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

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 pro...

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Bibliographic Details
Published in:Journal of optimization theory and applications 2019-08, Vol.182 (2), p.640-666
Main Authors: Adona, Vando A., Gonçalves, Max L. N., Melo, Jefferson G.
Format: Article
Language:English
Subjects:
Applications of Mathematics
Calculus of Variations and Optimal Control
Optimization
Complexity
Computational geometry
Convexity
Engineering
Error analysis
Iterative methods
Mathematics
Mathematics and Statistics
Multipliers
Operations Research/Decision Theory
Optimization
Theory of Computation
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Summary: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.
ISSN:0022-3239
1573-2878
DOI:10.1007/s10957-019-01525-8