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Dual decomposition coupled with the subgradient method has found application to optimal resource management in communication networks, as it can lead to distributed and scalable algorithms. Network entities-nodes or functional layers-exchange Lagrange multipliers and primal minimizers of the Lagrangian function towards optimizing a network-wide performance metric. It is of interest to study the performance of the resultant algorithms when such exchanges are delayed or lost. This paper deals with such asynchronous dual subgradient methods in separable convex programming. In this scenario, the subgradient vector is a sum of components, each possibly corresponding to an outdated Lagrange multiplier, and not the current one. A number of network entities is allowed to prematurely stop updating their corresponding variables, thereby effecting infinite delay between the current iterate and the multipliers used for a number of subgradient components. Conditions for convergence of the algorithm are developed. Specific applications include multipath routing in wireline networks and cross-layer optimization in wireless networks. Numerical tests for multipath routing in the Abilene network topology are presented.