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We study the solution of large-scale nonlinear optimization problems by methods which aim to exploit their inherent structure. In particular, we consider the all-pervasive property of partial separability, first studied by Griewank and Toint (1982b). A typical minimizationmethod for nonlinear optimization problems approximately solves a sequence of simplified linearized subproblems. In this paper, we explore how partial separability may be exploited by iterative methods for solving these subproblems. We particularly address the issue of computing effective preconditioners for such iterative methods. Numerical experiments indicate the effectiveness of these preconditioners on large-scale examples. Keywords: large-scale problems, unconstrained optimization, elememt-by-element preconditioners, conjugate-gradients. AMS(MOS) subject classifications: 65F05, 65F10, 65F15, 65F50, 65K05, 90C30. Also appeared as ENSEEIHT-IRIT report RT/APO/94/4. 1 Travel was funded, in part, by the ALLIANCE...

A nonmonotone strategy for solving nonlinear systems of equations is introduced. The idea consists of combining efficient local methods with an algorithm that reduces monotonically the squared norm of the system in a proper way. The local methods used are Newton's method and two quasiNewton algorithms. Global iterations are based on recently introduced boxconstrained minimization algorithms. We present numerical experiments. 1 INTRODUCTION Given F : IR n ! IR n ; F = (f 1 ; : : : ; f n ) T , our aim is to find solutions of F (x) = 0: (1) We assume that F is well defined and has continuous partial derivatives on an open set of IR n . J(x) denotes the Jacobian matrix of partial derivatives of F (x). We are mostly interested in problems where n is large and J(x) is structurally sparse. This means that most entries of J(x) are zero for all x in the domain of F . The package Nightingale has been developed at the Department of Applied Mathematics of the University of Campinas for...

In this paper we survey numerical methods for solving nonlinear systems of equations F (x) = 0, where F : IR n ! IR n . We are especially interested in large problems. We describe modern implementations of the main local algorithms, as well as their globally convergent counterparts. 1. INTRODUCTION Nonlinear systems of equations appear in many real - life problems. Mor'e [1989] has reported a collection of practical examples which include: Aircraft Stability problems, Inverse Elastic Rod problems, Equations of Radiative Transfer, Elliptic Boundary Value problems, etc.. We have also worked with Power Flow problems, Distribution of Water on a Pipeline, Discretization of Evolution problems using Implicit Schemes, Chemical Plant Equilibrium problems, and others. The scope of applications becomes even greater if we include the family of Nonlinear Programming problems, since the first-order optimality conditions of these problems are nonlinear systems. Given F : IR n ! IR n ; F = (...

In this paper, we propose a modication of the BFGS method for unconstrained optimization. A remarkable feature of the proposed method is that it possesses a global convergence property even without convexity assumption on the objective function. Under certain conditions, we also establish superlinear convergence of the method. Key words: BFGS method, global convergence, superlinear convergence 1 Present address (available before October, 1999): Department of Applied Mathematics and Physics, Graduate School of Engineering, Kyoto University, Kyoto 606, Japan, e-mail: lidh@kuamp.kyoto-u.ac.jp 1 Introduction Let f : R n ! R be continuously dierentiable. Consider the following unconstrained optimization problem: min f(x); x 2 R n : (1:1) Among numerous iterative methods for solving (1.1), quasi-Newton methods constitute particularly important class. Throughout the paper, we assume that f in (1.1) has Lipschitz continuous gradients, i.e. there is a constant L > 0 such kg(x) g(y)k ...

We consider so-called "matrix stretching" technique that make structured unassembled linear systems larger, but sparser. Our solution technique combines a direct factorization of the leading block diagonal submatrix of the stretched system, with a preconditioned conjugate gradient solution of the Schur complement system which results from the factorization of the diagonal blocks. We show that matrix stretching is an effective technique, particularly for ill-conditioned systems. The Schur complement is often considerably better conditioned than the whole system. The main challenge is to find a suitable preconditioner for this matrix. We consider a range of preconditioners, including those proposed by Chan, and band approximations. We also study the use of some Element-by-Element preconditioners such as EBE and the recently introduced Subspace-by-Subspace preconditioner. We report on experiments using structured problems and examples from the HarwellBoeing sparse matrix collection. We al...

Global Convergence of a Class of Collinear Scaling Algorithms with Inexact Line Searches on Convex Functions. Ariyawansa [2] has presented a class of collinear scaling algorithms for unconstrained minimization. A certain family of algorithms contained in this class may be considered as an extension of quasi-Newton methods with the Broyden family [11] of approximants of the objective function Hessian. Byrd, Nocedal and Yuan [7] have shown that all members except the DFP [11] method of the Broyden convex family of quasiNewton methods with Armijo [1] and Goldstein [12] line search termination criteria are globally and q-superlinearly convergent on uniformly convex functions. Extension of this result to the above class of collinear scaling algorithms of Ariyawansa [2] has been impossible because line search termination criteria for collinear scaling algorithms were not known until recently. Ariyawansa [4] has recently proposed such line search termination criteria. In this paper, we prove ...

In this paper, we propose a derivative-free line search suited to iterative methods for solving systems of nonlinear equations with symmetric Jacobian matrices. The proposed line search can be implemented conveniently by a backtracking process and has such an attractive property that any iterative method with this line search generates a sequence of iterates that is approximately norm descent. Moreover, if the Jacobian matrices are uniformly nonsingular, then the generated sequence converges to the unique solution. We incorporate this line search with a Gauss-Newton based DFP method for solving symmetric equations. Under appropriate conditions, we establish global and superlinear convergence of the proposed DFP method. The obtained results show, in particular, that the proposed DFP method with inexact line search converges globally and superlinearly even for nonconvex unconstrained optimization problems and equality constrained optimization problems.