Λεπτομέρειες βιβλιογραφικής εγγραφής
| Τίτλος: |
Nested Set-Covering/Packing Problem: Degeneracy Alleviation and Dual Stabilization. |
| Συγγραφείς: |
Guo, Siqi, Xiao, Fan, Liang, Zhe |
| Πηγή: |
Operations Research; Nov/Dec2025, Vol. 73 Issue 6, p2886-2913, 28p |
| Θεματικοί όροι: |
Column generation (Algorithms), Packing problem (Mathematics), Mathematical programming, Mathematical inequalities, Mathematical optimization |
| Περίληψη: |
Dual-optimal and deep dual-optimal inequalities (DOIs/DDOIs) are known to aid dual stabilization and accelerate convergence. In "Nested Set-Covering/Packing Problem: Degeneracy Alleviation and Dual Stabilization," Siqi Guo, Fan Xiao, and Zhe Liang observe that DOIs/DDOIs also increase primal degeneracy, thereby extra effort is needed to prove optimality. Therefore, when addressing linear programs, it is critical to stabilize the dual as well as reduce primal degeneracy. This paper studies a nested multistage set-covering problem and proposes a nested set-covering model, which can be viewed as a traditional set-covering problem with special DDOIs. The authors derive strengthened reduced costs that can prescreen and prune degenerate variables and a pruning-and-pricing mechanism that utilizes both standard and strengthened reduced costs to price out promising variables. An efficient nested column-and-row generation algorithm is developed to exploit the benefits of both dual stabilization and degeneracy alleviation. It is widely acknowledged that deep dual-optimal inequalities (DDOIs) can stabilize the dual of a linear programming problem and accelerate its convergence. However, we find that adding DDOIs is not free but always comes with a price; that is, it increases the number of primal degenerate bases, and extra effort might be needed to achieve dual feasibility and prove its optimality. As a result, when addressing a linear programming problem, it is critical to stabilize the dual on the one hand, reducing primal degeneracy on the other hand. In this paper, we study a nested multistage set-covering problem in which the results of the previous stage become the entities that make up the next stage decisions. We propose a nested set-covering model (NCM), which can be viewed as a traditional set-covering problem with DDOIs. Unlike the standard DDOIs, this new type of DDOI also increases the dimension of the dual problem, and we refer to it herein as Lift-DDOI. Furthermore, we demonstrate that introducing a large number of DDOIs/Lift-DDOIs may exacerbate the problem degeneracy. To resolve this issue, we derive strengthened reduced costs that can prescreen and prune degenerate variables and a pruning-and-pricing mechanism that utilizes both standard and strengthened reduced costs to price out promising variables. An efficient nested column-and-row generation algorithm is developed to exploit both the dual stabilization and degeneracy alleviation effects. We also demonstrate that the above findings and methods can be adapted to nested set-packing problems. A comprehensive computational study on the airline crew scheduling problem shows that the proposed Lift-DDOIs and the column-and-row generation algorithm with the pruning-and-pricing mechanism can reduce the solution time by an average of 71.43% compared with a standard column-generation approach for traditional set-covering problems. [ABSTRACT FROM AUTHOR] |
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