Samuel Rosat

Integral Simplex Using Decomposition with Primal Cuts

The integral simplex using decomposition (ISUD) algorithm [22] is a dynamic constraint reduction method that aims to solve the popular set partitioning problem (SPP). It is a special case of primal algorithms, i.e. algorithms that furnish an improving sequence of feasible solutions based on the resolution, at each iteration, of an augmentation problem that either determines an improving direction, or asserts that the current solution is optimal. To show how ISUD is related to primal algorithms, we introduce a new augmentation problem, MRA. We show that MRA canonically induces a decomposition of the augmentation problem and deepens the understanding of ISUD. We characterize cuts that adapt to this decomposition and relate them to primal cuts. These cuts yield a major improvement over ISUD, making the mean optimality gap drop from 33.92% to 0.21% on some aircrew scheduling problems.

Improved Primal Simplex: A More General Theoretical Framework and an Extended Experimental Analysis

In this article, we propose a general framework for an algorithm derived from the primal simplex that guarantees a strict improvement in the objective after each iteration. Our approach relies on the identification of compatible variables that ensure a nondegenerate iteration if pivoted into the basis. The problem of finding a strict improvement in the objective function is proved to be equivalent to two smaller problems, respectively, focusing on compatible and incompatible variables. We then show that the improved primal simplex (IPS) is a particular implementation of this generic theoretical framework. The resulting new description of IPS naturally emphasizes what should be considered as necessary adaptations of the framework versus specific implementation choices. This provides original insight into IPS that allows for the identification of weaknesses and potential alternative choices that would extend the efficiency of the method to a wider set of problems. We perform experimental tests on an extended collection of data sets including instances of Mittelmann’s benchmark for linear programming. The results confirm the excellent potential of IPS and highlight some of its limits while showing a path toward an improved implementation of the generic algorithm.

Dynamic penalization of fractional directions in the integral simplex using decomposition: Application to aircrew scheduling

To solve integer linear programs, primal algorithms follow an augmenting sequence of integer solutions leading to an optimal solution. In this work, we focus on a particular primal algorithm, the integral simplex using decomposition (ISUD). To find the next point, one solves a linear program to select an augmenting direction for the current point from a cone of feasible directions. To ensure that this linear program is bounded, a normalization constraint is added and the optimization is performed on a section of the cone. The solution of the linear program, i.e., the direction proposed by the algorithm, strongly depends on the chosen normalization weights, and so does the likelihood that the next solution is integer. We modify ISUD so that the normalization is dynamically updated whenever the direction leads to a fractional solution, to penalize that direction. We propose several update strategies, based on theoretical and experimental results. To prove the efficiency of our strategies, we show that our version of the algorithm yields better results than the former version and than classical branch-and-bound techniques on a benchmark of industrial aircrew scheduling instances. The benchmark that we propose here is, to the best of our knowledge, comparable to no other from the literature. It provides large-scale instances with up to 1,700 flights and 115,000 pairings, hence as many constraints and variables, and the instances are given in a set-partitioning form together with initial solutions that accurately mimic those of industrial applications. Our work shows the strong potential of primal algorithms for the crew scheduling problem, which is a key challenge for large airlines. Acknowledgments: This work was supported by a Collaborative research and development grant from NSERC and Kronos Inc. (RDCPJ 477127-14). Les Cahiers du GERAD G–2016–01 1