A New Implementation of the Generalized Basis Reduction Algorithm for Convex Integer Programming

This thesis is concerned with practical computation algorithms for solving general convex mixed-integer programming problems. Integer programming is an important mathematical model for economic decision making problems in the presence of indivisibilities. Integer programming problems are particularly difficult to solve because the traditional tests for optimality fail when some or all of the decision variables are discrete. In this thesis, we present a new implementation of the Lovasz and Scarf generalized basis reduction algorithm (Lovasz and Scarf, 1992) for convex mixed-integer programming. Our method generalizes branch and bound algorithms by branching on general linear integral functions rather than the coordinate variables. The generalized basis reduction algorithm is used to determine good integral linear functions that make the number of branches at each node small. Our method has been very successful in solving a set of linear and nonlinear testing problems.