Successive Convexification: A Superlinearly Convergent Algorithm for Non-convex Optimal Control Problems

Abstract

This paper presents the SCvx algorithm, a successive convexification algorithm designed to solve non-convex optimal control problems with global convergence and superlinear convergence-rate guarantees. The proposed algorithm handles nonlinear dynamics and non-convex state and control constraints by linearizing them about the solution of the previous iterate, and solving the resulting convex subproblem to obtain a solution for the current iterate. Additionally, the algorithm incorporates several safe-guarding techniques into each convex subproblem, employing virtual controls and virtual buffer zones to avoid artificial infeasibility, and a trust region to avoid artificial unboundedness. The procedure is repeated in succession, thus turning a difficult non-convex optimal control problem into a sequence of numerically tractable convex subproblems. Using fast and reliable Interior Point Method (IPM) solvers, the convex subproblems can be computed quickly, making the SCvx algorithm well suited for real-time applications. Analysis is presented to show that the algorithm converges both globally and superlinearly, guaranteeing the local optimality of the original problem. The superlinear convergence is obtained by exploiting the structure of optimal control problems, showcasing the superior convergence rate that can be obtained by leveraging specific problem properties when compared to generic nonlinear programming methods. Numerical simulations are performed for an illustrative non-convex quad-rotor motion planning example problem, and corresponding results obtained using Sequential Quadratic Programming (SQP) solver are provided for comparison. Our results show that the convergence rate of the SCvx algorithm is indeed superlinear, and surpasses that of the SQP-based method by converging in less than half the number of iterations.