A Long-Step, Cutting Plane Algorithm for Linear and Convex Programming

A cutting plane method for linear programming is described. This method is an extension of Atkinson and Vaidya’s algorithm, and uses the central trajectory. The logarithmic barrier function is used explicitly, motivated partly by the successful implementation of such algorithms. This makes it possible to maintain primal and dual iterates, thus allowing termination at will, instead of having to solve to completion. This algorithm has the same complexity (O(nL) iterations) as Atkinson and Vaidya’s algorithm, but improves upon it in that it is a ‘long-step’ version, while theirs is a ‘short-step’ one in some sense. For this reason, this algorithm is computationally much more promising as well. This algorithm can be of use in solving combinatorial optimization problems with large numbers of constraints, such as the Traveling Salesman Problem.