Regularized Decomposition of High-Dimensional Multistage Stochastic Programs with Markov Uncertainty

Abstract

We develop a quadratic regularization approach for the solution of high-dimensional multistage stochastic optimization problems characterized by a potentially large number of time periods/stages (e.g. hundreds), a high-dimensional resource state variable, and a Markov information process. The resulting algorithms are shown to converge to an optimal policy after a finite number of iterations under mild technical assumptions. Computational experiments are conducted using the setting of optimizing energy storage over a large transmission grid, which motivates both the spatial and temporal dimensions of our problem. Our numerical results indicate that the proposed methods exhibit significantly faster convergence than their classical counterparts, with greater gains observed for higher-dimensional problems.

A parallel Branch-and-Fix Coordination based matheuristic algorithm for solving large sized multistage stochastic mixed 0–1 problems

“…The methodology has been known regularized decomposition. The quadratic regularization for stochastic linear optimization was first developed in [63] for two-stage problems, see also [66] for twostage and multistage problems, and [6,7] for multistage problems, among others. The Stochastic Dual Dynamic Programming (SDDP) approach [56,57] is a decomposition methodology that has been most frequently used for testing different regularization mechanisms, since there it is assumed the stagewise independence of the random process and, then, the dimensions of the scenario tree could still be manageable.…”

“…The Stochastic Dual Dynamic Programming (SDDP) approach [56,57] is a decomposition methodology that has been most frequently used for testing different regularization mechanisms, since there it is assumed the stagewise independence of the random process and, then, the dimensions of the scenario tree could still be manageable. The mechanism that is considered in [6,63] for the stage-wise independence of the random process consists of appending to the objective function the (convex) quadratic regularization function. This function is based on the difference between the variables in the nodes of the tree and their values in the incumbent solution of the model.…”

“…However, for a high number of periods in the time horizon or, as it frequently happens in practice, the outcome of the random parameters at a given stage is not independent of the outcomes of the previous stages, then, the number of regularized quadratic terms could be unmanageable. For that type of problems, see in [6] its Algorithm 2 where the incumbent solutions are not indexed by the nodes of the scenario tree. So, the reference for the quadratic function could have different parts of the left hand side of constraints related to the stages and not to the nodes in the tree.…”

“…See [56,57,61]. Some works in the literature allow to consider Markovian multistage stochastic problems, where the uncertainty in each node of a given stage depends on the history of its ancestor nodes, see [6,30,75]. The treatment of the Conditional Value-at-Risk (CVaR) risk averse measure in SDP was introduced in [36,42,68].…”

“…Recently, parallel computing versions of the exact BFC methodology was presented in [4,55]. See also the parallel matheuristic bounding methods presented in [3,6,53,61,64,74] that also allow large sized instances to be solved.…”