Narrow valley effect

Although the solution of nonlinear systems results in the problem of finding a merit function that drives the search toward the solution of the problem (see Section 7.3), in the case of unconstrained minimization, this problem seems not to arise the unique function to be used is the obvious one it is the function to be minimized. In the previous sections, we saw how such a function can be used to accept or reject the iteration. Nevertheless, suppose we have a problem that can be 

In this case, a point at the bottom of the valley that is remote from the solution may have a function value better than one, which is very close to the solution but slightly outside the valley. 

This phenomenon is called the narrow mUey effect. 

Consider the following trivial example. Rosenbroke s function 

The prediction obtained with Newton s method is = 0.999999999999999 X2 = 0.973320053068153, which is closer to the solution. There, the objective function is F = 7.12 x 10 , worse than the previous one. 

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In constrained optimization, it is important to distinguish between the Maratos effect and the narrow valley effect (see Section 3.8). Both effects crop in deeming whether x +i is better or not than xj. 

The narrow valley effect is related to the fact that the isolines of the merit function could originate narrow valleys and not spheres ... 

The narrow valley effect could be handled using the same unconstrained minimization device showed in Section 3.8. 

Temperature Effects at Hli t- During the dn3 Time the temperatun limited area, say a square mile, is about the same unless there arc marked changes in topography or soil. On days wit h much sunshine and a low wind velocity, the low cr points, particularly those in narrow valleys, may be a few tenths of a degree warmer than the upper parts of the area. At night, the layer of air next to the ground grows colder and denser and drains like water into the valleys and places of low elevation. If the wind is unable to remove these pockets of cold air, a... 

The common options, described in the previous sections and adopted when Fi+i > Fi in Xi+i = Xi + di obtained with Newton methods, use Xi as the reference point to search for an improvement. None of these techniques is effective when x lies on the bottom of a narrow valley. 

Figure S illustrates the effect of the above procedure on the same rectilinear surface for two very different mass combinations. When << and the mass point is scarcely diverted from its original path, and little energy is released before it reaches the head of the narrow elongated exit valley. There the trajectory turns sharply and runs down the exit valley. Energy is released mainly as repulsion between the products and after AB has essentially formed this causes relatively modest vibrational excitation. The surface and dynamics are quite different when and > ntc- Now the entry vaUqy is long and narrow, whilst the exit valley is broad and falls steeply near its head. After crossing the barrier, the sliding...

Landslips are a common feature of valleys In mountainous areas, and large slips often cause narrowing of a valley that therefore looks topographically suitable for a dam. Unless they are shallow seated and can be removed or effectively drained. It Is prudent to avoid landslipped areas in dam location, because their unstable nature may result in movement during construction or, subsequently, on filling or drawdown of the reservoir. 

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