Robust combinatorial optimization with variable cost uncertainty

Michael Poss (1, 2)

1 Labex MS2T - Laboratoire d'Excellence "Maîtrise des Systèmes de Systèmes Technologiques"
2 Heudiasyc - Heuristique et Diagnostic des Systèmes Complexes [Compiègne]

We present in this paper a new model for robust combinatorial optimization with cost uncertainty that generalizes the classical budgeted uncertainty set. We suppose here that the budget of uncertainty is given by a function of the problem variables, yielding an uncertainty multifunction. The new model is less conservative than the classical model and approximates better Value-at-Risk objective functions, espe-cially for vectors with few non-zero components. An example of budget function is constructed from the probabilistic bounds computed by Bertsimas and Sim. We provide an asymptotically tight bound for the cost reduction obtained with the new model. We turn then to the tractability of the resulting opti-mization problems. We show that when the budget function is affine, the resulting optimization prob-lems can be solved by solving n þ 1 deterministic problems. We propose combinatorial algorithms to handle problems with more general budget functions. We also adapt existing dynamic programming algorithms to solve faster the robust counterparts of optimization problems, which can be applied both to the traditional budgeted uncertainty model and to our new model. We evaluate numerically the reduc-tion in the price of robustness obtained with the new model on the shortest path problem and on a sur-vivable network design problem.

Mots clés: Combinatorial optimization; Robust optimization; Dynamic programming; Price of robustness; Budgeted uncertainty

Domaines: Recherche opérationnelle [math.OC]

Michael Poss. Robust combinatorial optimization with variable cost uncertainty. European Journal of Operational Research, 2014, 237 (3), pp.836-845. ⟨10.1016/j.ejor.2014.02.060⟩. ⟨hal-01099569⟩