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

JOURNAL OF GLOBAL OPTIMIZATION 2019年第2期73卷 331-348页

作者： Adona, V. A. Goncalves, M. L. N. Melo, J. G. Univ Fed Goias IME Campus 2Caixa Postal 131 BR-74001970 Goiania Go Brazil

This paper analyzes the iteration-complexity of a generalized alternating direction method of multipliers (G-ADMM) for solving separable linearly constrained convex optimization problems. This ADMM variant, first proposed by Bertsekas and Eckstein, introduces a relaxation parameter into the second ADMM subproblem in order to improve its computational performance. It is shown that, for a given tolerance epsilon>0, the G-ADMM with (0,2) provides, in at most O(1/epsilon 2) iterations, an approximate solution of the Lagrangian system associated to the optimization problem under consideration. It is further demonstrated that, in at most O(1/epsilon) iterations, an approximate solution of the Lagrangian system can be obtained by means of an ergodic sequence associated to a sequence generated by the G-ADMM with (0,2]. Our approach consists of interpreting the G-ADMM as an instance of a hybrid proximal extragradient framework with some special properties. Some preliminary numerical experiments are reported in order to confirm that the use of >1 can lead to a better numerical performance than =1 (which corresponds to the standard ADMM).

关键词： Generalized alternating direction method of multipliers Hybrid extragradient method convex program Pointwise iteration-complexity Ergodic iteration-complexity

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

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2019年第2期182卷 640-666页

作者： Adona, Vando A. Goncalves, Max L. N. Melo, Jefferson G. Univ Fed Goias IME BR-74001970 Goiania Go Brazil

This paper proposes a partially inexact proximal alternating direction method of multipliers for computing approximate solutions 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 iteration-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 alternating direction method of multipliers with relative error criteria have been analyzed. Some preliminary numerical experiments are reported to illustrate the advantages of the new method.

关键词： Alternating direction method of multipliers Relative error criterion Hybrid extragradient method convex program Pointwise iteration-complexity Ergodic iteration-complexity

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INEXACT VARIANTS OF THE ALTERNATING DIRECTION METHOD OF MULTIPLIERS AND THEIR ITERATION-COMPLEXITY ANALYSES

INEXACT VARIANTS OF THE ALTERNATING DIRECTION METHOD OF MULT...

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作者： Vando Antonio Adona UNIVERSIDADE FEDERAL DE GOIAS

学位级别：博士

This thesis proposes and analyzes some variants of the alternating direction method of multipliers (ADMM) for solving separable linearly constrained convex optimization problems. This thesis is divided into three parts. First, we establish the iteration-complexity of a proximal generalized ADMM. This ADMM variant, proposed by Bertsekas and Eckstein, introduces a relaxation parameter α into the second ADMM subproblem in order to improve its computational performance. We show that, for a given tolerance ρ > 0, the proximal generalized ADMM with α ∈ (0, 2) provides, in at most O(1/ρ2) iterations, an approximate solution of the Lagrangian system associated to the optimization problem under consideration. It is further demonstrated that, in at most O(1/ρ) iterations, an approximate solution of the Lagrangian system can be obtained by means of an ergodic sequence associated to a sequence generated by the proximal generalized ADMM with α ∈ (0, 2]. Second, we propose and analyze an inexact variant of the aforementioned proximal generalized ADMM. In this variant, the first subproblem is approximately solved using a relative error condition whereas the second one is assumed to be easy to solve. It is important to mention that in many ADMM applications one of the subproblems has a closed-form solution; for instance, (cid:96)1-regularized convex composite optimization problems. We show that the proposed method possesses iteration-complexity bounds similar to its exact version. Third, we develop an inexact proximal ADMM whose first subproblem is inexactly solved using an approximate relative error criterion similar to the aforementioned inexact proximal generalized ADMM. Pointwise and ergodic iteration-complexity bounds for the proposed method are established. Our approach consists of interpreting these ADMM variants as an instance of a hybrid proximal extragradient framework with some special properties. Finally, in order to show the applicability and advantage of the inexact A

关键词： Alternating direction method of multipliers convex program Hybrid extragradient method Relative error criterion Pointwise iteration-complexity Ergodic iteration-complexity

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Robust Nonnegative Sparse Recovery and the Nullspace Property of 0/1 Measurements

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IEEE TRANSACTIONS ON INFORMATION THEORY 2018年第2期64卷 689-703页

作者： Kueng, Richard Jung, Peter Free Univ Berlin Dahlem Ctr Complex Quantum Syst D-14195 Berlin Germany Tech Univ Berlin Commun & Informat Theory Grp D-10587 Berlin Germany

We investigate recovery of nonnegative vectors from non-adaptive compressive measurements in the presence of noise of unknown power. In the absence of noise, existing results in the literature identify properties of the measurement that assure uniqueness in the non-negative orthant. By linking such uniqueness results to nullspace properties, we deduce uniform and robust compressed sensing guarantees for nonnegative least squares. No l(1)-regularization is required. As an important proof of principle, we establish that m x n random i.i.d. 0/1-valued Bernoulli matrices obey the required conditions with overwhelming probability provided that m = O(s log(n/s)). We achieve this by establishing the robust nullspace property for random 0/1-matrices-a novel result in its own right. Our analysis is motivated by applications in wireless network activity detection.

关键词： Compressed sensing convex program sparse representation nonnegative binary matrix Bernoulli nullspace property basis pursuit activitity detection

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Multi-Agent Optimization for Residential Demand Response under Real-Time Pricing

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ENERGIES 2019年第15期12卷 2867-2867页

作者： Wang, Zhanle Paranjape, Raman Chen, Zhikun Zeng, Kai Univ Regina Fac Engn & Appl Sci Regina SK S4S 0A2 Canada North China Univ Sci & Technol Dept Elect Engn Tangshan 063210 Peoples R China

Demand response (DR) programs encourage consumers to adapt the time of using electricity based on certain factors, such as cost of electricity, renewable energy availability, and ancillary request. It is one of the most economical methods to improve power system stability and energy efficiency. Residential electricity consumption occupies approximately one-third of global electricity usage and has great potential in DR applications. In this study, we propose a multi-agent optimization approach to incorporate residential DR flexibility into the power system and electricity market. The agents collectively optimize their own interests;meanwhile, the global optimal solution is achieved. The agent perceives its environment, predicts electricity consumption, and forecasts electricity price, based on which it takes intelligent actions to minimize electrical energy cost and time delay of using household appliances. The decision-making action is formulated into a convex program (CP) model. A distributed heuristic algorithm is developed to solve the proposed multi-agent optimization model. Case studies and numerical analysis show promising results with low variation of the aggregated load profile and reduction of electrical energy cost. The proposed approaches can be utilized to investigate various emerging technologies and DR strategies.

关键词： demand response multi-agent optimization convex program electric vehicle heuristic algorithm real-time pricing

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Inferring large graphs using l₁-penalized likelihood (vol 28, pg 905, 2018)

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STATISTICS AND COMPUTING 2018年第6期28卷 1231-1231页

作者： Champion, Magali Picheny, Victor Vignes, Matthieu Univ Paris 05 Lab MAP5 45 Rue St Peres F-75270 Paris 06 France Univ Paul Sabatier Inst Math Toulouse UMR5219 118 Route Narbonne F-31062 Toulouse 9 France Univ Toulouse INRA MIAT F-31326 Castanet Tolosan France Massey Univ Inst Fundamental Sci Palmerston North New Zealand

We address the issue of recovering the structure of large sparse directed acyclic graphs from noisy observations of the system. We propose a novel procedure based on a specific formulation of the \(\ell _1\)-norm regularized maximum likelihood, which decomposes the graph estimation into two optimization sub-problems: topological structure and node order learning. We provide convergence inequalities for the graph estimator, as well as an algorithm to solve the induced optimization problem, in the form of a convex program embedded in a genetic algorithm. We apply our method to various data sets (including data from the DREAM4 challenge) and show that it compares favorably to state-of-the-art methods. This algorithm is available on CRAN as the R package GADAG.

关键词： Directed acyclic graphs Lasso convex program Optimization Genetic Algorithm (GA) alachlor algorithms Likelihood Dataset convex programming maximum likelihood

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

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2018年第2期177卷 448-478页

作者： Goncalves, Max L. N. Alves, Maicon Marques Melo, Jefferson G. Univ Fed Goias IME BR-74001970 Goiania Go Brazil Univ Fed Santa Catarina Dept Matemat BR-88040900 Florianopolis SC Brazil

In this paper, we obtain global pointwise and ergodic convergence rates for a variable metric proximal alternating direction method of multipliers for solving linearly constrained convex optimization problems. We first propose and study nonasymptotic convergence rates of a variable metric hybrid proximal extragradient framework for solving monotone inclusions. Then, the convergence rates for the former method 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 variable metric proximal alternating direction method of multipliers (resp. variable metric hybrid proximal extragradient framework).

关键词： Alternating direction method of multipliers Variable metric Pointwise and ergodic convergence rates Hybrid proximal extragradient method convex program

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Iterative convex Optimization of Multi-Beam Directional Modulation With Artificial Noise

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IEEE COMMUNICATIONS LETTERS 2018年第8期22卷 1712-1715页

作者： Christopher, Ryan M. Borah, Deva K. New Mexico State Univ Dept Elect & Comp Engn Las Cruces NM 88003 USA

A joint optimization algorithm for designing multi-beam directional modulation (MBDM) symbols with artificial noise (AN) is proposed for physical layer security in wireless communication systems. The proposed approach guarantees specific minimum error probabilities along the given eavesdropper directions. The algorithm is implemented by iterating between a convex program that optimizes symbols with AN and a convex (linear) program optimizing a set of weight vectors to guarantee specific error probabilities along the eavesdropper directions. The proposed method is compared against several MBDM algorithms, and is found to provide excellent bit error rate performance results. A hardware experimental test bed is described and is used to demonstrate the feasibility of the proposed solution.

关键词： Directional modulation artificial noise physical layer security convex program

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IMPROVED POINTWISE ITERATION-COMPLEXITY OF A REGULARIZED ADMM AND OF A REGULARIZED NON-EUCLIDEAN HPE FRAMEWORK

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SIAM JOURNAL ON OPTIMIZATION 2017年第1期27卷 379-407页

作者： Goncalves, Max L. N. Melo, Jefferson G. Monteiro, Renato D. C. Univ Fed Goias Inst Math & Stat Campus 2Caixa Postal 131 BR-74001970 Goiania Go Brazil Georgia Inst Technol Sch Ind & Syst Engn Atlanta GA 30332 USA

This paper describes a regularized variant of the alternating direction method of multipliers (ADMM) for solving linearly constrained convex programs. It is shown that the pointwise iteration-complexity of the new variant is better than the corresponding one for the standard ADMM method and that, up to a logarithmic term, is identical to the ergodic iteration-complexity of the latter method. Our analysis is based on first presenting and establishing the pointwise iteration-complexity of a regularized non-Euclidean hybrid proximal extragradient framework whose error condition at each iteration includes both a relative error and a summable error. It is then shown that the new ADMM variant is a special instance of the latter framework where the sequence of summable errors is identically zero when the ADMM stepsize is less than one or a nontrivial sequence when the stepsize is in the interval [1, (1 + root 5)/2).

关键词： alternating direction method of multipliers hybrid proximal extragradient method non-Euclidean Bregman distances convex program pointwise iteration-complexity first-order methods inexact proximal point method regularization

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A convex Formulation for Path Analysis in Structural Equation Modeling 55

A Convex Formulation for Path Analysis in Structural Equatio...

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55th Annual Conference of the Society of Instrument and Control Engineers of Japan (SICE)

作者： Pruttiakaravanich, Anupon Songsiri, Jitkomut Chulalongkorn Univ Dept Elect Engn Bangkok Thailand

ISBN: (纸本)9784907764500

Path analysis is a special problem in Structural Equation Modeling (SEM) where its model describes causal relations among measured variables in a form of multiple linear regression. This work presents an alternative estimation formulation for a special case problem in path analysis as a convex framework by relaxing an equality constraint of the original problem. Under a condition on problem parameters, we show that, our optimal solution is low rank and provides an estimate of the path matrix of the original problem. To solve our estimation problem in a convex framework, we apply the alternating direction method of multipliers (ADMM) algorithm which is suitable for large-scale implementation. The performance of this algorithm is demonstrated in numerical experiments.

关键词： path analysis confirmatory SEM convex program

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