Abstract
In nonlinear optimization it is often important to estimate large sparse Hessian or Jacobian matrices, to be used for example in a trust region method. We propose an algorithm for computing a matrix B with a given sparsity pattern from a bundle of the m most recent difference vectors
where B should approximately map △ into Г. In this paper B is chosen such that it satisfies m quasi—Newton conditions B△ = Г in the least squares sense.
We show that B can always be computed by solving a positive semi—definite system of equations in the nonzero components of B. We give necessary and sufficient conditions under which this system is positive definite and indicate how B can be computed efficiently using a conjugate gradient method.
In the case of unconstrained optimization we use the technique to determine a Hessian approximation which is used in a trust region method. Some numerical results are presented for a range of unconstrained test problems.
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Fletcher, R., Grothey, A., Leyffer, S. (1997). Computing Sparse Hessian and Jacobian Approximations with Optimal Hereditary Properties. In: Biegler, L.T., Coleman, T.F., Conn, A.R., Santosa, F.N. (eds) Large-Scale Optimization with Applications. The IMA Volumes in Mathematics and its Applications, vol 93. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1960-6_3
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DOI: https://doi.org/10.1007/978-1-4612-1960-6_3
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