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Quasi-Newton methods are well known techniques for large-scale numericaloptimization. They use an approximation of the Hessian in optimization problemsor the Jacobian in system of nonlinear equations. In the Interior Pointcontext, quasi-Newton algorithms compute low-rank updates of the matrixassociated with the Newton systems, instead of computing it from scratch atevery iteration. In this work, we show that a simplified quasi-Newtonprimal-dual interior point algorithm for linear programming enjoys polynomialworst-case iteration complexity. Feasible and infeasible cases of the algorithmare considered and the most common neighborhoods of the central path areanalyzed. To the best of our knowledge, this is the first attempt to deliverpolynomial worst-case iteration complexity bounds for these methods.Unsurprisingly, the worst-case complexity results obtained when quasi-Newtondirections are used are worse than their counterparts when Newton directionsare employed. However, quasi-Newton updates are very attractive for large-scaleoptimization problems where the cost of factorizing the matrices is much higherthan the cost of solving linear systems.

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