Tools for primal degenerate linear programs: IPS, DCA, and PE

The Capacitated Vehicle Routing Problem: Stronger bounds in pseudo-polynomial time

“…The most pressing one is whether the new relaxations can somehow be solved more quickly, so that they can be used to compute strong bounds for instances of larger size (such as those described in [34]). Perhaps one of the recentlydeveloped techniques for alleviating primal degeneracy, surveyed in [9], could be useful. Alternatively, one could apply Lagrangian relaxation instead of Dantzig-Wolfe decomposition to MCF2b, and then solve the Lagrangian dual via a method with proven fast convergence (see, e.g., [14,21]).…”

“…That is, each member of Π is a subset of V c whose total demand does not exceed Q. Observe that each member of Π corresponds to an extreme point of the knapsack polytope (9). For each (i, j) ∈ A and each P ∈ Π, let σ P ij be a binary variable indicating whether a vehicle departs from the depot with loading pattern P and traverses the arc (i, j) at some point.…”

Stabilized column generation for the temporal knapsack problem using dual-optimal inequalities

“…In order to explicitly stabilize the dual values, algorithmic techniques like the box step method (Marsten et al, 1975), bundle methods (Hiriart-Urruty and Lemaréchal, 1993), and tailored stabilization approaches have been proposed (du Merle et al, 1999;Rousseau et al, 2007;Lee and Park, 2011). Furthermore, some recently proposed techniques can help overcome or even benefit from primal degeneracy when solving huge LPs (Gauthier et al, 2014;Desrosiers et al, 2014).…”