On diagonally-preconditioning the 2-step BFGS method with accumulated steps for linearly constrained nonlinear programming |
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Authors: | LF Escudero |
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Institution: | IBM Madrid Scientific Center, Paseo de la Castellana, 4, Madrid 1, Apartado 179, Spain |
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Abstract: | We present an algorithm for very large-scale linearly constrained nonlinear programming (LCNP) based on a Limited-Storage Quasi-newton method. In large-scale programming solving the reduced Newton equation at each iteration can be expensive and may not be justified when far from a local solution; besides, the amount of storage required by the reduced Hessian matrix, and even the computing time for its Quasi-Newton approximation, may be prohibitive. An alternative based on the reduced Truncated-Newton methodology, that has proved to be satisfactory for large-scale problems, is not recommended for very large-scale problems since it requires an additional gradient evaluation and the solving of two systems of linear equations per each minor iteration. We recommend a 2-step BFGS approximation of the inverse of the reduced Hessian matrix that does not require to store any matrix since the product matrix-vector is the vector to be approximated; it uses the reduced gradient and information from two previous iterations and the so-termed restart iteration. A diagonal direct BFGS preconditioning is used. |
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Keywords: | Preconditioning limited storage quasi-Newton methods de-activating strategies |
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