Dual Decomposition Optimisation for sensor networks design

Due to the deployment of large scale sensor networks, there has been a lot of interest in finding distributed methods for solving complex optimisation problems. Indeed, due to the absence of a central entity which would coordinate network sensor node operations, each node needs to operate using information of local nature only. Recently, Markov Random Fields (MRF) theory has been applied to provide distributed methods for the problem of a global network cost function [3], [1], [2]. While MRF based techniques scale very well with the size of the network, their main drawback is the fact that they rely on stochastic methods (simulated annealing and gibbs sampler) which convergence properties are problematic: indeed, in order to ensure convergence to the global minimum of the cost function, a very large number of iterations is required, hence making these techniques impractical for sensor network optimisation problems. The other fundamental issue is that these techniques are designed for discrete optimisation problems, which is not always appropriate for networking problems. In order to circumvene the issue raised by stochastic methods, a new and deterministic technique for the optimisation of MRFs problem based on a dual decomposition optimisation technique was proposed in [6] in the context of computer vision. The application of this technique to the issue of resource allocation in sensor network was presented in [2] and showed very good performance in terms of convergence time. However, no formal proof of convergence was feasible due to the non modular property of the cost function. Moreover, even when dealing with modular cost functions, these techniques only apply to discrete optimisation problems. In this paper, we propose an extension of the method proposed in [6] to continuous optimisation problems and we show that the distributed solution to the generic problem does converge to global minimum of the cost function. Moreover, we provide the optimum values for parameters that drive the convergence speed properties of our adaptive algorithm. We then show how this generic formulation can be applied to the problem of resource allocation and general cross layer design problems in sensor networks.