Virtual Library of Simulation Experiments:

Test Functions and Datasets

Rosenbrock Function


Dimensions: d

The Rosenbrock function, also referred to as the Valley or Banana function, is a popular test problem for gradient-based optimization algorithms. It is shown in the plot above in its two-dimensional form.

The function is unimodal, and the global minimum lies in a narrow, parabolic valley. However, even though this valley is easy to find, convergence to the minimum is difficult (Picheny et al., 2012).

Input Domain:

The function is usually evaluated on the hypercube xi ∈ [-5, 10], for all i = 1, …, d, although it may be restricted to the hypercube xi ∈ [-2.048, 2.048], for all i = 1, …, d.

Global Minimum:

Modifications and Alternate Forms:

Picheny et al. (2012) use the following rescaled form of the function, with d = 4, on [0, 1]2:

This rescaled form of the function has a mean of zero and a variance of one. The authors also add a small Gaussian error term to the output.



Dixon, L. C. W., & Szego, G. P. (1978). The global optimization problem: an introduction. Towards global optimization, 2, 1-15.

Global Optimization Test Problems. Retrieved June 2013, from

Molga, M., & Smutnicki, C. Test functions for optimization needs (2005). Retrieved June 2013, from

Picheny, V., Wagner, T., & Ginsbourger, D. (2012). A benchmark of kriging-based infill criteria for noisy optimization.

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