SEQUOIA: A sequential algorithm providing feasibility guarantees for constrained optimization*

Abstract

Real-time optimal control applications rely on quickly solving nonlinear programs (NLPs) with many constraints. It is usually challenging to design a safe controller that guarantees feasibility of the returned points even in the case of early termination. We present a novel algorithm called SEQUOIA that allows users to choose the maximum optimality gap tolerated and implement sub-optimal, feasible decisions faster in time-critical scenarios. Typical constrained optimization methods approach the feasible set from its exterior, hence the point returned in the case of early termination is not guaranteed to be fit for purpose because critical constraints might be violated. Our algorithm guarantees a feasible point is returned after any iteration and an incrementally better point can be easily constructed. Additionally, we developed the theoretical framework behind our algorithm and show some key mathematical properties. The core idea is to store exact upper and lower bounds for the optimal cost and iteratively solve feasibility problems in order to tighten these bounds. To solve the feasibility problems we use a residual-based approach and we can provide an infeasibility detection feature during the first iteration. We illustrate our implementation on a constrained optimization benchmark as well as a maximum radius orbit transfer optimal control problem. Solving this type of dynamic optimization problems is difficult because of the high number of equality constraints generated from the discretization of dynamic and algebraic path constraints. The numerical experiments show that one can reduce the computational time by more than 40% if the allowed optimality gap is increased from 10−15 to 10−10 and the overall solve time can be decreased by up to two orders of magnitude when compared to a quadratic penalty method (QPM).