## Sphere Function

### Description:

*Dimensions: d*

The Sphere function has d local minima except for the global one. It is continuous, convex and unimodal. The plot shows its two-dimensional form.

### Input Domain:

The function is usually evaluated on the hypercube x_{i}∈ [-5.12, 5.12], for all i = 1, …, d.

### Global Minimum:

### Modifications and Alternate Forms:

Picheny et al. (2012) use the following, slightly different, version of the Sphere function, with d = 6, on [0, 1]^{6}:

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

### Code:

### References:

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

Molga, M., & Smutnicki, C. Test functions for optimization needs (2005). Retrieved June 2013, from http://www.zsd.ict.pwr.wroc.pl/files/docs/functions.pdf.

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

For questions or comments, please email Derek Bingham at: dbingham@stat.sfu.ca.

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