A stabilized column generation scheme for the traveling salesman subtour problem

Abstract

Given an undirected graph with edge costs and both revenues and weights on the vertices, the traveling salesman subtour problem is to find a subtour that includes a depot vertex, satisfies a knapsack constraint on the vertex weights, and that minimizes edge costs minus vertex revenues along the subtour.We propose a decomposition scheme for this problem. It is inspired by the classic side-constrained 1-tree formulation of the traveling salesman problem, and uses stabilized column generation for the solution of the linear programming relaxation. Further, this decomposition procedure is combined with the addition of variable upper bound (VUB) constraints, which improves the linear programming bound. Furthermore, we present a heuristic procedure for finding feasible subtours from solutions to the column generation problems. An extensive experimental analysis of the behavior of the computational scheme is presented.

Power efficient uplink scheduling in SC-FDMA: benchmarking by column generation

“…Hence, the dual solutions obtained in consecutive iterations tend to vary significantly, which also leads to slow convergence in the primal problem. To improve the stability, the column generation scheme presented in Section 3 is modified to include a boxstep restriction (Marsten et al, 1975) on the individual dual variables, so that each new dual point is kept close to the previous one (see also, e.g., du Merle et al, 1999;Larsson et al, 2004;Westerlund et al, 2006). For the dual problem of the RLPM presented in Section 3.1, we choose to include box constraints only for the dual variables y nt , since the values of the dual variables v u are uniquely determined by the values of y nt due to the structure of the constraints.…”

Subgradient Optimization Prior to Column Generation --- A Means for Predicting Optimal Columns

“…This stabilization has been used with good results by e.g. Larsson et al [16] and Westerlund et al [27]. A similar stabilization technique is proposed by du Merle et al [7].…”

A branch‐and‐price approach for the stochastic generalized assignment problem

“…Moreover, if the parameter values pertaining to the assignment of jobs to machines are not much different, the SGAP ends up generating solutions having almost identical objective function values. Both of these phenomena can potentially adversely impact the convergence behavior of B&P. To ameliorate these impediments, as well as to curtail fluctuations in the dual‐solutions generated, we used a modified form of the Dual Slablization Technique (DST) suggested by Westerlund et al , which was adapted to suit our problem as discussed next.…”