The Projection Technique for Two Open Problems of Unconstrained Optimization Problems

There are two problems for nonconvex functions under the weak Wolfe–Powell line search in unconstrained optimization problems. The first one is the global convergence of the Polak–Ribière–Polyak conjugate gradient algorithm and the second is the global convergence of the Broyden–Fletcher–Goldfarb–Shanno quasi-Newton method. Many scholars have proven that the two problems do not converge, even under an exact line search. Two circle counterexamples were proposed to generate the nonconvergence of the Polak–Ribière–Polyak algorithm for the nonconvex functions under the exact line search, which inspired us to define a new technique to jump out of the circle point and obtain the global convergence. Thus, a new Polak–Ribière–Polyak algorithm is designed by the following steps. (i) Given the current point and a parabolic surface is designed; (ii) An assistant point is defined based on the current point; (iii) The assistant point is projected onto the surface to generate the next point; (iv) The presented algorithm has the global convergence for nonconvex functions with the weak Wolfe–Powell line search. A similar technique is used for the quasi-Newton method to get a new quasi-Newton algorithm and to establish its global convergence. Numerical results show that the given algorithms are more competitive than other similar algorithms. Meanwhile, the well-known hydrologic engineering application problem, called parameter estimation problem of nonlinear Muskingum model, is also done by the proposed algorithms.