Potential Reduction Algorithms

Potential reduction algorithms have a distinguished role in the area of interior point methods for mathematical programming. Karmarkar’s [44] algorithm for linear programming, whose announcement in 1984 initiated a torrent of research into interior point methods, used three key ingredients: a non–standard linear programming formulation, projective transformations, and a potential function with which to measure the progress of the algorithm. It was quickly shown that the non–standard formulation could be avoided, and eventually algorithms were developed that eliminated the projective transformations, but retained the use of a potential function. It is then fair to say that the only really essential element of Karmarkar’s analysis was the potential function. Further modifications to Karmarkar’s original potential function gave rise to potential reduction algorithms having the state–of–the– art theoretical complexity of O( √ nL) iterations, to solve a standard form linear program with n variables, and integer data with total bit size L. In the classical optimization literature, potential reduction algorithms are most closely related to Huard’s [39] “method of centres,” see also Fiacco and McCormick [21, Section 7.2]. However, Karmarkar’s use of a potential function to facilitate a complexity, as opposed to convergence analysis, was completely novel. The purpose of this article is to give a comprehensive survey of potential reduction algorithms for linear programming. (In the final section we will also briefly describe the extension of potential reduction algorithms to more general problems.) The major algorithms that are discussed, and analyzed, are Karmarkar’s algorithm, the affine potential reduction method, and the primal–dual potential reduction algorithm. The different algorithms are all