Weak Loss Convexification for Sequential Convex Programming

This article presents a methodology to reduce the convexification loss in sequential convex programming for a class of nonconvex optimal control problems. The nonconvexity arises from the presence of smooth concave inequality constraints or polyhedral exclusion constraints. The concave inequality constraints are approximated by successive linearization, with the convexification loss reduced either by reconstructing concave functions or revising expansion points. The expansion point revision results in a weak loss convexification. Moreover, polyhedral exclusion constraints are approximated by separating hyperplanes. The convexification loss can be reduced by solving additional dual problems or identifying the supporting hyperplanes, both of which result in weak loss convexifications. The four methods are demonstrated to have no impact on the iterative feasibility and convergence of sequential convex programming. Simulation demonstrates that the reduction of convexification loss can reduce the conservatism during the iteration process, which in turn has the potential to reduce the number of iterations and computation time.