Polynomial worst-case iteration complexity of quasi-Newton primal-dual interior point algorithms for linear programming

Abstract

Quasi-Newton methods are well known techniques for large-scale numerical optimization. They use an approximation of the Hessian in optimization problems or the Jacobian in system of nonlinear equations. In the Interior Point context, quasi-Newton algorithms compute low-rank updates of the matrix associated with the Newton systems, instead of computing it from scratch at every iteration. In this work, we show that a simplified quasi-Newton primal-dual interior point algorithm for linear programming, which alternates between Newton and quasi-Newton iterations, enjoys polynomial worst-case iteration complexity. Feasible and infeasible cases of the algorithm are considered and the most common neighborhoods of the central path are analyzed. To the best of our knowledge, this is the first attempt to deliver polynomial worst-case iteration complexity bounds for these methods. Unsurprisingly, the worst-case complexity results obtained when quasi-Newton directions are used are worse than their counterparts when Newton directions are employed. However, quasi-Newton updates are very attractive for large-scale optimization problems where the cost of factorizing the matrices is much higher than the cost of solving linear systems.