Interior-point Algorithms: First Year Progress Report on Nsf Grant Ddm-8922636 1.1. Column Generation 1.2. Algorithm Termination

1. Introduction This document is the rst year progress report on the optimization projects funded by NSF Grant DDM-8922636. The projects principally include the interior-point algorithms for linear programming (LP), quadratic programming (QP), linear complementarity problem (LCP), and nonlinear programming (NP). The anticipated discoveries and advances resulting from the project include the following. While various interior point algorithms have been developed for LP (e.g., 12]]15]]17]]23]), they all share a common drawback: they require the complete knowledge of the full constraint system. In contrast, the (revised) simplex and ellipsoid methods do not require the complete knowledge of the system in advance. These methods allow for column generation|the addition of constraints to the system only when needed. This technique permits a great deal of exibility for solving optimization problems in which the number of constraints is either very large or not explicitly known. One main topic of the project is to develop interior-point algorithms that allow column generation and are implementable with practical eeciency. This will advance the practical solvability for a large class of optimization problems, such as semi-innnite programming, convex nonlinear programming, and combinatorial optimization. Unlike the simplex method for LP which terminates in nite time 5], interior-point algorithms are continuous optimization algorithms that generate a solution sequence converging to the optimal solution facet in innnite time. If the data for the LP is integral or rational, an argument is made that after a worst-case time bound the solution can be rounded from the latest approximate solution. However, several questions naturally arise. First, if the data is real, how do we achieve nite convergence (i.e., nd a solution in nite time)? Second, regardless of the data's status, can we utilize a practical, cost-eeective test to identify an solution so that the algorithm can be terminated before the worse-case time bound? Here, the solution may be given as a mathematical closed form using basic arithmetic 1