## Warm start and $\varepsilon$-subgradients in the cutting plane scheme for block-angular linear programs

### J. Gondzio and J.-P. Vial

This paper addresses the issues involved with an interior point-based decomposition applied to the solution of linear programs with a block-angular structure. Unlike classical decomposition schemes that use the simplex method to solve subproblems, the approach presented in this paper employs a primal-dual infeasible interior point method. The above-mentioned algorithm offers perfect measure of the distance to optimality, which is exploited to terminate the algorithm earlier (with a rather loose optimality tolerance) and to generate $\epsilon$-subgradients. In the decomposition scheme, subproblems are sequentially solved for varying objective functions. It is essential to be able to exploit the optimal solution of the previous problem when solving a subsequent one (with a modified objective). Warm start routine is described that deals with this problem.

The approach proposed has been implemented within the context of two optimization codes freely available for research use: the Analytic Center Cutting Plane Method (ACCPM) - interior point based decomposition algorithm and the Higher Order Primal-Dual Method (HOPDM) - general purpose interior point LP solver. Computational results are given to illustrate the potential advantages of the approach applied to the solution of very large structured linear programs.

Logilab Technical Report 97.1 Section of Management Studies, University of Geneva, 102 Bd Carl Vogt, CH-1211 Geneva 4, Switzerland, June 1997.

Contact: gondzio@divsun.unige.ch