Pareto optimality on compact spaces in a preference-based setting under incompleteness

We characterize the existence of Pareto optimal elements for a family of not necessarily total preorders on a compact topological space. We identify a rather general semicontinuity assumption, called weak upper semicontinuity, under which there exist Pareto optimal elements. We also show that weak upper semicontinuity of each individual preorder is a necessary and sufficient condition for determining the Pareto optimal elements by solving the classical multi-objective optimization problem in case that each function is upper semicontinuous and order-preserving for the respective preorder, and each preorder satisfies a condition of weak separability.