Jean Bertrand Gauthier

Row-reduced column generation for degenerate master problems

Column generation for solving linear programs with a huge number of variables alternates between solving a master problem and a pricing subproblem to add variables to the master problem as needed. The method is known to often suffer from degeneracy in the master problem. Inspired by recent advances in coping with degeneracy in the primal simplex method, we propose a row-reduced column generation method that may take advantage of degenerate solutions. The idea is to reduce the number of constraints to the number of strictly positive basic variables in the current master problem solution. The advantage of this row-reduction is a smaller working basis, and thus a faster re-optimization of the master problem. This comes at the expense of a more involved pricing subproblem, itself eventually solved by column generation, that needs to generate weighted subsets of variables that are said compatible with the row-reduction, if possible. Such a subset of variables gives rise to a strict improvement in the objective function value if the weighted combination of the reduced costs is negative. We thus state, as a by-product, a necessary and sufficient optimality condition for linear programming.

About the minimum mean cycle-canceling algorithm

This paper focuses on the resolution of the capacitated minimum cost flow problem on a network comprising? n nodes and? m arcs. We present a method that counts imperviousness to degeneracy among its strengths, namely the minimum mean cycle-canceling algorithm?(MMCC). At each iteration, primal feasibility is maintained and the objective function strictly improves. The goal is to write a uniform and hopefully more accessible paper which centralizes the ideas presented in the seminal work of Goldberg and Tarjan (1989) as well as the additional paper of Radzik and Goldberg (1994) where the complexity analysis is refined. Important properties are proven using linear programming rather than constructive arguments.We also retrieve Cancel-and-Tighten from the former paper, where each so-called phase which can be seen as a group of iterations requires O ( m log n ) time. MMCC turns out to be a strongly polynomial algorithm which runs in? O ( m n ) phases, hence in? O ( m 2 n log n ) time. This new complexity result is obtained with a combined analysis of the results in both papers along with original contributions which allows us to enlist Cancel-and-Tighten as an acceleration strategy.

Operating room management under uncertainty

The operating room management problems are legion. This paper tackles the scheduling of surgical procedures in an operating theatre containing up to two operating rooms and two surgeons. We first solve a deterministic version that uses the constraint programming paradigm and then a stochastic version which embeds the former in a sample average approximation scheme. The latter produces more robust schedules that cope better with the surgeries’ time variability

Vector Space Decomposition for Solving Large-Scale Linear Programs

We develop an algorithmic framework for linear programming guided by dual optimality considerations. The solution process moves from one feasible solution to the next according to an exchange mecha...

Linear fractional approximations for master problems in column generation

Abstract In the context of large-scale linear programs solved by a column generation algorithm, we present a primal algorithm for handling the master problem. Successive approximations of the latter are created to converge to optimality. The main properties are that, for every approximation except the last one, the cost of the solution decreases whereas the sum of the variable values increases. Moreover, the minimum reduced cost of the variables also increases and converges to zero with a super-geometric growth rate.