Updating quasi newton matrices with limited storage
In Section 3, a Newton-like trust region method for large-scale unconstrained nonconvex minimization is proposed and the convergence property is proved under some reasonable assumptions. In this section, we deduce a straightforward limited memory quasi-Newton updating based on the modified quasi-Newton equation, which employs both the gradients and function values to construct the approximate Hessian and is a compensation for the missing data in limited memory techniques.
And then we apply the derived formula in trust region method.
Solving-large scale problems needs expensive computation and storage.
Liu and Nocedal [15, 16] proposed a limited memory BFGS method (L-BFGS) for solving unconstrained optimization and proved its global convergence.  gave the compact representations of the limited memory BFGS and SR1 formula, which made it possible for combining limited memory techniques with trust region method.
Considering that the L-BFGS updating formula used the gradient information merely and ignored the available function value information, Yang and Xu  deduced modified quasi-Newton formula with limited memory compact representation based on the modified quasi-Newton equation with a vector parameter .
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.
Search for updating quasi newton matrices with limited storage:
Newton’s method has been efficiently safeguarded to ensure its global convergence to first- and even second-order critical points, in the presence of local nonconvexity of the objective using line search , trust region , or other regularization techniques [9, 13].