The progressive hedging algorithm minimizes an expected "cost" by iteratively decomposing into separate subproblems for each scenario. Up to now it has depended on convexity of the underlying "cost" function with respect to the decision variables and the constraints on them. However, a new advance makes it possible to obtain convergence to a locally optimal solution when the procedure is executed close enough to it and a kind of second-order local sufficiency condition is satisfied. Besides applications in which costs and associated constraints may directly be nonconvex, there are applications to stochastic programming problems in which those are convex but the probabilities for the scenarios may be decision-dependent. For example, in a two-stage problem the probabilities in the recourse stage could be influenced by the first-stage decision.