(7.1) |

The way of generating the parameters of is taken from [RP63], and is as follows. The elements of the matrices and are random integers from the interval , and a vector is chosen whose components are random numbers from . Then, the parameters are defined by the equation , and the starting vector is formed by adding random perturbations of to the components of . All distributions of random numbers are uniform. There are two remarks to do on this objective function:

- Because the number of terms in the sum of squares is equals to the number of variables, it happens often that the Hessian matrix is ill-conditioned around .
- Because is periodic, it has many saddle points and maxima.

We will compare the CONDOR algorithm with an older algorithm: ''CFSQP''. CFSQP uses line-search techniques. In CFSQP, the Hessian matrix of the function is reconstructed using a update, the gradient is obtained by finite-differences.

Parameters of CONDOR: .

Parameters of : . The algorithm stops when the step size is smaller than .

Recalling that , we will say that we have a success when the value of the objective function at the final point of the optimization algorithm is lower then .

We obtain the following results, after 100 runs of both algorithms:

Mean number of | Mean best value of | |||||

Dimension | function evaluations | Number of success | the objective function | |||

of the space | CONDOR | CFSQP | CONDOR | CFSQP | CONDOR | CFSQP |

3 | 44.96 | 246.19 | 100 | 46 | 3.060873e-017 | 5.787425e-011 |

5 | 99.17 | 443.66 | 99 | 27 | 5.193561e-016 | 8.383238e-011 |

10 | 411.17 | 991.43 | 100 | 14 | 1.686634e-015 | 1.299753e-010 |

20 | 1486.100000 | -- | 100 | -- | 3.379322e-016 | -- |

We can now give an example of execution of the algorithm to illustrate the discussion of Section 6.2:

Rosenbrock's function () | ||

function evaluations | Best Value So Far | |

33 | ||

88 | ||

91 | ||

94 | ||

97 | ||

100 | ||

101 | ||

103 |

With the

We will use the same choice of parameters (for and ) as before. The starting point is .

As you can see, the number of evaluations performed when is reduced is far inferior to .