Stabilized column generation for the temporal knapsack problem using dual-optimal inequalities
We present two new methods to stabilize column-generation algorithms for the temporal knapsack problem (TKP). Caprara et al. (INFORMS J Comp 25(3):560–571, 2013] were the first to suggest the use of branch-and-price algorithms for Dantzig–Wolfe reformulations of the TKP. Herein, the respective pricing problems are smaller-sized TKP that can be solved with a general-purpose MIP solver or by dynamic programming. Our stabilization methods are tailored to the TKP as they use (deep) dual-optimal inequalities, that is, inequalities known to be fulfilled by all (at least some) optimal dual solutions to the linear relaxation. Extensive computational tests reveal that both new stabilization techniques are helpful. Several previously unsolved instances are now solved to proven optimality.
KeywordsColumn generation Dual inequalities Stabilization
This research was funded by the Deutsche Forschungsgemeinschaft (DFG) under Grant No. IR 122/6-1.
- Bartlett M, Frisch A, Hamadi Y, Miguel I, Tarim S, Unsworth C (2005) The temporal knapsack problem and its solution. In: Barták R, Milano M (eds) Integration of AI and OR techniques in constraint programming for combinatorial optimization problems, Lecture notes in computer science, vol 3524, Springer, Berlin, pp 34–48, doi: 10.1007/11493853_5
- Desaulniers G, Desrosiers J, Solomon M (eds) (2005) Column generation. Springer, New YorkGoogle Scholar
- Gauthier JB, Desrosiers J, Lübbecke ME (2016) Tools for primal degenerate linear programs. EURO J Transport Logist 5(2):161–204. doi: 10.1007/s13676-015-0077-5
- Hiriart-Urruty JB, Lemaréchal C (1993) Convex analysis and minimization algorithms, part 2: advanced theory and bundle methods, Grundlehren der mathematischen Wissenschaften, vol 306. Springer, BerlinGoogle Scholar
- Poggi de Aragao M, Uchoa E (2003) Integer program reformulation for robust branch-and-cut-and-price algorithms. In: Proc. Conf. Math. Program in Rio: A Conference in Honour of Nelson Maculan, Rio de Janeiro, Brazil, pp 56–61Google Scholar