Progressive hedging for stochastic programs with cross-scenario inequality constraints

In this paper, we show how progressive hedging may be used to solve stochastic programming problems that involve cross-scenario inequality constraints. In contrast, standard stochastic programs involve cross-scenario equality constraints that describe the non-anticipative nature of the optimal solution. The standard progressive hedging algorithm (PHA) iteratively manipulates the objective function coefficients of the scenario subproblems to reflect the costs of non-anticipativity and penalize deviations from a non-anticipative, aggregated solution. Our proposed algorithm follows the same principle, but works with cross-scenario inequality constraints. Specifically, we focus on the problem of determining optimal bids for hydropower producers that participate in wholesale electricity auctions. The cross-scenario inequality constraints arise from the fact that bids are required to be non-decreasing. We show that PHA for inequality constraints have the same convergence properties as standard PHA, and illustrate our algorithm with results for an instance of the hydropower bidding problem.