11: Modified Shekel's Foxholes function [6]

Minimize:

$$f_{11}(\vec x)= - \sum_{i=1}^{30} \frac{1}{\sum_{j=1}^{D} (x_j-a_{ij})^2 + c_i}$$

Where:

$$A = \left( \begin{array}{cccccccccc} 9.681 & 0.667 & 4.783 & 9.09... ...32 & 9.661 & 5.611 & 5.500 & 6.886 & 2.341 & 9.699 & 6.500 \end{array} \right)$$

$$\begin{array}{ccccccccccccc} \vec c= & \Big( & 0.806 & 0.51... ...2 & 0.817 & 0.632 & 0.883 & 0.608 & 0.326 & \Big) \end{array}$$

With constraints:

$$Unconstrained$$

Global optimum in 2 dimensional case:

$$f_{11}(\vec x^*)=-12.1190083797535965715042038937$$

$${x_1}^*=8.0240653, {x_2}^*=9.1465340$$

Global optimum in 5 dimensional case:

$$f_{11}(\vec x^*)=-10.40561723899245$$

$${x_1}^*=8.02491489, {x_2}^*=9.15172576, {x_3}^*=5.11392781, {x_4}^*=7.62086096, {x_5}^*=4.56408839$$

Features:

Multimodal, Non-separable

Figure 22: Modified Shekel's Foxholes function