Hatem Ben Amor

On the choice of explicit stabilizing terms in column generation

Column generation algorithms are instrumental in many areas of applied optimization, where linear programs with an enormous number of columns need to be solved. Although successfully employed in many applications, these approaches suffer from well-known instability issues that somewhat limit their efficiency. Building on the theory developed for nondifferentiable optimization algorithms, a large class of stabilized column generation algorithms can be defined which avoid the instability issues by using an explicit stabilizing term in the dual; this amounts at considering a (generalized) augmented Lagrangian of the primal master problem. Since the theory allows for a great degree of flexibility in the choice and in the management of the stabilizing term, one can use piecewise-linear or quadratic functions that can be efficiently dealt with using off-the-shelf solvers. The practical effectiveness of this approach is demonstrated by extensive computational experiments on large-scale Vehicle and Crew Scheduling problems. Also, the results of a detailed computational study on the impact of the different choices in the stabilization term (shape of the function, parameters), and their relationships with the quality of the initial dual estimates, on the overall effectiveness of the approach are reported, providing practical guidelines for selecting the most appropriate variant in different situations.

Dual-Optimal Inequalities for Stabilized Column Generation

Column generation is one of the most successful approaches for solving large-scale linear programming problems. However, degeneracy difficulties and long-tail effects are known to occur as problems become larger. In recent years, several stabilization techniques of the dual variables have proven to be effective. We study the use of two types of dual-optimal inequalities to accelerate and stabilize the whole convergence process. Added to the dual formulation, these constraints are satisfied by all or a subset of the dual-optimal solutions. Therefore, the optimal objective function value of the augmented dual problem is identical to the original one. Adding constraints to the dual problem leads to adding columns to the primal problem, and feasibility of the solution may be lost. We propose two methods for recovering primal feasibility and optimality, depending on the type of inequalities that are used. Our computational experiments on the binary and the classical cutting-stock problems, and more specifically on the so-called triplet instances, show that the use of relevant dual information has a tremendous effect on the reduction of the number of column generation iterations.

Stabilized column generation for highly degenerate multiple-depot vehicle scheduling problems

Column generation has proven to be efficient in solving the linear programming relaxation of large scale instances of the multiple-depot vehicle scheduling problem (MDVSP). However difficulties arise when the instances are highly degenerate. Recent research has been devoted to accelerate column generation while remaining within the linear programming framework. This paper presents an efficient approach to solve the linear relaxation of the MDVSP. It combines column generation, preprocessing variable fixing, and stabilization. The outcome shows the great potential of such an approach for degenerate instances.

A proximal trust-region algorithm for column generation stabilization

This paper proposes a generalization of the proximal point algorithm using both penalty and trust-region concepts. Finite convergence is established while assuming the trust regions are of full dimension and never shrink to a single point. The approach is specialized to the cutting plane/column generation context. The resulting algorithm ensures convergence to a pair of primal and dual optimal solutions. Computational experiments carried over multi-depot vehicle scheduling instances show a great stabilizing and accelerating effect on the column generation method.

Recovering an optimal LP basis from an optimal dual solution

Given a linear program, we describe an approach for crossing over from an optimal dual solution to an optimal basic primal solution. It consists in restricting the dual problem to a small box around the available optimal dual solution then, resolving the associated modified primal problem.