-Complexity Analysis

Adona

Gonçalves

Melo

2019

J Optim Theory Appl

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This paper proposes a partially inexact alternating direction method of multipliers for computing approximate solution of a linearly constrained convex optimization problem. This method allows its first subproblem to be solved inexactly using a relative approximate criterion, whereas a proximal term is added to its second subproblem in order to simplify it. A stepsize parameter is included in the updating rule of the Lagrangian multiplier to improve its computational performance. Pointwise and ergodic interation-complexity bounds for the proposed method are established. To the best of our knowledge, this is the first time that complexity results for an inexact ADMM with relative error criteria has been analyzed. Some preliminary numerical experiments are reported to illustrate the advantages of the new method.2000 Mathematics Subject Classification: 47H05, 49M27, 90C25, 90C60, 65K10. and jefferson@ufg.br). The work of these authors was supported in part by CAPES, CNPq Grants 302666/2017-6 and 406975/2016-7.2) to provide pointwise and ergodic iteration-complexity bounds for the proposed method;3) to illustrate, by means of numerical experiments, the efficiency of the new method for solving some real-life applications.Iteration-complexity results have been considered in the literature for most of exact ADMM variants. Paper [20] presented an ergodic iteration-complexity analysis of the ADMM. Subsequently, [6] and [7] analyzed ergodic and pointwise iteration-complexities of a partially proximal ADMM, respectively. We refer the reader to [21,22,23,24,25,26,27,28] where iteration-complexities of other ADMM variants have been considered. The complexity analyses of the present paper are based on showing that the proposed method falls within the setting of a hybrid proximal extragradient framework whose iteration-complexity bounds were established in [9]. To the best of our knowledge, this work is the first one to present iteration-complexity results for an inexact ADMM with relative error.This paper is organized as follows. Section 2 contains some preliminary results and it is divided into two subsections. The first subsection presents our notation and basic definitions while the second one recalls a modified HPE framework and its basic iteration-complexity results. Section 3 introduces the partially inexact proximal ADMM and establishes its iteration-complexity bounds. Section 4 is devoted to the numerical experiments. Preliminary ResultsThis section is divided into two subsections. The first one presents our notation and basic results. The second subsection recalls a modified HPE framework and its iteration-complexity bounds. Notation and Basic DefinitionsThis section presents some definitions, notation and basic results used in this paper.The p−norm (p ≥ 1) and maximum norm of z ∈ R n are denoted, respectively, by z p = ( n i=1 |z i | p )

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“…where f : R n → (−∞, ∞] and g : R p → (−∞, ∞] are proper, closed and convex functions, A ∈ R m×n , B ∈ R m×p , and b ∈ R m . Convex optimization problems with a separable structure such as (1) appear in many applications areas such as machine learning, compressive sensing and image processing. The augmented Lagrangian method (see, e.g., [7]) attempts to solve (1) directly without taking into account its particular structure.…”

Section: Introductionmentioning

confidence: 99%

“…• the generalized proximal ADMM (for short G-P-ADMM) with the relaxation factor α := τ + 1 when θ = 1; see [1,11]. The proof of the latter fact can be found, for example, in [23,Remark 5.8];…”

Section: Introductionmentioning

confidence: 99%

An inexact version of the symmetric proximal ADMM for solving separable convex optimization

Adona¹,

Gonçalves²

2020

Preprint

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In this paper, we propose and analyze an inexact version of the symmetric proximal alternating direction method of multipliers (ADMM) for solving linearly constrained optimization problems. Basically, the method allows its first subproblem to be solved inexactly in such way that a relative approximate criterion is satisfied. In terms of the iteration number k, we establish global O(1/ √ k) pointwise and O(1/k) ergodic convergence rates of the method for a domain of the acceleration parameters, which is consistent with the largest known one in the exact case. Since the symmetric proximal ADMM can be seen as a class of ADMM variants, the new algorithm as well as its convergence rates generalize, in particular, many others in the literature. Numerical experiments illustrating the practical advantages of the method are reported. To the best of our knowledge, this work is the first one to study an inexact version of the symmetric proximal ADMM.

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

Cited by 13 publications

References 38 publications

Optimal Power Allocation in Battery/Supercapacitor Electric Vehicles Using Convex Optimization

Optimal Power Allocation in Battery/Supercapacitor Electric Vehicles Using Convex Optimization

A Partially Inexact Proximal Alternating Direction Method of Multipliers and Its Iteration-Complexity Analysis

An inexact version of the symmetric proximal ADMM for solving separable convex optimization

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