Locality in Network Optimization

We develop the first results on the local sensitivity analysis for the solution of a broad class of equality-constrained convex network optimization problems when perturbations are made on the constraints. In the context of the minimum-cost network flow problem — a paradigm in the theory of optimization — these results suggest a notion of decay of correlation for constrained optimization procedures that can be captured via the spectral analysis of either random walks or graph Laplacians. This decay of correlation property, which relies on an interplay between the structure of the Hessian of convex functions and the sparsity of the constraint matrices, can be exploited to localize canonical optimization procedures — we consider the projected gradient descent algorithm — and provide lower computational complexity than centralized algorithms, as required in modern largescale applications. To the best of our knowledge, our results represent the first systematic investigation of (non-random) decay of correlation for constrained optimization procedures.