Interior point methods are not worse than Simplex

Xavier Allamigeon, Daniel Dadush, Georg Loho, Bento Natura, László A. Végh

Abstract: We develop a new interior point method for solving linear programs. Our algorithm is universal in the sense that it matches the number of iterations of any interior point method that uses a self-concordant barrier function up to a factor $O(n^{1.5}\log n)$ for an $n$-variable linear program in standard form. The running time bounds of interior point methods depend on bit-complexity or condition measures that can be unbounded in the problem dimension. This is in contrast with the simplex method that always admits an exponential bound. Our algorithm also admits a combinatorial upper bound, terminating with an exact solution in $O(2^{n} n^{1.5}\log n)$ iterations. This complements previous work by Allamigeon, Benchimol, Gaubert, and Joswig (SIAGA 2018) that exhibited a family of instances where any path-following method must take exponentially many iterations.