Discrete-Space Lagrangian Optimization for Multi-Objective Temporal Planning

In this paper, we study multi-objective temporal planning problems in discrete time and space formulated as single-objective dynamic optimization problems with a minimax objective function. We propose efficient node-dominance relations for pruning states that will not lead to locally optimal plans. Based on the theory of Lagrange multipliers in discrete space, we present the necessary and sufficient conditions for locally optimal plans, partition the Lagrangian function into distributed Lagrangian functions, one for each stage, and show the distributed necessary conditions in the form of local saddle-point conditions in each stage. By utilizing these dominance relations, we present efficient search algorithms whose complexity, despite exponential, has a much smaller base as compared to that without using the relations, and that can converge asymptotically to Pareto optimal plans. Finally, we demonstrate the performance of our approach by integrating it in the ASPEN planner and show significant improvements in CPU time and solution quality on some spacecraft scheduling and planning benchmarks.