Reactive trajectory optimization for robotics presents formidable challenges, demanding the rapid generation of purposeful robot motion in complex and swiftly changing dynamic environments. While much existing research predominantly addresses robotic motion planning with predefined objectives, emerging problems in robotic trajectory optimization frequently involve dynamically evolving objectives and stochastic motion dynamics. However, effectively addressing such reactive trajectory optimization challenges for robot manipulators proves difficult due to inefficient, high-dimensional trajectory representations and a lack of consideration for time optimization. In response, we introduce a novel trajectory optimization framework called RETRO. RETRO employs adaptive optimization techniques that span both spatial and temporal dimensions. As a result, it achieves a remarkable computing complexity of $O(T^{2.4}) + O(Tn^{2})$, a significant improvement over the traditional application of DDP, which leads to a complexity of $O(n^{4})$ when reasonable time step sizes are used. To evaluate RETRO's performance in terms of error, we conducted a comprehensive analysis of its regret bounds, comparing it to an Oracle value function obtained through an Oracle trajectory optimization algorithm. Our analytical findings demonstrate that RETRO's total regret can be upper-bounded by a function of the chosen time step size. Moreover, our approach delivers smoothly optimized robot trajectories within the joint space, offering flexibility and adaptability for various tasks. It can seamlessly integrate task-specific requirements such as collision avoidance while maintaining real-time control rates. We validate the effectiveness of our framework through extensive simulations and real-world robot experiments in closed-loop manipulation scenarios.