Updating quasi newton matrices with limited storage

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We give an update formula which generates matrices using information from the last m iterations, where m is any number supplied by the user.

We also present a compact representation of the matrices generated by Broyden's update for solving systems of nonlinear equations.

We derive compact representations of BFGS and symmetric rank-one matrices for optimization.

These representations allow us to efficiently implement limited memory methods for large constrained optimization problems.

And a new straightforward limited memory quasi-Newton updating based on the modified quasi-Newton equation is deduced to construct the trust region subproblem, in which the information of both the function value and gradient is used to construct approximate Hessian. Numerical results indicate that the proposed method is competitive and efficient on some classical large-scale nonconvex test problems. Trust region methods [1–14] are robust, can be applied to ill-conditioned problems, and have strong global convergence properties.

Another advantage of trust region methods is that there is no need to require the approximate Hessian of the trust region subproblem to be positive definite.

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