Economic Information

I. Introduction: Andy Keane (1994) designed the "bump" function to test the performance of(constrained) optimization methods. The optimization problem of Keane's bump function may be presented (Keane, 1994) as follows:

Fig.-1: Keane's Bump Function in 2 Dimensions	A visual appreciation of Keane's two-dimensional (m=2) bump function may be obtained from the graphical presentation (Fig.-1; Hacker et al., 2002). As the dimension (m) grows larger, the optimum value of the function becomes more and more difficult to obtain. Keane (1994) observed that for m=20 the value of min[f(x)] could be about -0.76 and for m=50 it could be about -0.835 but did not know this to be the case.
Source: Hacker, Eddy and Lewis (2002)

Keane's bump function is considered as a standard benchmark for nonlinear constrained optimization. It is highly multi-modal and its optimum is located at the non-linear constrained boundary. Emmerich (2005, p. 116) noted that the true minimum of this function is unknown. Using their various hybridized Genetic Algorithms Hacker et al. (2002) obtained min[f(x)] = -0.365 for a 2-dimensional and -0.6737 for a 10-dimensional Keane's problem. They also found that the Genetic Algorithms (without hybridization) perform worse than their hybridized Genetic Algorithms.

read more

---

The author is thankful to Dr. Kenneth L Judd of the Hoover Institution, Stanford University, USA for sending the paper of Hacker et al. (2002) that led to the present work on optimization of Keane’s function, and his constructive suggestions to consider odd and oddly spaced dimensions, etc.