Space splitting convexification: a local solution method for nonconvex optimal control problems

Abstract

Many optimal control tasks for engineering problems require the solution of nonconvex optimisation problems, which are rather hard to solve. This article presents a novel iterative optimisation procedure for the fast solution of such problems using successive convexification. The approach considers two types of nonconvexities. Firstly, equality constraints with possibly multiple univariate nonlinearities, which can arise for nonlinear system dynamics. Secondly, nonconvex sets comprised of convex subsets, which may occur for semi-active actuators. By introducing additional decision variables and constraints, the decision variable space is decomposed into affine segments yielding a convex subproblem which is solved efficiently in an iterative manner. Under certain conditions, the algorithm converges to a local optimum of a scalable, piecewise linear approximation of the original problem. Furthermore, the algorithm tolerates infeasible initial guesses. Using a single-mass oscillator application, the procedure is compared with a nonlinear programming algorithm, and the sensitivity regarding initial guesses is analysed.