Exact and Heuristic Solution Techniques for Mixed-Integer Quantile Minimization Problems

Diego Cattaruzza (1) , Martine Labbé (1, 2) , Matteo Petris (1) , Marius Roland (3) , Martin Schmidt (3)

1 INOCS - Integrated Optimization with Complex Structure
2 ULB - Université libre de Bruxelles
3 Trier University

We consider mixed-integer linear quantile minimization problems that yield large-scale problems that are very hard to solve for real-world instances. We motivate the study of this problem class by two important realworld problems: a maintenance planning problem for electricity networks and a quantile-based variant of the classic portfolio optimization problem. For these problems, we develop valid inequalities and present an overlapping alternating direction method. Moreover, we discuss an adaptive scenario clustering method for which we prove that it terminates after a finite number of iterations with a global optimal solution. We study the computational impact of all presented techniques and finally show that their combination leads to an overall method that can solve the maintenance planning problem on large-scale real-world instances provided by the EURO/ROADEF challenge 2020 1 and that they also lead to significant improvements when solving a quantile-version of the classic portfolio optimization problem.

Mots clés: Quantile Minimization; Value-at-Risk (VaR); Mixed-Integer Optimization; Valid Inequalities; Adaptive Clustering

Domaines: Recherche opérationnelle [math.OC]; Combinatoire [math.CO]

Diego Cattaruzza, Martine Labbé, Matteo Petris, Marius Roland, Martin Schmidt. Exact and Heuristic Solution Techniques for Mixed-Integer Quantile Minimization Problems. INFORMS Journal on Computing, inPress. ⟨hal-03665771⟩