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Representations of quasi-Newton matrices and their use in limited memory methods. (English) Zbl 0809.90116

The authors derive new representations of limited memory quasi-Newton matrices and show how to use them efficiently in the kind of matrix computations required in constrained optimization methods. They present new expressions for both the BFGS and symmetric rank-one formulae for optimization and also derive a compact expression for Broyden’s method for solving systems of nonlinear equations.

These representations allow us to efficiently implement limited memory methods for large constrained optimization problems. In particular, it is discussed how to compute projections of limited memory matrices onto subspaces.

90C30Nonlinear programming
90C06Large-scale problems (mathematical programming)
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