Dennis, J.E. Jr.Vu, Phuong2018-06-182018-06-181983-04Dennis, J.E. Jr. and Vu, Phuong. "Toward Direct Sparse Updates of Cholesky Factors." (1983) <a href="https://hdl.handle.net/1911/101554">https://hdl.handle.net/1911/101554</a>.https://hdl.handle.net/1911/101554A very important problem in numerical optimization is to find a way to update a sparse Hessian approximation so that it will be positive definite under reasonable circumstances. This problem has motivated research, which is yet to show much progress, toward a "sparse BFGS method." In this paper, we suggest a different approach to the problem based on using a sparse Broyden, or Schubert, update directly on the Cholesky factor of the current Hessian approximation to define the next Hessian approximation implicitly in terms of its Cholesky factorization. This approach has the added advantage of being able to cheaply find the Newton step, since no factorization step is required. The difficulty with our approach is in finding a satisfactory secant or quasi-Newton condition to use in the update.18 ppengTechnical reportTR83-13