Decomposition methods for monotone two-time-scale stochastic optimization problems

It is common that strategic investment decisions are made at a slow time-scale, whereas operational decisions are made at a fast time-scale. Hence, the total number of decision stages may be huge. In this paper, we consider multistage stochastic optimization problems with two time-scales, and we propose a time block decomposition scheme to address them numerically. More precisely, (i) we write recursive Bellman-like equations at the slow time-scale and (ii), under a suitable monotonicity assumption, we propose computable upper and lower bounds—relying respectively on primal and dual decomposition—for the corresponding slow time-scale Bellman functions. With these functions, we are able to design policies. We assess the methods tractability and validate their efficiency by solving a battery management problem where the fast time-scale operational decisions have an impact on the storage current capacity, hence on the strategic decisions to renew the battery at the slow time-scale.