Description Usage Arguments Details Value References Examples

Simple and often used test function defined in higher dimensions and with
analytical gradients, especially suited for performance tests. Analytical
gradients, where existing, are provided with the `gr`

prefix.
The dimension is determined by the length of the input vector.

1 2 3 4 5 6 7 8 9 10 11 | ```
fnRosenbrock(x)
grRosenbrock(x)
fnRastrigin(x)
grRastrigin(x)
fnNesterov(x)
grNesterov(x)
fnNesterov1(x)
fnHald(x)
grHald(x)
fnShor(x)
grShor(x)
``` |

`x` |
numeric vector of a certain length. |

**Rosenbrock** – Rosenbrock's famous valley function from 1960. It can
also be regarded as a least-squares problem:

*∑_{i=1}^{n-1} (1-x_i)^2 + 100 (x_{i+1}-x_i^2)^2*

No. of Vars.: | n >= 2 |

Bounds: | -5.12 <= xi <= 5.12 |

Local minima: | at f(-1, 1, ..., 1) for n >= 4 |

Minimum: | 0.0 |

Solution: | xi = 1, i = 1:n |

**Nesterov** – Nesterov's smooth adaptation of Rosenbrock, based on the
idea of Chebyshev polynomials. This function is even more difficult to
optimize than Rosenbrock's:

*(x_1 - 1)^2 / 4 + ∑_{i=1}^{n-1} (1 + x_{i+1} - 2 x_i^2)*

No. of Vars.: | n >= 2 |

Bounds: | -5.12 <= xi <= 5.12 |

Local minima: ? | |

Minimum: | 0.0 |

Solution: | xi = 1, i = 1:n |

**Rastrigin** – Rastrigin's function is a famous, non-convex example from 1989 for global optimization. It is a typical example of a multimodal function with many local minima:

*10 n + ∑_1^n (x_i^2 - 10 \cos(2 π x_i))*

No. of Vars.: | n >= 2 |

Bounds: | -5.12 <= xi <= 5.12 |

Local minima: | many |

Minimum: | 0.0 |

Solution: | xi = 0, i = 1:n |

**Hald** – Hald's function is a typical example of a non-smooth test
function, from Hald and Madsen in 1981.

*\max_{1 ≤ i ≤ n} \frac{x_1 + x_2 t_i}{1 + x_3 t_i + x_4 t_i^2 + x_5 t_i^3} - \exp(t_i)*

where *t_i = -1 + (i - 1)/10* for *1 ≤ i ≤ 21*.

No. of Vars.: | n =5 |

Bounds: | -1 <= xi <= 1 |

Local minima: | ? |

Minimum: | 0.0001223713 |

Solution: | (0.99987763, 0.25358844, -0.74660757, 0.24520150, -0.03749029) |

**Shor** – Shor's function is another typical example of a non-smooth test
function, a benchmark for Shor's R-algorithm.

Returns the values of the test function resp. its gradient at that point. If an analytical gradient is not available, a function computing the gradient numerically will be provided.

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