摘要:

The trust region problem requires the global minimum of a general quadratic function subject to an ellipsoidal constraint. The development of algorithms for the solution of this problem has found applications in nonlinear and combinatorial optimization. In this paper we generalize the trust region problem by allowing a general quadratic constraint. The main results are a characterization of the global minimizer of the generalized trust region problem, and the development of an algorithm that finds an approximate global minimizer in a finite number of iterations.

ISSN:1055-6788    

DOI:10.1080/10556789308805542

出版商:Gordon and Breach Science Publishers    

年代:1993

数据来源: Taylor

摘要:

Second- and higher-order derivatives are required by applications in scientific computation, especially for optimization algorithms. The two complementary ideas of interpolating partial derivatives from univariate Taylor series and preaccumulating of “local” derivatives form the mathematical foundations for accurate, efficient computation of second-and higher-order partial derivatives for large codes. We compute derivatives in a fashion that parallelizes well, exploits sparsity or other structure frequently found in Hessian matrices, can compute only selected elements of a Hessian matrix, and computes Hessian × vector products.

ISSN:1055-6788    

DOI:10.1080/10556789308805543

出版商:Gordon and Breach Science Publishers    

年代:1993

数据来源: Taylor

摘要:

This paper is concerned with the development of the method of multipliers combined with the conjugate gradient algorithm for solving the special linear systemwhere A is a real symmetric non negative definite (n×n) matrixBis a real (n×m) matrix with full column rank (m<n) and A and BThave no nontrivial null vector in common. We assume that A and BT are large and not extremely sparse. The proposed method is well suitable for parallel implementation on a multiprocessor system that can execute concurrently different tasks on a few vector processors with shared central memory, such as the CRAY Y-MP. The results of an extensive computer experimentation, which is aimed at evaluating the effectiveness of the method, are reported.

ISSN:1055-6788    

DOI:10.1080/10556789308805544

出版商:Gordon and Breach Science Publishers    

年代:1993

数据来源: Taylor

摘要:

In this paper we describe the effects of an inexact implementation of Newton's method on the behavior of the iteration for certain nonlinear equations in Banach space for which the Fréchet derivative is singular at the solution. We give a termination criterion for the inner iteration that preserves not only the q-linear convergence of the Newton iterates but also the fine structure required for an acceleration method.

ISSN:1055-6788    

DOI:10.1080/10556789308805545

出版商:Gordon and Breach Science Publishers    

年代:1993

数据来源: Taylor

摘要:

Block bordered systems of nonlinear equations are systems whose Jacobian matrix consists of a series of diagonal blocks, plus a border of possibly dense rows and columns. Large systems of this type occur in many applications in science and engineering, and are attractive candidates for solution on parallel computers. Recently, Zhang, Byrd, and Schnabel developed a new class of algorithms for solving such systems that has significant computational advantages over previous methods on sequential computers, and even greater advantages on parallel computers. Its main feature is that the methods perform multiple inner iterations on the diagonal blocks for each outer iteration on the overall system, in a new way that retains fast local convergence. This paper investigates whether one can develop related algorithms that retain these advantages of parallelizability and fast local convergence, and are also globally convergent and computationally robust on a broad class of problems, including those where the Jacolbian matrix or any diagonal block is singular or ill-conditioned. We introduce related new algorithms for solving such systems, and show that they have strong global convergence properties under very mild assumptions, and retain parallelizability and fast local convergence.

ISSN:1055-6788    

DOI:10.1080/10556789308805546

出版商:Gordon and Breach Science Publishers    

年代:1993

数据来源: Taylor

摘要:

In [7], we proposed row update ABS methods for solving systems of nonlinear equations. A numerical comparison showed that the new methods are competitive with other methods. In this paper, we extend the row update ABS methods to sparse nonlinear systems of equations. We prove the local superlinear convergence of the proposed method.

ISSN:1055-6788    

DOI:10.1080/10556789308805547

出版商:Gordon and Breach Science Publishers    

年代:1993

数据来源: Taylor

摘要:

We consider the problem of solving the Quasi-Newton equation under the additional condition that some elements of the solution take prescribed values. The general solution is obtained in a straightforward way by using the Huang algorithm of the ABS class. It is given in explicit form without the need of solving intermediate linear systems. Particular cases include the general symmetric or unsymmetric full or sparse update. In the sparse symmetric case we show that while positive definiteness cannot generally be forced a weaker condition called quasi positive definiteness can be satisfied under a very mild condition. This condition guarantees that all eigenvalues, except at most two, are positive.

ISSN:1055-6788    

DOI:10.1080/10556789308805548

出版商:Gordon and Breach Science Publishers    

年代:1993

数据来源: Taylor

摘要:

When nonlinear equation solvers are applied to parameter-dependent problems, their iterates can be interpreted as functions of these variable parameters. The derivatives (if they exist) of these iterated functions can be recursively evaluated by the forward mode of automatic differentiation. Then one may ask whether and how fast these derivative values converge to the derivative of the implicit solution function, which may be needed for parameter identification, sensitivity studies, or design optimization.

ISSN:1055-6788    

DOI:10.1080/10556789308805549

出版商:Gordon and Breach Science Publishers    

年代:1993

数据来源: Taylor

摘要:

In a previous paper, Ford and Moghrabi [7] introduced a new, generalized approach to quasi-Newton methods, based on employing interpolatory polynomials which utilize infortion from the m most recent steps (where standard quasi-Newton methods correspond to m=1, working only with the latest step). In these new methods, the iterates where interpolated by a curve in such a way that consecutive points corresponeded to a unit-spacing of the parameter defining the curve. In this paper we derive and evalutate some alternative choices for defining the parameter-values which correspond to the iterates on the curve. The experimental results show clearly that such methods can give substantial gains in performance (by comparisaon with the “unit-spaced” method, itself yields improvements over standard quasi-Newton methods

ISSN:1055-6788    

DOI:10.1080/10556789308805550

出版商:Gordon and Breach Science Publishers    

年代:1993

数据来源: Taylor

ISSN:1055-6788    

DOI:10.1080/10556789308805541

出版商:Gordon and Breach Science Publishers    

年代:1993

数据来源: Taylor