Progressive hedging innovations for a class of stochastic mixed-integer resource allocation problems

Abstract

Numerous planning problems can be formulated as multi-stage stochastic programs and many possess key discrete (integer) decision variables in one or more of the stages. Progressive hedging (PH) is a scenario-based decomposition technique that can be leveraged to solve such problems. Originally devised for problems possessing only continuous variables, PH has been successfully applied as a heuristic to solve multi-stage stochastic programs with integer variables. However, a variety of critical issues arise in practice when implementing PH for the discrete case, especially in the context of very difficult or large-scale mixed-integer problems. Failure to address these issues properly results in either non-convergence of the heuristic or unacceptably long run-times. We investigate these issues and describe algorithmic innovations in the context of a broad class of scenario-based resource allocation problem in which decision variables represent resources available at a cost and constraints enforce the need for sufficient combinations of resources. The necessity and efficacy of our techniques is empirically assessed on a two-stage stochastic network flow problem with integer variables in both stages.

Assignment of Electrical Properties to Power Grid Topologies

“…The Alternating Direction Method of Multipliers (ADMM) as described in [16] is adopted, where β is the communicated variable. This approach is also very similar to the Progressive Hedging method as described in [17]. Instead of updating the dual variable based on the difference between the copies of each element in β as done in (13), the update is with respect to the average value,…”

“…The goal of the forced consensus problem is to alter the the power injection vector as little as possible, while still satisfying all the constraints. This is achieved by a the quadratic objective in the (17). Since the generation and load are now handled individually, instead of parameter p we have the nodal generation P g and nodal consumption P d .…”

Toward scalable, parallel progressive hedging for stochastic unit commitment

“…In particular, the presence of integer decision variables can induce cycling behavior. However, effective techniques for detecting and breaking cycles have been recently introduced (e.g., see [9]). Further, accelerators are typically necessary to improve PH convergence (even in the linear case), as the linear convergence rates are typically too slow to achieve practical run-times.…”

“…Specifically, we employ variable fixing (freezing the values of variables that have converged over the past n PH iterations) and slamming (forcing early convergence of specific variables that have minimal impact on the objective). Both of these techniques are described fully in [9].…”

“…In our PH configuration, we use variable-specific ρ values for unit on/off variables (the only first stage variables), which we compute based on the LMPs (Locational Marginal Prices) of the economic dispatches of scenario sub-problems following PH iteration 0. This strategy is a case of "costproportional" rho setting, which has been shown to be an effective technique for PH parameterization [3,9].…”