A type of modified BFGS algorithm with any rank defects and the local Q-superlinear convergence properties
Tóm tắt
A modified BFGS algorithm for solving the unconstrained optimization, whose Hessian matrix at the minimum point of the convex function is of rank defects, is presented in this paper.The main idea of the algorithm is first to add a modified term to the convex function for obtain an equivalent model, then simply the model to get the modified BFGS algorithm. The superlinear convergence property of the algorithm is proved in this paper. To compared with the Tensor algorithms presented by R. B. Schnabel (seing [4],[5]), this method is more efficient for solving singular unconstrained optimization in computing amount and complication.
Tài liệu tham khảo
Dennis J J, Morè,Quasi-Newton Methods, Motivation and Theory, SIAM REIEW19 (1977), 46–89.
Powell M J D,Some Global Convergence Properties of a Variable Metric Algorithm for Minimization without Exact Line Searches, Nonlinear Programming, SIAM-AMS proceedings, Cottle R W and Lemke C E, eds.9 (1976), 53–72.
Werner J,Uber die global konvergenze von variablemetric verfahren mit nichtexakter schrittweitenbestimmong, Numer. Math.31 (1978), 321–334.
R B Schnabel and Ta-Tung Chow,Tensor Methods for Unconstrained Optimization Using Second Derivatives, SIAM J. Optimization1(3) (1991), 293–315.
Dan Feng and R B Schnabel,Tensor Methods for Equality Constrained Optimization, SIAM J. Optimization6(3) (1996), 653–673.
Ali Bouaricha,Tensor Methods for Large Sparse Unconstrained Optimization, SIAM J. Optimization7(3) (1997), 732–756.
Jie Han and Guanghui Liu,The general linear search model and global convergence of BFGS algorithm for unconstrained optimization, Acta Mathematicae Applicatae Sinica1 (1995),112–122.
Yaxiang Yuan and Wenyu Sun,The Theory and Methods of Optimization, Science Press, Beijing, 2001.
J M Ortega and W C Rheinboldt,Iterative Solution of nonlinear Equations in several Variables, Academic Press, London, 1971.
J J More, B S Garbow and K E Hillstrom,Testing Unconstrained Optimization Software, ACM Trans on Mathematical Software7(1) (1981), 19–31.
Dong-Hui Li and Masao Fukushima,On the Global Convergence of the BFGS Method for Nonconvex Unconstrained Optimization Problems, SIAM Journal on Optimization11(4) (2001), 1054–1064.