A soft approach to multi-objective optimization

Many combinatorial optimization problems require the assignment of a set of variables in such a way that an objective function is optimized. Often, the objective function involves different criteria, and it may happen that the requirements are in conflict: assignments that are good wrt. one objective may behave badly wrt. another. An optimal solution wrt. all criteria may not exist, and either the efficient frontier (the set of best incomparable solutions, all equally relevant in the absence of further information) or an approximation has to be looked after. The paper shows how the soft constraints formalism based on semirings, so far exploited for finding approximations, can embed also the computation of the efficient frontier in multi-objective optimization problems. The main result is the proof that the efficient frontier of a multi-objective problem can be obtained as the best level of consistency distilled from a suitable soft constraint problem.