Levenberg-Marquardt trajectories

Figure 3 Levenberg-Marquardt trajectories on the surface + obtained by expansion around (—15,10). The trajectories intersect the trust-region boundaries at their stationary points. The concentric circles represent boundaries of trust regions of radius 8, 16.13, and 24.

It is of some interest to compare the steepest-descent path Eq. (46) with the Levenberg-Marquardt trajectory Eq. (35) which may be rewritten as... 

This expression highlights the similarities between the second-order steepest-descent trajectory Eq. (46) and the Levenberg-Marquardt trajectory. In particular, we notice that both trajectories end at the minimum of the second-order surface for t = oo. For small steps, we expand around t = 0 and obtain... 

Fig. 12.1. The tmst-region method for a two-dimensional function. On the left, we have plotted the function + lOy with superimposed Levenberg-Marquardt trajectories and concentric circles representing three different trust regions centred at (—15, 10). On the right, we have plotted the corresponding step-length function (12.3.15) with superimposed horizontal lines representing the radii of the three trust regions.

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