Scenarios in which multiple agents interact in a goal-directed manner can be modeled as differential games. Here each agent aims to optimize an associated cost function. Systems ranging from classical economic models to emerging applications in autonomous vehicles can fit into this framework. In scenarios such as analysis of human or animal behaviors, the cost-functions are not known. In order to design control strategies that interact with such agents, the objectives must be modeled or identified. This paper presents a methodology for identifying cost functions for interacting agents. In particular, we identify costs that lead to open-loop Nash equilibria for nonzero-sum constrained differential games. To solve this problem, we extend an inverse optimal control method, known as residual optimization, to the case of differential games. In this paper, we show that residual optimization is tractable for constrained differential games. Specifically, the residual optimization problems turn out to be fully decoupled. Numerical examples indicate that accurate costs can be learned from observing trajectories.