New Quasi-Newton Equation And Method Via Higher Order Tensor Models

This thesis introduces a general approach by proposing a new quasi-Newton (QN) equation via fourth order tensor model. To approximate the curvature of the objective function, more available information from the function-values and gradient is employed. The efficiency of the usual QN methods is improved by accelerating the performance of the algorithms without causing more storage demand. The presented equation allows the modification of several algorithms involving QN equations for practical optimization that possess superior convergence prop- erty. By using a new equation, the BFGS method is modified. This is done twice by employing two different strategies proposed by Zhang and Xu (2001) and Wei et al. (2006) to generate positive definite updates. The superiority of these methods compared to the standard BFGS and the modification proposed by Wei et al. (2006) is shown. Convergence analysis that gives the local and global convergence property of these methods and numerical results that shows the advantage of the modified QN methods are presented. Moreover, a new limited memory QN method to solve large scale unconstrained optimization is developed based on the modified BFGS updated formula. The comparison between this new method with that of the method developed by Xiao et al. (2008) shows better performance in numerical results for the new method. The global and local convergence properties of the new method on uniformly convex problems are also analyzed. The compact limited memory BFGS method is modified to solve the large scale unconstrained optimization problems. This method is derived from the proposed new QN update formula. The new method yields a more efficient algorithm compared to the standard limited memory BFGS with simple bounds (L-BFGS-B) method in the case of solving unconstrained problems. The implementation of the new proposed method on a set of test problems highlights that the derivation of this new method is more efficient in performing the standard algorithm

On limited-memory quasi-Newton methods for minimizing a quadratic function

The main focus in this paper is exact linesearch methods for minimizing a quadratic function whose Hessian is positive definite. We give two classes of limited-memory quasi-Newton Hessian approximations that generate search directions parallel to those of the method of preconditioned conjugate gradients, and hence give finite termination on quadratic optimization problems. The Hessian approximations are described by a novel compact representation which provides a dynamical framework. We also discuss possible extensions of these classes and show their behavior on randomly generated quadratic optimization problems. The methods behave numerically similar to L-BFGS. Inclusion of information from the first iteration in the limited-memory Hessian approximation and L-BFGS significantly reduces the effects of round-off errors on the considered problems. In addition, we give our compact representation of the Hessian approximations in the full Broyden class for the general unconstrained optimization problem. This representation consists of explicit matrices and gradients only as vector components

Limited-memory BFGS Systems with Diagonal Updates

In this paper, we investigate a formula to solve systems of the form (B + {\sigma}I)x = y, where B is a limited-memory BFGS quasi-Newton matrix and {\sigma} is a positive constant. These types of systems arise naturally in large-scale optimization such as trust-region methods as well as doubly-augmented Lagrangian methods. We show that provided a simple condition holds on B_0 and \sigma, the system (B + \sigma I)x = y can be solved via a recursion formula that requies only vector inner products. This formula has complexity M^2n, where M is the number of L-BFGS updates and n >> M is the dimension of x

The LBFGS Quasi-Newtonian Method for Molecular Modeling Prion AGAAAAGA Amyloid Fibrils

Experimental X-ray crystallography, NMR (Nuclear Magnetic Resonance) spectroscopy, dual polarization interferometry, etc are indeed very powerful tools to determine the 3-Dimensional structure of a protein (including the membrane protein); theoretical mathematical and physical computational approaches can also allow us to obtain a description of the protein 3D structure at a submicroscopic level for some unstable, noncrystalline and insoluble proteins. X-ray crystallography finds the X-ray final structure of a protein, which usually need refinements using theoretical protocols in order to produce a better structure. This means theoretical methods are also important in determinations of protein structures. Optimization is always needed in the computer-aided drug design, structure-based drug design, molecular dynamics, and quantum and molecular mechanics. This paper introduces some optimization algorithms used in these research fields and presents a new theoretical computational method - an improved LBFGS Quasi-Newtonian mathematical optimization method - to produce 3D structures of Prion AGAAAAGA amyloid fibrils (which are unstable, noncrystalline and insoluble), from the potential energy minimization point of view. Because the NMR or X-ray structure of the hydrophobic region AGAAAAGA of prion proteins has not yet been determined, the model constructed by this paper can be used as a reference for experimental studies on this region, and may be useful in furthering the goals of medicinal chemistry in this field