Modifications for Quasi-Newton Method and Its Spectral Algorithm for Solving Unconstrained Optimization Problems

Abstract

In this paper, two modifications for spectral quasi-Newton algorithm of type BFGS are imposed. In the first algorithm, named SQNEI, a certain spectral parameter is used in such a step for BFGS algorithm differs from other presented algorithms. The second algorithm, SQNEv-Iv, has both new parameter position and value suggestion. In SQNEI and SQNEv-Iv methods, the parameters are involved in a search direction after an approximated Hessian matrix is updated. It is provided that two methods are effective under some assumptions. Moreover, the sufficient descent property is proved as well as the global and superlinear convergence for SQNEv-Iv and SQNEI.  Both of them are superior the standard BFGS (QNBFGS) and previous spectral quasi-Newton (SQNLC). However, SQNEv-Iv is outstanding SQNEI if it is convergent to the solution. This means that, two modified methods are in the race for the more efficiency method in terms less iteration numbers and consuming time in running CPU. Finally, numerical results are presented for the four algorithms by running list of test problems with inexact line search satisfying Armijo condition.