报告摘要：There are two open problems for nonconvex functions under the weak Wolfe-Powell (WWP) line search technique in unconstrained optimization problems. The first one is the global convergence of the Polak-Ribi\`{e}re-Polyak (PRP) conjugate gradient algorithm and the second one is the global convergence of the BFGS (Broyden, Fletcher, Goldfarb, and Shanno) quasi-Newton method. Many scholars have proven that these two problems do not converge, even if an exact line search is used. Two circle counterexamples were proposed to generate the nonconvergence of the PRP algorithm for the nonconvex functions under the exact line search (see Powell, Lecture Notes in Math. 1066(1984) and Dai, SIAM J. Optim. 13(2003) in detail), which inspired us to define a new technique to jump out of the circle point and obtain the global convergence. Thus, a new PRP conjugate gradient algorithm is designed by the following steps. (i) The current point $x_k$ is defined, and a parabolic surface $P_k$ is designed; (ii) an assistant point $\kappa_k$ is defined by the PRP formula based on $x_k$; (iii) $\kappa_k$ is projected onto the parabolic surface $P_k$ to generate the next point $x_{k+1}$; (iv) the presented PRP conjugate gradient algorithm has the global convergence for nonconvex functions with the WWP line search; (v) a similar technique is used for the BFGS quasi-Newton method to get a new BFGS algorithm and establish its global convergence; and (vi) The numerical results show that the given algorithms are more competitive than those of other similar algorithms. And the well-known hydrologic engineering application problem called parameter estimation problem of nonlinear Muskingum model is also done by the proposed algorithms.
