Decomposition Methods for Monotone Two-Time-Scale Stochastic Optimization Problems

It is common that strategic investment decisions are made at a slow time-scale, whereasoperational decisions are made at a fast time-scale. Hence, the total number of decisionstages may be huge. In this paper, we consider multistage stochastic optimization problemswith two time-scales, and we propose a time block decomposition scheme to addressthem numerically. More precisely, i) we write recursive Bellman-like equations at the slowtime-scale and ii), under a suitable monotonicity assumption, we propose computable upperand lower bounds — relying respectively on primal and dual decomposition — forthe corresponding slow time-scale Bellman functions. With these functions, we are ableto design policies. We assess the methods tractability and validate their efficiency bysolving a battery management problem where the fast time-scale operational decisionshave an impact on the storage current capacity, hence on the strategic decisions to renewthe battery at the slow time-scale