Complementarity, not Optimization, is the Language of Markets

Each market agent (producer or consumer) in a power market pursues its own objective, typically to maximize its own profit. As such, the specific behavior of each agent in the market is conveniently formulated as a bi-level optimization problem whose upper-level problem represents the profit seeking behavior of the agent and whose lower-level problem represents the clearing of the market. The objective function and the constraints of this bi-level problem depend on the agent's own decision variables and on those of other agents as well. Understanding the outcomes of the market requires considering and solving jointly the interrelated bi-level problems of all market agents, which is beyond the purview of optimization. Solving jointly a set of bi-level (or single-level) optimization problems that are interrelated is the purview of complementarity. In this paper and in the context of power markets, we review complementarity using a tutorial approach.