Nonmonotone Modified BFGS Algorithm For Unconstrained Optimization

Quasi-Newton method is a kind of efficient methods to solve the general unconstrained optimization. The update of B_k is important for the convergence of algorithms.One property of the BFGS method is its self-correcting mechanism [19]. Within the framework of Byrd and Nocedalthe [5], BFGS method is better at correcting small eigenvalues than large ones. To find a method with better ability to correct large eigenvalues, Aiping Liao proposed a BFGS~L method [17]. Yuan [26] proposed a BFGS~Y Algorithm, in which quasi-Newton condition can be interpreted as the interpolation condition that the gradient value of the local quadratic model matches that of the objective function at the previous iterate. The BFGS~Y algorithm requires that the function value of the local quadratic model matches that of the objective function at the previous iterate. This algorithm preserves the global and local superlinear convergence properties of the BFGS algorithm.In this work, we combine the BFGS~L algorithm [17] with the popular non-monotone technique . we also combine the BFGS~Y algorithm [17] with the inexact line search and the nonmonotone technique. Numerical results to compare the behavior of this methods with modified the BFGS algorithms are presented. These results indicate that the nonmonotone BFGS~L algorithm,the non-monotone BFGS~Y is efficient.