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Examples of non-convergence of conjugate descent algorithms with exact line-searches

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Abstract

Examples are given in which quasi-Newton and other conjugate descent algorithms fail to converge to a minimum of the object function. In the first there is convergence to a point where the gradient is infinite; in the second, a region characterized by a fine terraced structure causes the iterates to spiral indefinitely. A modification of the second construction gives convergence to a point where the gradient is non-zero, but the gradient is not continuous at this point. The implication of these examples is discussed.

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Reference

  1. M.J.D. Powell, “Some properties of the variable Metric Algorithm”, in: F.A. Lootsma, ed.,Numerical Methods for non-linear optimization (Academic Press, New York 1972) pp. 1–17.

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Thompson, J.R. Examples of non-convergence of conjugate descent algorithms with exact line-searches. Mathematical Programming 12, 356–360 (1977). https://doi.org/10.1007/BF01593803

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  • DOI: https://doi.org/10.1007/BF01593803

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