10: Modified Langerman function [6]

Minimize:

$$f_{10}(\vec x)=- \sum_{i=1}^{5} c_i e^{-\frac{1}{\pi}\sum_{j=1}^{D} (x_j-a_{ij})^2} \cos \left( \pi \sum_{j=1}^{D} (x_j-a_{ij})^2 \right)$$

Where:

$$A = \left( \begin{array}{cccccccccc} 9.681 & 0.667 & 4.783 & 9.09... ...67 & 1.863 & 6.708 & 6.349 & 4.534 & 0.276 & 7.633 & 1.567 \end{array} \right)$$

$$\vec c = \left( \begin{array}{ccccc} 0.806 & 0.517 & 0.100 & 0.908 & 0.965 \end{array} \right)$$

With constraints:

$$Unconstrained$$

Global optimum in 2 dimensional case:

$$f_{10}(\vec x^*)=-1.08093846723926811925764468469$$

$${x_1}^*=9.6810707, {x_2}^*=0.6666515$$

Global optimum in 5 dimensional case:

$$f_{10}(\vec x^*)=-0.964999919793332$$

$${x_1}^*=8.074000, {x_2}^*=8.777001, {x_3}^*=3.467004, {x_4}^*=1.863013, {x_5}^*= 6.707995$$

Features:

Multimodal, Non-separable

Figure 20: Modified Langerman function: Large scale

Figure 21: Modified Langerman function: The area near optimum