The procedure for the complete updating of immersive sets in one cutting plane method

Abstract

In order to improve the practical implementation of the previously proposed cutting plane method, we have developed a procedure for updating immersive polyhedral sets in it. These sets successively approximate the epigraphs of some auxiliary functions that are being built at each step of the method. Each of the approximating sets is obtained by cutting off the current iterative point from the previous set by the support hyperplane. In connection with this, additional linear inequalities that form approximating sets are accumulated with an increase in the number of steps. This accumulation of additional constraints leads to an increase in the complexity of solving linear programming problems, which are set up in the method for finding iterative points. The procedure for updating approximating sets developed and included in the method allows periodically discarding accumulating cutting planes, which greatly simplifies the solution of auxiliary problems. This procedure is based on the introduced criterion for estimating the quality of the approximation of epigraphs. At those iterations, where the quality of the approximation becomes good enough, the mentioned updates occur. The convergence of the cutting plane method with the updating procedure included in it is proved. The implementation of the method is discussed.