Decomposition for adjustable robust linear optimization subject to uncertainty polytope

• Published in: Computational management science Vol. 13; no. 2; pp. 219 - 239
• Main Authors: Ayoub, Josette, Poss, Michael
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.04.2016; Springer Nature B.V; Springer Verlag
• Subjects: Adjustable; Algorithms; Approximation; Budgeting; Business and Management; Computer Science; Decomposition; Linear programming; Operations Research; Operations Research/Decision Theory; Optimization; Optimization techniques; Polytopes; Separation; Studies; Uncertainty; United States > US; Adjustable robust optimization; Network design; Uncertainty polytope; Benders decomposition; Mixed-integer linear programming
• ISSN: 1619-697X; 1619-6988
• DOI: 10.1007/s10287-016-0249-2

We present in this paper a general decomposition framework to solve exactly adjustable robust linear optimization problems subject to polytope uncertainty. Our approach is based on replacing the polytope by the set of its extreme points and generating the extreme points on the fly within row generation or column-and-row generation algorithms. The novelty of our approach lies in formulating the separation problem as a feasibility problem instead of a max–min problem as done in recent works. Applying the Farkas lemma, we can reformulate the separation problem as a bilinear program, which is then linearized to obtained a mixed-integer linear programming formulation. We compare the two algorithms on a robust telecommunications network design under demand uncertainty and budgeted uncertainty polytope. Our results show that the relative performance of the algorithms depend on whether the budget is integer or fractional.