A Bi-dimensional Search-based Algorithm for Unconstrained Optimization Problems

In this paper, a bi-dimensional search method is proposed for unconstrained optimization problems. Traditional methods for solving the unconstrained optimization problems are usually based on one-dimensional search (or line search) to determine the step size as these methods only use one descent direction in each iteration. When two search directions are available in each iteration, conventional one-dimensional search method is not applicable. In view of this, a bi-dimensional search method is needed to determine the step size vector. To do so, a mathematical formulation of the bi-dimensional search problem is presented. Then, using the second-order Taylor expansion the bi-dimensional search problem is approximately transformed into a least squares problem, which is easy to solve. The convergence of the proposed method is proved. Finally, numerical examples are given to demonstrate the efficiency of the proposed method.