2018 Optimization Days
HEC Montréal, Québec, Canada, May 7 — 9, 2018
WA8 Primal and dual methods for integer programming
May 9, 2018 10:30 AM – 12:10 PM
Location: Metro inc. (72)
Chaired by Issmail El Hallaoui
4 Presentations

10:30 AM  10:55 AM
Primal integer optimization: A new paradigm
After some very interesting progress in solving the set partitioning problems (clustering problems for example), we introduce in this presentation a new primal paradigm, called Primal Integer Optimization (PIO), for the exact solution of general ILP, thus opening up a huge field of applications. These primal methods make it possible to go from an integer solution to a better one until the optimality is proven. To do this, they use a new method of decomposition that benefits, rather than suffer, from the degeneracy inherent in these problems. This translates in practice by two great advantages: i) this paradigm makes it possible to generate quite quickly several highquality solutions, contrary to the "dual" methods of the Branch & Cut type. The other major advantage of this paradigm is its great flexibility: we can easily integrate the metaheuristics, often used in practice, without sacrificing the exactness (optimality), which is the major drawback of the latter.

10:55 AM  11:20 AM
A multidirectional dynamic programming algorithm for the shortest path problem with resource constraints
The shortest path problem with resource constraints finds the least cost path between two nodes in a network while respecting constraints on resource consumption. The problem is mainly used as a subproblem inside column generation for crew scheduling and vehicle routing problems. The standard approach for the subproblems is based on dynamic programming. This class of methods is generally effective in practice when there are only a few resources, but it seems to be timeconsuming for huge instances with many resources. To handle this problem, we propose a new exact primal algorithm called the multidirectional dynamic programming algorithm (MDDPA). The proposed approach splits the state space into small disjoint subspaces. These subspaces are sequentially explored in several iterations, where each iteration builds on the previous ones, to reduce the dimension of the subspaces to explore and to quickly generate better paths. Computational experiments on vehicle and crew scheduling instances show the excellent performance of our approach compared to the standard dynamic programming method. In particular, MDDPA is able to generate feasible paths with up to 90% of the optimal cost in less than 10% of the time required by standard dynamic programming. This feature is useful in column generation and may greatly reduce the computational effort, because we can stop the MDDPA solution process once columns with sufficiently negative reduced costs are obtained.

11:20 AM  11:45 AM
The shortest path problem with congestion and waiting times
We focus on a shortest path problem in a congested network where the driver can wait at vertices to avoid wasting time in traffic. The objective is to minimize the driving time. We study the time complexity of the problem. Then, we describe two approaches by integer programming and dynamic programming.

11:45 AM  12:10 PM
A new Lagrangian relaxation for multicommodity capacitated network design problem
The usual Lagrangian relaxations for multicommodity capacitated network design are the socalled shortest path and knapsack relaxations, which are obtained, respectively, by relaxing linking constraints and flow conservation equations. We present a new reformulation and Lagrangian relaxation for the problem. We show that the Lagrangian dual bound improves upon the socalled strong LP bound (known to be equal to the Lagrangian dual bounds of the shortest path and knapsack relaxations).