Optimizing over the Efficient Set of a Multi-Objective Discrete Optimization Problem

Optimizing over the efficient set of a discrete multi-objective problem is a challenging issue. The main reason is that, unlike when optimizing over the feasible set, the efficient set is implicitly characterized. Therefore, methods designed for this purpose iteratively generate efficient solutions by solving appropriate single-objective problems. However, the number of efficient solutions can be quite large and the problems to be solved can be difficult practically. Thus, the challenge is both to minimize the number of iterations and to reduce the difficulty of the problems to be solved at each iteration. In this paper, a new enumeration scheme is proposed. By introducing some constraints and optimizing over projections of the search region, potentially large parts of the search space can be discarded, drastically reducing the number of iterations. Moreover, the single-objective programs to be solved can be guaranteed to be feasible, and a starting solution can be provided allowing warm start resolutions. This results in a fast algorithm that is simple to implement. Experimental computations on two standard multi-objective instance families show that our approach seems to perform significantly faster than the state of the art algorithm.