Ladislav Lukšan Simple modifiations of the limited-memory BFGS method (L-BFGS) for large scale unconstrained optimization are considered, which consist in corrections (derived from the idea of conjugate directions) of the used difference vectors, utilizing information from the preceding iteration. In case of quadratic objective functions, the improvement of convergence is the best one in some sense and … Read more
In this report we propose a new recursive matrix formulation of limited memory variable metric methods. This approach can be used for an arbitrary update from the Broyden class (and some other updates) and also for the approximation of both the Hessian matrix and its inverse. The new recursive formulation requires approximately $4 m n$ … Read more
This report is devoted to preconditioning techniques for the matrix-free truncated Newton method. After a review of basic known pproaches, we propose ew results concerning tridiagonal and pentadiagonal preconditioners based on the standard BFGS updates and on numerical differentiation. Furthermore, we present results of extensive numerical experiments serving for the careful comparison of suitable preconditioning … Read more
Two families of limited-memory variable metric or quasi-Newton methods for unconstrained minimization based on quasi-product form of update are derived. As for the first family, four variants how to utilize the Strang recurrences for the Broyden class of variable metric updates are investigated; three of them use the same number of stored vectors as the … Read more
The Broyden class of quasi-Newton updates for inverse Hessian approximation are transformed to the formal BFGS update, which makes possible to generalize the well-known Nocedal method based on the Strang recurrences to the scaled limited-memory Broyden family, using the same number of stored vectors as for the limited-memory BFGS method. Two variants are given, the … Read more
A new family of limited-memory variable metric or quasi-Newton methods for unconstrained minimization is given. The methods are based on a positive definite inverse Hessian approximation in the form of the sum of identity matrix and two low rank matrices, obtained by the standard scaled Broyden class update. To reduce the rank of matrices, various … Read more
In this report, several modifications of the nonlinear conjugate gradient method are described and investigated. Theoretical properties of these modifications are proved and their practical performance is demonstrated using extensive numerical experiments. Citation Technical report No. 1038, Institute of Computer Science, Pod Vodarenskou Vezi 2, 18207 Praha 8. December 2008 Article Download View Computational experience … Read more
In this report, we propose an algorithm for solving nonlinear programming problems with com-plementarity constraints, which is based on the interior-point approach. Main theoretical results concern direction determination and step-length selection. We use an exact penalty function to remove complementarity constraints. Thus a new indefinite linear system is defined with a tridiagonal low-right submatrix. Inexact … Read more
In this report, we propose a primal interior-point method for large sparse generalized minimax optimization. After a short introduction, where the problem is stated, we introduce the basic equations of the Newton method applied to the KKT conditions and propose a primal interior-point method. Next we describe the basic algorithm and give more details concerning … Read more
We present fourteen basic FORTRAN subroutines for large-scale unconstrained and box constrained optimization and large-scale systems of nonlinear equations. Subroutines {\tt PLIS} and {\tt PLIP}, intended for dense general optimization problems, are based on limited-memory variable metric methods. Subroutine {\tt PNET}, also intended for dense general optimization problems, is based on an inexact truncated Newton … Read more