  In probability theory and statistics notions of correlation among random
variables, decay of correlation, and bias-variance trade-off are fundamental.
In this work we introduce analogous notions in optimization, and we show their
usefulness in a concrete setting. We propose a general notion of correlation
among variables in optimization procedures that is based on the sensitivity of
optimal points upon (possibly finite) perturbations. We present a canonical
instance in network optimization (the min-cost network flow problem) that
exhibits locality, i.e., a setting where the correlation decays as a function
of the graph-theoretical distance in the network. In the case of warm-start
reoptimization, we develop a general approach to localize a given optimization
routine in order to exploit locality. We show that the localization mechanism
is responsible for introducing a bias in the original algorithm, and that the
bias-variance trade-off that emerges can be exploited to minimize the
computational complexity required to reach a prescribed level of error
accuracy. We provide numerical evidence to support our claims.
