Calculus of variations in discrete space for constrained nonlinear dynamic optimization

We propose new dominance relations that can speed up significantly the solution process of nonlinear constrained dynamic optimization problems in discrete time and space. We first show that path dominance in dynamic programming cannot be applied when there are general constraints that span across multiple stages, and that node dominance, in the form of Euler-Lagrange conditions developed in optimal control theory in continuous space, cannot be extended to that in discrete space. This paper is the first to propose efficient dominance relations, in the form of local saddle-point conditions in each stage of a problem, for pruning states that will not lead to locally optimal paths. By utilizing these dominance relations, we develop efficient search algorithms whose complexity, despite exponential, has a much smaller base as compared to that without using the relations. Finally, we demonstrate the performance of our algorithms on some spacecraft planning and scheduling benchmarks and show significant improvements in CPU time and solution quality as compared to those obtained by the existing ASPEN planner.