F. N. C. Sobral

J. Gondzio, F. N. C. Sobral

Interior Point Methods (IPM) rely on the Newton method for solving 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 efficiently solving linear systems, IPMs enjoy fast convergence 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 directly 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.

E. V. Castelani, R. Lopes, W. V. I. Shirabayashi et al.

The robust adjustment of nonlinear models to data is considered in this paper. When data comes from real experiments, it is possible that measurement errors cause the appearance of discrepant values, which should be ignored when adjusting models to them. This work presents a Lower Order-value Optimization (LOVO) version of the Levenberg-Marquardt algorithm, which is well suited to deal with outliers in fitting problems. A general algorithm is presented and convergence to stationary points is demonstrated. Numerical results show that the algorithm is successfully able to detect and ignore outliers without too many specific parameters. Parallel and distributed executions of the algorithm are also possible, allowing for the use of larger datasets. Comparison against publicly available robust algorithms shows that the present approach is able to find better adjustments in well known statistical models.