A class of convergent algorithms for resource allocation in wireless fading networks
IEEE Transactions on Wireless Communications
On dual decomposition and linear programming relaxations for natural language processing
EMNLP '10 Proceedings of the 2010 Conference on Empirical Methods in Natural Language Processing
Dual decomposition for parsing with non-projective head automata
EMNLP '10 Proceedings of the 2010 Conference on Empirical Methods in Natural Language Processing
Ergodic stochastic optimization algorithms for wireless communication and networking
IEEE Transactions on Signal Processing
Dual decomposition for natural language processing
HLT '11 Proceedings of the 49th Annual Meeting of the Association for Computational Linguistics: Tutorial Abstracts of ACL 2011
Cross-layer designs in coded wireless fading networks with multicast
IEEE/ACM Transactions on Networking (TON)
Exact decoding of phrase-based translation models through Lagrangian relaxation
EMNLP '11 Proceedings of the Conference on Empirical Methods in Natural Language Processing
Journal of Artificial Intelligence Research
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In this paper, we study methods for generating approximate primal solutions as a byproduct of subgradient methods applied to the Lagrangian dual of a primal convex (possibly nondifferentiable) constrained optimization problem. Our work is motivated by constrained primal problems with a favorable dual problem structure that leads to efficient implementation of dual subgradient methods, such as the recent resource allocation problems in large-scale networks. For such problems, we propose and analyze dual subgradient methods that use averaging schemes to generate approximate primal optimal solutions. These algorithms use a constant stepsize in view of its simplicity and practical significance. We provide estimates on the primal infeasibility and primal suboptimality of the generated approximate primal solutions. These estimates are given per iteration, thus providing a basis for analyzing the trade-offs between the desired level of error and the selection of the stepsize value. Our analysis relies on the Slater condition and the inherited boundedness properties of the dual problem under this condition. It also relies on the boundedness of subgradients, which is ensured by assuming the compactness of the constraint set.