Stochastic optimization by message passing

Most optimization problems in applied sciences realistically involve uncertainty in the parameters defining the cost function, of which only statistical information is known beforehand. Here we provide an in-depth discussion of how message passing algorithms for stochastic optimization based on the cavity method of statistical physics can be constructed. We focus on two basic problems, namely the independent set problem and the matching problem, for which we display the general method and caveats for the case of the so called two-stage problem with independently distributed stochastic parameters. We compare the results with some greedy algorithms and briefly discuss the extension to more complicated stochastic multi-stage problems.