Compact versus Noncompact LP Formulations for minimizing Convex Choquet Integrals
Résumé
We address here the problem of minimizing Choquet Integrals (also known as “Lovász Extensions”) over solution sets which can be either polyhedra or (mixed) integer sets. Typical applications of such problems concern the search of compromise solutions in multicriteria optimization. We focus here on the case where the Choquet Integrals to be minimized are convex, implying that the set functions (or “capacities”) underlying the Choquet Integrals considered are submodular. We first describe an approach based on a large scale LP formulation, and show how it can be handled via the so-called column-generation technique. We next investigate alternatives based on compact LP formulations , i.e. featuring a polynomial number of variables and constraints. Various potentially useful special cases corresponding to well-identified subclasses of underlying set functions are considered: quadratic and cubic submodular functions, and a more general class including set functions which, up to a sign, correspond to capacities which are both (k+1)−additive and kk-monotone for k≥3. Computational experiments carried out on series of test instances, including transportation problems and knapsack problems, clearly confirm the superiority of compact formulations. As far as we know, these results represent the first systematic way of practically solving Choquet minimization problems on solution sets of significantly large dimensions.