Global maxima

Fig. 11.12 Three sample fitness landscapes (a) has a single smooth maximum, (b) has many equivalent local maxima and one global maximum, but is circularly symmetric (c.) has many irregularly spaced local maxima, and is a good example of a rugged landscape.

One strategy that has often been used is to proceed along the path of steepest ascent until a maximum is reached. Then another search is made. A path of steepest ascent is determined and followed until another maximum is reached. This is continued until the climber thinks he is in the vicinity of the global maximum. To aid in reaching the maximum, the technique of using three points to estimate a quadratic surface, as was done previously, may be used. 

Analogously, a local maximum is the global maximum of/(x) if the objective function is concave and the constraints form a convex set. 

Key a—inflection point (scalar equivalent to a saddle point) b—global maximum (and local maximum) c—local minimum d—local maximum... 

Figure 2.7 Response surface exhibiting two local maxima, one at x, = 2, the other (the global maximum) at = 7.

Locate the three local minima in Figure 2.7. Which is the global minimum Locate the four local maxima in Figure 2.8. Which is the global maximum ... 

A term used to describe the behavior of a curvilinear function by specifying that, over a given interval, the first derivative increases, reaches a local or global maximum, and then decreases. 

The average local electrostatic potential V(r)/p(r), introduced by Pohtzer [57], led Sen and coworkers [58] to conjecture that the global maximum in V(r)/p(r) defines the location of the core-valence separation in ground-state atoms. Using this criterion, one finds N values [Eq. (3.1)] of 2.065 and 2.112 e for carbon and neon, respectively, and 10.073 e for argon, which are reasonable estimates in light of what we know about the electronic shell structure. Politzer [57] also made the significant observation that V(r)/p(r) has a maximum any time the radial distribution function D(r) = Avr pir) is found to have a minimum. 

Figure 3.3. Examples of local and global optima, (a) One factor (pH) with the response (yield) giving a maximum, but with a wide acceptable range, (b) Function of two variables giving a number of maxima. The global maximum and one local maximum are shown, (c) The response surface of graph b as a contour map.

A global maximum or minimum which has a value greater or smaller than all other points within the domain of the function. 

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