Second-Order Partial Outer Convexification for Switched Dynamical Systems

Mixed-integer optimal control problems arise in many practical applications combining nonlinear, dynamic, and combinatorial features. To cope with the resulting complexity, several approaches have been suggested in the past. Some of them rely on solving a reformulated and relaxed control problem, referred to as partial outer convexification (POC). Inspired by an efficient algorithm for switching time optimization by Stellato and coworkers, SwitchTimeOpt.jl, we developed an algorithmic approach for POC implemented in a Julia package. Both approaches are based on linearization and exponential integration to obtain second derivatives. We show the efficiency and applicability of the novel approach by comparing it to SwitchTimeOpt.jl, by extending the concept and calculations to the treatment of constraints, and by investigating warm starting of switching time optimization. An additional comparison to a CasADi-based standard single shooting approach shows a significant reduction in computational time despite an increase in iterations. The new solver facilitates the reliable and fast solution of mixed-integer optimal control problems.