Convergence Properties Of Multi-Directional Parallel Algorithms For Unconstrained Minimization
Cheng-xian Xu 1, Yue-ting Yang 11 Faculty of Sciences, Xi'an Jiaotong University, Xi'an 710049, China
Convergence properties of a class of multi-directional parallel quasi-Newton algorithms for the solution of unconstrained minimization problems are studied in this paper. At each iteration these algorithms generate several different quasi-Newton directions, and then apply line searches to determine step lengths along each direction, simultaneously. The next iterate is obtained among these trail points by choosing the lowest point in the sense of function reductions. Different quasi-Newton updating formulas from the Broyden family are used to generate a main sequence of Hessian matrix approximations. Based on the BFGS and the modified BFGS updating formulas, the global and superlinear convergence results are proved. It is observed that all the quasi-Newton directions asymptotically approach the Newton direction in both direction and length when the iterate sequence converges to a local minimum of the objective function, and hence the result of superlinear convergence follows.
Key words: Unconstrained minimization; Multi-directional parallel quasi-Newton method; Global convergece; Superlinear convergence.