## Abstract

Interior Point Methods (IPM) rely on the Newton method for solv-

ing systems of nonlinear equations. Solving the linear systems which arise from

this approach is the most computationally expensive task of an interior point

iteration. If, due to problem's inner structure, there are special techniques

for eciently solving linear systems, IPMs demonstrate a reduced computing

time and are able to solve large scale optimization problems. It is tempting to

try to replace the Newton method by quasi-Newton methods. Quasi-Newton

approaches to IPMs either are built to approximate the Lagrangian function

for nonlinear programming problems or provide an inexpensive preconditioner.

In this work we study the impact of using quasi-Newton methods applied di-

rectly to the nonlinear system of equations for general quadratic programming

problems. The cost of each iteration can be compared to the cost of computing

correctors in a usual interior point iteration. Numerical experiments show that

the new approach is able to reduce the overall number of matrix factorizations

and is suitable for a matrix-free implementation.

ing systems of nonlinear equations. Solving the linear systems which arise from

this approach is the most computationally expensive task of an interior point

iteration. If, due to problem's inner structure, there are special techniques

for eciently solving linear systems, IPMs demonstrate a reduced computing

time and are able to solve large scale optimization problems. It is tempting to

try to replace the Newton method by quasi-Newton methods. Quasi-Newton

approaches to IPMs either are built to approximate the Lagrangian function

for nonlinear programming problems or provide an inexpensive preconditioner.

In this work we study the impact of using quasi-Newton methods applied di-

rectly to the nonlinear system of equations for general quadratic programming

problems. The cost of each iteration can be compared to the cost of computing

correctors in a usual interior point iteration. Numerical experiments show that

the new approach is able to reduce the overall number of matrix factorizations

and is suitable for a matrix-free implementation.

Original language | English |
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Pages (from-to) | 93-120 |

Journal | Computational optimization and applications |

Volume | 74 |

Issue number | 1 |

Early online date | 2 May 2019 |

DOIs | |

Publication status | Published - 30 Sep 2019 |