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 constrained optimal control problems with global convergence and superlinear convergence-rate guarantees. The proposed algorithm can handle nonlinear dynamics and non-convex state and control constraints. It solves the original problem to optimality by successively linearizing non-convex dynamics and constraints about the solution of the previous iteration. The resulting convex subproblems are numerically tractable, and can be computed quickly and reliably using convex optimization solvers, making the SCvx algorithm well suited for real-time applications. Analysis is presented to show that the algorithm converges both globally and superlinearly, guaranteeing i) local optimality recovery: if the converged solution is feasible with respect to the original problem, then it is also a local optimum; ii) strong convergence: if the Kurdyka-Lojasiewicz (KL) inequality holds at the converged solution, then the solution is unique. The superlinear rate of convergence is obtained by exploiting the structure of optimal control problems, showcasing that faster rate of convergence can be achieved by leveraging specific problem properties when compared to generic nonlinear programming methods. Numerical simulations are performed for a non-convex quad-rotor motion planning problem, and corresponding results obtained using Sequential Quadratic Programming (SQP) and general purpose Interior Point Method (IPM) solvers are provided for comparison. The results show that the convergence rate of the SCvx algorithm is indeed superlinear, and that SCvx outperforms the other two methods by converging in less number of iterations.