Decomposition Algorithms for Some Deterministic and Two-Stage Stochastic Single-Leader Multi-Follower Games

We consider a certain class of hierarchical decision problems that can be viewed as single-leader multi-follower games, and be represented by a virtual market coordinator trying to set a price system for traded goods, according to some criterion that balances supply and demand. The objective function of the market coordinator involves the decisions of many agents, which are taken independently by solving convex optimization problems that depend on the price configuration and on realizations of future states of the economy. One traditional way of solving this problem is via a mixed complementarity formulation. However, this approach can become impractical when the numbers of agents and/or scenarios become large. This work concerns agent-wise and scenario-wise decomposition algorithms to solve the equilibrium problems in question, assuming that the solutions of the agents’ problems are unique, which is natural in many applications (when solutions are not unique, the approximating problems are still well-defined, but the convergence properties of the algorithm are not established). The algorithm is based on a previous work of the authors, where a suitable regularization of solution mappings of fully parameterized convex problems is developed. Here, we show one specific strategy to manage the regularization parameter, extend some theoretical results to the current setting, and prove that the smooth approximations of the market coordinator’s problem converge epigraphically to the original problem. Numerical experiments and some comparisons with the complementarity solver PATH are shown for the two-stage stochastic Walrasian equilibrium problem.