FOR INTEGER PROGRAMS

Recent work by Fisher and Shapiro has used Gomory's group theoretic methods together with Lagrange multipliers to obtain bounds for the optimal value of integer programs. Here it is shown how an extension of the associated Abelian group can improve the bound without the artificiality of adding a cut. It is proved that there exists a finite group for which the dual ascent procedure of Fisher and Shapiro converges to the optimal integer solution. Constructive methods are given for finding this group. A worked example is included. A primal-dual ascent algorithm of Fisher and Shapiro (3) uses Lagrange multipliers with Gomory's group problems (5) to give bounds on the optimal value of integer programs. If the ascent procedure does not discover the optimal solution, a cut is available to add to the original I.P. problem after which the whole process of forming the group problem and applying the ascent procedure may be repeated. This paper gives a method of improving the bound without the need of adding a cut by using a sequence of group relaxations. It is proved that a finite group exists for which the ascent procedure gives the optimal solution, and a constructive method for finding this group is given. The first section outlines the basic ideas of the primal-dual ascent method of Fisher and Shapiro. 1. The primal-dual ascent algorithm. The general linear integer pro-