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Matrix algebras in quasi-Newton methods for unconstrained minimization. (English) Zbl 1034.65045

The authors describe a new class of quasi-Newton methods for unconstrained minimization problems. The matrix used in the recursion process is defined as the best Frobenius norm least-squares fit of the classical well known choice in quasi-Newton methods, by a special algebra of matrices simultaneously diagonalized by a unitary fast Fourier transform. The method requires \({\mathcal{O}}(n\log n)\) flops per iteration and \({\mathcal{O}}(n)\) memory allocations. Convergence and computational complexity are also analysed for the classes of methods described in the paper.

MSC:

65K05 Numerical mathematical programming methods
90C30 Nonlinear programming
90C53 Methods of quasi-Newton type

Software:

L-BFGS-B; L-BFGS
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