Convex–Concave Decomposition of Nonlinear Equality Constraints in Optimal Control

In optimal control and optimization problems where partial convexity already exists, it is highly desirable to reformulate the nonconvex parts of the problem in a way amendable to the use of a convex optimization algorithm. The biggest challenge in doing so lies with nonlinear equality constraints. Traditional linearization-based techniques have inherent shortcomings that can cause them to be ineffective in some routine applications. In this paper a novel method is developed to treat a class of nonlinear equality constraints by a convex–concave decomposition. In a relaxed problem each nonlinear equality constraint of the class is represented by three inequality constraints that are either convex or concave. It is theoretically established that the relaxed problem can be set up to have the same solution as the original one. A convergent successive solution approach is designed to find the solution of the relaxed problem by solving a sequence of convex optimization problems. The application of the proposed method in a fuel-optimal finite-thrust spacecraft circumnavigation problem demonstrates the effectiveness of the approach where the conventional linearization method fails under most conditions.