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.

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