Conditioning and Hessians in analytical and numerical optimization - Some illustrations

This note illustrates the connections between the Hessians of numerical optimization problems, variance-covariance matrices for parameter vectors, and the influence that data mismeasurement may have on parameter estimates. Condition numbers provide a central guide to the sensitivity of common numerical problems to data mismeasurement. Examples are provided that clarify their importance. Two simple prescriptions arise from this analysis. First, data must be of an ‘appropriate’ scale. In some cases this means that the data need similar means and similar variances. Second, in numerical algorithms it is desirable to ascertain the condition number of the Hessian implied by the initial parameter values used for numerical optimisation algorithms. Condition numbers are easy to compute and indicate whether the updates from an initial starting value are likely to be poor.