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.

我们考虑一类层次决策问题，这些问题可以看作是单领导者多跟随者游戏，并由虚拟市场协调员代表，他们试图根据某种平衡供需的标准为交易商品建立价格系统。市场协调员的目标功能涉及许多代理商的决策，这些决策是通过解决凸优化问题（取决于价格配置和未来经济状况的实现）而独立采取的。解决此问题的一种传统方法是通过混合互补公式。但是，当代理和/或场景的数量变大时，此方法可能变得不切实际。这项工作涉及智能体和情景智能分解算法，以解决相关的平衡问题，假设代理问题的解决方案是唯一的，这在许多应用程序中都是很自然的（当解决方案不是唯一的时，近似问题仍然定义明确，但是算法的收敛性尚未建立）。该算法基于作者的先前工作，其中开发了全参数化凸问题的解映射的合适正则化方法。在这里，我们展示了一种管理正则化参数的特定策略，将一些理论结果扩展到当前设置，并证明市场协调员问题的平滑逼近在文字上收敛于原始问题。给出了两阶段随机Walrasian平衡问题的数值实验和与互补求解器PATH的一些比较。