Iterative Optimizers

4LandGener

Problems of controllable difficulty can be directly built by defining the positions and sizes of plateaus and basins of attraction of local minima. To this end, I have applied the method described by Algorithm 4.1 (others are obviously possible), in which all the domains of basins and plateaus are D-triangles.

It should be noted that despite the fact that the midpoint has a smaller value than those of the vertices of its D-triangle, this does not guarantee that it will be the local minimum because of the way in which the local basin is defined (polynomial interpolation), even if it is often well the case.

In the Matlab^® coding of the LandGener algorithm, I have added a little subtlety so that adjacent basins connect without discontinuity. On the other hand, plateaus may define “cliffs”. This is merely for aesthetics, because as such, in dimension 2, quite realistic landscapes can then be obtained. The code is freely available for download. The program possesses a graphical interface, but the various modules (creation, display, calculations of difficulties, etc.) can also be launched at the command line.

1. 1) Place N points in the definition space. To simplify, this space is the D-square [0, 1]^D.
2. 2) Option (for small dimension): add the 2^D vertices of the D-square.
3. 3) Define a Delaunay triangulation on the set of these points. Each D-triangle is a domain. This is an interval in dimension 1, a true triangle in dimension 2, a tetrahedron in dimension 3, etc.
4. 4) Indicate ...