Differentiable-by-design Nonlinear Optimization for Model Predictive Control

Nonlinear optimization-based control policies, such as those those arising in nonlinear Model Predictive Control, have seen remarkable success in recent years. These policies require solving computationally demanding nonlinear optimization programs online at each time-step. The resulting solution map, viewed as a function of the measured state of the system and design parameters, may not be differentiable, which poses significant challenges if the control policy is embedded in a gradient-based policy optimization scheme. We propose a principled way to regularize the nonlinear optimization problem, obtaining a surrogate derivative even if when the original problem is not differentiable. The surrogate problem is differentiable by design and its solution map coincides with the solution of the unregularized problem. We demonstrate the effectiveness of our approach in a free-final-time optimal control problem and a receding-horizon nonlinear MPC example.