Deriving time complexities for a class of distributed gradient projection-based optimal routing algorithms
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An upper bound is derived for the time complexity of the gradient projection-based optimal routing algorithm of D. P. Bertsekas et al. (MIT Tech. Rep. LIDS-P-1364, Feb. 1984). The overall time complexity of the algorithm is given by the product of the complexity of each iteration and the complexity of the number of iterations needed to converge. It turns out that the complexity of each iteration is dominated by the time required to solve shortest path problems, and is therefore straightforward to estimate. On the other hand, estimating a meaningful bound for the number of iterations needed for convergence presents a formidable challenge, and is therefore the main focus of this study. Classical results related to convergence rates of gradient projection-type algorithms require precise knowledge of the spectral content of Hessian matrix. Unfortunately, such classical techniques are not readily applicable to the problem at hand since the eigenvalues of the Hessian depend on factors such as network topology and traffic demand patterns in a very non-tractable way. In this paper, an alternate analysis technique is developed which yields an upper bound for the number of iterations needed for convergence to within a small neighborhood of the optimal solution.