Using central prices in the decomposition of linear programs

Abstract

This paper deals with a decomposition technique for linear programs which proposes a new treatment of the master program in the classical Dantzig-Wolfe algorithm. This approach exploits (a) the relationship between the master program and the minimization of a convex piecewise linear function and (b) a recently proposed cutting plane algorithm for convex nondifferentiable optimization. The new algorithm seeks to achieve ‘deep’ cuts by generating them at, or near to, the analytic centre of a set of localization containing the solution. Therefore each iteration entails a sizeable decrease of the set of localization and the overall algorithm shows fast convergence. This algorithm has been used to solve reputedly difficult convex nondifferentiable optimization problems. Its implementation, as a decomposition algorithm for large-scale structured linear program, seems to be a promising alternative to the standard Dantzig-Wolfe approach which suffers from very slow convergence in many practical problems. The new approach differs from the Dantzig-Wolfe algorithm through the fact that it does not use prices corresponding to an optimal combination of the past proposals of the subproblem, but rather ‘central’ prices (i.e. analytic centres) which balance them all. This property seems to explain the good behaviour of the new algorithm.