Quasi-newton Method. Family Extended Its Global Convergence,

The Quasi-Newton methods is one of the most effective mothods for solving the unconstrained optimization problems, whose basic idea is to estimate the second order derivatives with the first order derivatives.The main differences between different types of the Quasi-Newton methods are: the change way of estimate of the second order derivatives from one iterate to another iterate, and the types and accuracy of line search. so it produces a series of approximation matrix B_k+1 to the second order derivatives of the objective function. The nature of B_k+1 is B_k+1 = B_k + A_k, in which B_k is the approximation matrix at the last iteration, A_k is some matrix. In this paper, the writer first propose the conditionwhich can satisfy A_k is s_k^TA_ks_k =θ_k, in whichθ_k = 2(f_k-f_k+1) + s_k^T(g_k+1+g_k).And then three formulae to satisfy A_k are given: (1)(?); (2)(?); (3)(?), in whichu_k,u_k∈Rⁿ, and satisfy s_k^Tu_k≠0,s_k^Tu_k≠0. From which six reasonable choices are gotten: (1)(?)Based on those choices, the three corresponding algorithms are proved to possess global convergence property. At last eighteen popular test functions have conducted which show that the proposed algorithms are very encouraging.