Currently displaying 1 – 20 of 20

Showing per page

Order by Relevance | Title | Year of publication

A generalized limited-memory BNS method based on the block BFGS update

Vlček, JanLukšan, Ladislav — 2017

Programs and Algorithms of Numerical Mathematics

A block version of the BFGS variable metric update formula is investigated. It satisfies the quasi-Newton conditions with all used difference vectors and gives the best improvement of convergence in some sense for quadratic objective functions, but it does not guarantee that the direction vectors are descent for general functions. To overcome this difficulty and utilize the advantageous properties of the block BFGS update, a block version of the limited-memory BNS method for large scale unconstrained...

Modifications of the limited-memory BFGS method based on the idea of conjugate directions

Vlček, JanLukšan, Ladislav — 2013

Programs and Algorithms of Numerical Mathematics

Simple modifications of the limited-memory BFGS method (L-BFGS) for large scale unconstrained optimization are considered, which consist in corrections of the used difference vectors (derived from the idea of conjugate directions), utilizing information from the preceding iteration. For quadratic objective functions, the improvement of convergence is the best one in some sense and all stored difference vectors are conjugate for unit stepsizes. The algorithm is globally convergent for convex sufficiently...

Application of the infinitely many times repeated BNS update and conjugate directions to limited-memory optimization methods

Vlček, JanLukšan, Ladislav — 2019

Programs and Algorithms of Numerical Mathematics

To improve the performance of the L-BFGS method for large scale unconstrained optimization, repeating of some BFGS updates was proposed e.g. in [1]. Since this can be time consuming, the extra updates need to be selected carefully. We show that groups of these updates can be repeated infinitely many times under some conditions, without a noticeable increase of the computational time; the limit update is a block BFGS update [17]. It can be obtained by solving of some Lyapunov matrix equation whose...

A hybrid method for nonlinear least squares that uses quasi-Newton updates applied to an approximation of the Jacobian matrix

Lukšan, LadislavVlček, Jan — 2019

Programs and Algorithms of Numerical Mathematics

In this contribution, we propose a new hybrid method for minimization of nonlinear least squares. This method is based on quasi-Newton updates, applied to an approximation A of the Jacobian matrix J , such that A T f = J T f . This property allows us to solve a linear least squares problem, minimizing A d + f instead of solving the normal equation A T A d + J T f = 0 , where d R n is the required direction vector. Computational experiments confirm the efficiency of the new method.

A modified limited-memory BNS method for unconstrained minimization derived from the conjugate directions idea

Vlček, JanLukšan, Ladislav — 2015

Programs and Algorithms of Numerical Mathematics

A modification of the limited-memory variable metric BNS method for large scale unconstrained optimization of the differentiable function f : N is considered, which consists in corrections (based on the idea of conjugate directions) of difference vectors for better satisfaction of the previous quasi-Newton conditions. In comparison with [11], more previous iterations can be utilized here. For quadratic objective functions, the improvement of convergence is the best one in some sense, all stored corrected...

New quasi-Newton method for solving systems of nonlinear equations

Ladislav LukšanJan Vlček — 2017

Applications of Mathematics

We propose a new Broyden method for solving systems of nonlinear equations, which uses the first derivatives, but is more efficient than the Newton method (measured by the computational time) for larger dense systems. The new method updates QR or LU decompositions of nonsymmetric approximations of the Jacobian matrix, so it requires O ( n 2 ) arithmetic operations per iteration in contrast with the Newton method, which requires O ( n 3 ) operations per iteration. Computational experiments confirm the high efficiency...

Recursive form of general limited memory variable metric methods

Ladislav LukšanJan Vlček — 2013

Kybernetika

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 multiplications and additions per iteration, so it is comparable with other efficient limited memory variable metric methods. Numerical experiments concerning Algorithm...

Primal interior point method for minimization of generalized minimax functions

Ladislav LukšanCtirad MatonohaJan Vlček — 2010

Kybernetika

In this paper, 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. (i. e. interior point method that uses explicitly computed approximations of Lagrange multipliers instead of their updates). Next we describe the basic algorithm and give more details concerning its implementation...

Primal interior-point method for large sparse minimax optimization

Ladislav LukšanCtirad MatonohaJan Vlček — 2009

Kybernetika

In this paper, we propose a primal interior-point method for large sparse minimax optimization. After a short introduction, the complete algorithm is introduced and important implementation details are given. We prove that this algorithm is globally convergent under standard mild assumptions. Thus the large sparse nonconvex minimax optimization problems can be solved successfully. The results of extensive computational experiments given in this paper confirm efficiency and robustness of the proposed...

Page 1 Next

Download Results (CSV)