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. [17] 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 [19] deduced modified quasi-Newton formula with limited memory compact representation based on the modified quasi-Newton equation with a vector parameter [18].

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:

updating quasi newton matrices with limited storage-14updating quasi newton matrices with limited storage-31

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 [3], trust region [4], or other regularization techniques [9, 13].

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