The Homotopy Principle and Algorithms for Linear Programming

Linear programming techniques are formulated within the unifying framework of the homotopy principle and Newton’s method. Key strategies that determine the effectiveness of an implementation are considered in detail. A complexity analysis is developed for an elevator predictor, Newton corrector algorithm started at an arbitrary interior primal-dual feasible point. This analysis is based on a fundamental theorem of Smale [“Algorithms for solving equations,” Proc. Internat. Congress of Mathematicians, Universityof California, Berkeley, CA, 1986], in the form given by Renegar and Shub [Report No. 807, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY, 1988].