Generalized Barzilai and Borwein method for large-scale unconstrained optimization.

The focus of the thesis is on finding the unconstrained minimizer of a function by using the fixed steps gradient method. Specifically, we will focus on the Barzilai and Borwein (BB) method. In this thesis, we propose a generalized Barzilai and Borwein (GBB) method that can overcome some disadvantages of the standard BB methods. The generalized Barzilai and Borwein method presented a special choice of steplength for the gradient method, which is a convex combination of two standard BB methods. Generally, the standard BB method does not guarantee a descent in the objective function at each iteration. We choose different scalar of combination between 0 and 1 to ensure that are descending in function value. This property is shown to be able to reduce the number of iteration in obtaining an approximate minimizer. The relationship between any gradient method and the shifted power method is considered. This relationship allows us to establish the convergence of the generalized Barzilai and Borwein method when applied to the problem of minimizing any strictly convex quadratic function. To highlight the performance of the generalized Barzilai and Borwein method, we applied them in solving the convex quadratic problem for the cases n = 2, 3, and 4. The results shown the number of iteration is decreasing for almost all cases. Finally, we concluded the achievements in our research and some future extensions are given at the end of thesi