A subspace adaptation of the Coleman-Li trust region and interior method is proposed for solving large-scale bound-constrained minimization problems. This method can be implemented with either sparse Cholesky factorization or conjugate gradient computation. Under reasonable conditions the convergence properties of this subspace trust region method are as strong as those of its full-space version.

. The search for low energy states of molecular clusters is associated with the study of molecular conformation and especially protein folding. This paper describes a new global minimization algorithm which is effective and efficient for finding low energy states and hence stable structures of molecular clusters. The algorithm combines simulated annealing with a class of effective energy functions which are transformed from the original energy function based on the theory of renormalization groups. The algorithm converges to low energy states asymptotically, and is more efficient than a general simulated annealing method. Abbreviated title: Effective Energy Simulated Annealing for Molecular Conformation Key words: global/local minimization, simulated annealing, renormalization group, parallel computation, protein folding AMS (MOS) subject classification: 49M37, 68Q22, 92C40 y Department of Computer Science and Advanced Computing Research Institute, Cornell University, Ithaca, NY 148...

We consider three parametric relaxations of the 0-1 quadratic programming problem. These relaxations are to: quadratic maximization over simple box constraints, quadratic maximization over the sphere, and the maximum eigenvalue of a bordered matrix. When minimized over the parameter, each of the relaxations provides an upper bound on the original discrete problem. Moreover, these bounds are efficiently computable. Our main result is that, surprisingly, all three bounds are equal. This author would like to thank the Department of Civil Engineering and Operations Research, Princeton University, for their support during his research leave. Key words: quadratic boolean programming, bounds, quadratic programming, trust region subproblems, minmax eigenvalue problems. AMS 1991 Subject Classification: Primary: 90C09, 90C25; Secondary: 90C27, 90C20. 1 INTRODUCTION Consider the \Sigma1 quadratic programming problem (P ) ¯ := max q(x) := x t Qx + c t x; x 2 F := f\Gamma1; 1g n ; ...

We propose a "layered-step" interior point (LIP) algorithm for linear programming. This algorithm follows the central path, either with short steps or with a new type of step called a "layered least squares" (LLS) step. The algorithm returns the exact global minimum after a finite number of steps---in particular, after O(n 3:5 c(A)) iterations, where c(A) is a function of the coefficient matrix. The LLS steps can be thought of as accelerating a path-following interior point method whenever near-degeneracies occur. One consequence of the new method is a new characterization of the central path: we show that it composed of at most n 2 alternating straight and curved Department of Computer Science, Upson Hall, Cornell University, Ithaca, NY 14853. Email: vavasis@cs.cornell.edu. This work is supported in part by the National Science Foundation, the Air Force Office of Scientific Research, and the Office of Naval Research, through NSF grant DMS-8920550. Also supported in part by an ...

This paper reviews advances in Newton, quasi-Newton and conjugate gradient methods for large scale optimization. It also describes several packages developed during the last ten years, and illustrates their performance on some practical problems. Much attention is given to the concept of partial separabilitywhich is gaining importance with the arrival of automatic differentiation tools and of optimization software that fully exploits its properties.

In nonlinear optimization it is often important to estimate large sparse Hessian or Jacobian matrices, to be used for example in a trust region method. We propose an algorithm for computing a matrix B with a given sparsity pattern from a bundle of the m most recent difference vectors \Delta = h ffi k\Gammam+1 : : : ffi k i ; \Gamma = h fl k\Gammam+1 : : : fl k i ; where B should approximately map \Delta into \Gamma. In this paper B is chosen such that it satisfies m quasi--Newton conditions B\Delta = \Gamma in the least squares sense. We show that B can always be computed by solving a positive semi--definite system of equations in the nonzero components of B. We give necessary and sufficient conditions under which this system is positive definite and indicate how B can be computed efficiently using a conjugate gradient method. In the case of unconstrained optimization we use the technique to determine a Hessian approximation which is used in a trust region method. Some n...

Abstract As cyberspace becomes an integral part of our daily life, its mastering becomes harder. To help, cyberspace can be represented by resources arranged in a multidimensional space. With geographical maps to exhibit the topology of this virtual space, people can have a better visual understanding. In this paper, methods focusing on the construction of lower dimension representations of this space are examined and illustrated with the World-Wide Web. It is expected that this work will contribute to addressing issues of navigation in cyberspace and, especially, avoiding the lost-in-cyberspace syndrome. Résumé Alors que le cyberspace envahit notre vie quotidienne, sa maîtrise devient de plus en plus complexe. On peut l’imaginer comme un ensemble de ressources arrangées dans un espace multi-dimensionnel. En utilisant des cartes géographiques pour représente la topologie virtuelle de cet espace, on arrive à mieux le comprendre, le cerner. Dans ce papier, des méthodes se concentrant sur la construction de représentations à dimensions réduites sont étudiées en les appliquant au World-Wide Web. On espère que ce travail contribuera à résoudre les problèmes de navigation dans ce monde virtuel et en particulier à éviter de s’y perdre. Ubersicht In einer Zeit, in der der Cyberspace ein integraler Bestandteil unseres täglichen Lebens wird, wird seine Beherrschung zunehmend schwieriger. Zur Erleichterung kann Cyberspace anhand von Quellen, angeordnet in einem multidimensionalen Raum, dargestellt werden. Mit geographischen Karten, die die Topologie dieses künstlichen Raumes aufzeigen, kann das visuelle Verständnis verbessert werden. In dieser Arbeit werden Methoden zur Konstruktion von Darstellungen mit niedriger Dimension dieses Raumes untersucht und anhand des World-Wide Web verdeutlicht.

In this survey paper, an overview on the different approaches for solving nonlinear optimization problems is given. The presentation includes theory, numeric, interval and symbolic methods.

This report consists of two parts. The first part contains a short summary of the progress made in the last six months on each of the four main projects: parallelizing compilers, computational linear algebra, computational optimization, and numerical methods for partial differential equations. Included also are a list of ACRI researchers and their research interests, a list of technical reports produced in the last six months, and a list of ACRI seminars. In the second part we highlight one of the projects, the parallelizing compiler work, where we give a more detailed introduction into this area and sketch our novel approach. Contents

In nonlinear optimization it is often important to estimate large sparse Hessian or Jacobian matrices, to be used for example in a trust region method. We propose an algorithm for computing a matrix B with a given sparsity pattern from a bundle of the m most recent difference vectors \Delta = h ffi k\Gammam+1 : : : ffi k i ; \Gamma = h fl k\Gammam+1 : : : fl k i ; where B should approximately map \Delta into \Gamma. In this paper B is chosen such that it satisfies m quasi--Newton conditions B\Delta = \Gamma in the least squares sense. We show that B can always be computed by solving a positive semi--definite system of equations in the nonzero components of B. We give necessary and sufficient conditions under which this system is positive definite and indicate how B can be computed efficiently using a conjugate gradient method. In the case of unconstrained optimization we use the technique to determine a Hessian approximation which is used in a trust region method. Some numerical results are presented for a range of unconstrained test problems. Keywords: Sparse nonlinear equations, sparse Hessian, limited memory, Procrustes Problems 1