Trajectory optimization is an important tool for control and planning of\ncomplex, underactuated robots, and has shown impressive results in real world\nrobotic tasks. However, in applications where the cost function to be optimized\nis non-smooth, modern trajectory optimization methods have extremely slow\nconvergence. In this work, we present TRON, an iterative solver that can be\nused for efficient trajectory optimization in applications with non-smooth cost\nfunctions that are composed of smooth components. TRON achieves this by\nexploiting the structure of the objective to adaptively smooth the cost\nfunction, resulting in a sequence of objectives that can be efficiently\noptimized. TRON is provably guaranteed to converge to the global optimum of the\nnon-smooth convex cost function when the dynamics are linear, and to a\nstationary point when the dynamics are nonlinear. Empirically, we show that\nTRON has faster convergence and lower final costs when compared to other\ntrajectory optimization methods on a range of simulated tasks including\ncollision-free motion planning for a mobile robot, sparse optimal control for\nsurgical needle, and a satellite rendezvous problem.\n