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The convergence of variable metric matrices in unconstrained optimization

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Abstract

It is proved that, if the DFP or BFGS algorithm with step-lengths of one is applied to a functionF(x) that has a Lipschitz continuous second derivative, and if the calculated vectors of variables converge to a point at which ∇F is zero and ∇2 F is positive definite, then the sequence of variable metric matrices also converges. The limit of this sequence is identified in the case whenF(x) is a strictly convex quadratic function.

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Ren-pu, G., Powell, M.J.D. The convergence of variable metric matrices in unconstrained optimization. Mathematical Programming 27, 123–143 (1983). https://doi.org/10.1007/BF02591941

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