Research on Three-Dimensional Extension of Barzilai-Borwein-like Method

The Barzilai-Borwein (BB) method usually uses BB stepsize for iteration so as to eliminate the line search step in the steepest descent method. In this paper, we modify the BB stepsize and extend it to solve the optimization problems of three-dimensional quadratic functions. The discussion is divided into two cases. Firstly, we study the case where the coefficient matrix of the quadratic term of quadratic function is a special third-order diagonal matrix and prove that using the new modified stepsize, this case is R-superlinearly convergent. In addition to that, we extend it to n-dimensional case and prove the rate of convergence is R-linear. Secondly, we analyze that the coefficient matrix of the quadratic term of quadratic function is a third-order asymmetric matrix, that is, when the matrix has a double characteristic root and prove the global convergence of this case. The results of numerical experiments show that the modified method is effective for the above two cases.