Evolution and Optimum Seeking

124 Evolution Strategies for Numerical Optimization
askew distribution this is not the case. Perhaps, however, the skewness is only slight, so
that one can determine at least approximately the expectation value from the position of
the maximum.
Before treating the sphere and corridor models in this way,wewillcheck the usefulness
of the scheme with an even simpler objective function.
5.2.2.1 The Linear Model (Inclined Plane)
The simplest way the objective function can depend on the variables is linearly. Imagining
the function to be a terrain in the (n + 1)-dimensional space, it appears as an inclined
plane. In the two dimensional projection the contours are straight, parallel lines in this
case. Without loss of generality one can orient the coordinate system so that the plane only
slopes in the direction of one axis x1 and the starting point or parent under consideration
lies at the origin (Fig. 5.5).
The useful distance s` towards the objective thatiscovered by descendant N` of the
parent E is just the part of the random vector z lying along the x1 axis. Since the
components zi of z are independent, we have
and
p(s`