Convergence of the cyclical relaxation method for linear inequalities

The relaxation method for linear inequalities is studied and new bounds on convergence obtained. An asymptotically tight estimate is given for the case when the inequalities are processed in a cyclical order. An improvement of the estimate by an order of magnitude takes place if strong underrelaxation is used. Bounds on convergence usually involve the so-called condition number of a system of linear inequalities, which we estimate in terms of their coefficient matrix. Paper presented at the XI. International Symposium on Mathematical Programming, Bonn, August 23–27, 1982.