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Rastrigin function
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<p>In <a href="/facts/Mathematical_optimization/oRn8Iv5I">mathematical optimization</a>, the Rastrigin function is a non-<a href="/facts/Convex_function/IbzG5SLF">convex function</a> used as a performance test problem for <a href="/facts/Optimization_algorithm/oRn8Iv5I">optimization algorithms</a>. It is a typical example of non-linear multimodal function. It was first proposed in 1974 by Rastrigin as a 2-dimensional function and has been generalized by Rudolph. The generalized version was popularized by Hoffmeister & Bäck and Mühlenbein et al. Finding the minimum of this function is a fairly difficult problem due to its large search space and its large number of <a href="/facts/Local_minimum/ADlgnyzV">local minima</a>. </p><p>On an n {\displaystyle n} -dimensional domain it is defined by: </p> f ( x ) = A n + ∑ i = 1 n [ x i 2 − A cos ( 2 π x i ) ] {\displaystyle f(\mathbf {x} )=An+\sum _{i=1}^{n}\left[x_{i}^{2}-A\cos(2\pi x_{i})\right]} <p>where A = 10 {\displaystyle A=10} and x i ∈ [ − 5.12 , 5.12 ] {\displaystyle x_{i}\in [-5.12,5.12]} . There are many extrema: </p> <ul><li>The global minimum is at x = 0 {\displaystyle \mathbf {x} =\mathbf {0} } where f ( x ) = 0 {\displaystyle f(\mathbf {x} )=0} .</li> <li>The maximum function value for x i ∈ [ − 5.12 , 5.12 ] {\displaystyle x_{i}\in [-5.12,5.12]} is located around x i ∈ [ ± 4.52299366... , . . . , ± 4.52299366... ] {\displaystyle x_{i}\in [\pm 4.52299366...,...,\pm 4.52299366...]} :</li></ul> <table><tbody><tr><th>Number of dimensions</th><th>Maximum value at ± 4.52299366 {\displaystyle \pm 4.52299366} </th></tr><tr><td>1</td><td>40.35329019</td></tr><tr><td>2</td><td>80.70658039</td></tr><tr><td>3</td><td>121.0598706</td></tr><tr><td>4</td><td>161.4131608</td></tr><tr><td>5</td><td>201.7664509</td></tr><tr><td>6</td><td>242.1197412</td></tr><tr><td>7</td><td>282.4730314</td></tr><tr><td>8</td><td>322.8263216</td></tr><tr><td>9</td><td>363.1796117</td></tr></tbody></table> <p>Here are all the values at 0.5 interval listed for the 2D Rastrigin function with x i ∈ [ − 5.12 , 5.12 ] {\displaystyle x_{i}\in [-5.12,5.12]} : </p> <table><tbody><tr><th rowspan="2" colspan="2"> f ( x ) {\displaystyle f(x)} </th><th colspan="12"> x 1 {\displaystyle x_{1}} </th></tr><tr><td> 0 {\displaystyle 0} </td><td> ± 0.5 {\displaystyle \pm 0.5} </td><td> ± 1 {\displaystyle \pm 1} </td><td> ± 1.5 {\displaystyle \pm 1.5} </td><td> ± 2 {\displaystyle \pm 2} </td><td> ± 2.5 {\displaystyle \pm 2.5} </td><td> ± 3 {\displaystyle \pm 3} </td><td> ± 3.5 {\displaystyle \pm 3.5} </td><td> ± 4 {\displaystyle \pm 4} </td><td> ± 4.5 {\displaystyle \pm 4.5} </td><td> ± 5 {\displaystyle \pm 5} </td><td> ± 5.12 {\displaystyle \pm 5.12} </td></tr><tr><td rowspan="12"> x 2 {\displaystyle x_{2}} </td><td> 0 {\displaystyle 0} </td><td>0</td><td>20.25</td><td>1</td><td>22.25</td><td>4</td><td>26.25</td><td>9</td><td>32.25</td><td>16</td><td>40.25</td><td>25</td><td>28.92</td></tr><tr><td> ± 0.5 {\displaystyle \pm 0.5} </td><td>20.25</td><td>40.5</td><td>21.25</td><td>42.5</td><td>24.25</td><td>46.5</td><td>29.25</td><td>52.5</td><td>36.25</td><td>60.5</td><td>45.25</td><td>49.17</td></tr><tr><td> ± 1 {\displaystyle \pm 1} </td><td>1</td><td>21.25</td><td>2</td><td>23.25</td><td>5</td><td>27.25</td><td>10</td><td>33.25</td><td>17</td><td>41.25</td><td>26</td><td>29.92</td></tr><tr><td> ± 1.5 {\displaystyle \pm 1.5} </td><td>22.25</td><td>42.5</td><td>23.25</td><td>44.5</td><td>26.25</td><td>48.5</td><td>31.25</td><td>54.5</td><td>38.25</td><td>62.5</td><td>47.25</td><td>51.17</td></tr><tr><td> ± 2 {\displaystyle \pm 2} </td><td>4</td><td>24.25</td><td>5</td><td>26.25</td><td>8</td><td>30.25</td><td>13</td><td>36.25</td><td>20</td><td>44.25</td><td>29</td><td>32.92</td></tr><tr><td> ± 2.5 {\displaystyle \pm 2.5} </td><td>26.25</td><td>46.5</td><td>27.25</td><td>48.5</td><td>30.25</td><td>52.5</td><td>35.25</td><td>58.5</td><td>42.25</td><td>66.5</td><td>51.25</td><td>55.17</td></tr><tr><td> ± 3 {\displaystyle \pm 3} </td><td>9</td><td>29.25</td><td>10</td><td>31.25</td><td>13</td><td>35.25</td><td>18</td><td>41.25</td><td>25</td><td>49.25</td><td>34</td><td>37.92</td></tr><tr><td> ± 3.5 {\displaystyle \pm 3.5} </td><td>32.25</td><td>52.5</td><td>33.25</td><td>54.5</td><td>36.25</td><td>58.5</td><td>41.25</td><td>64.5</td><td>48.25</td><td>72.5</td><td>57.25</td><td>61.17</td></tr><tr><td> ± 4 {\displaystyle \pm 4} </td><td>16</td><td>36.25</td><td>17</td><td>38.25</td><td>20</td><td>42.25</td><td>25</td><td>48.25</td><td>32</td><td>56.25</td><td>41</td><td>44.92</td></tr><tr><td> ± 4.5 {\displaystyle \pm 4.5} </td><td>40.25</td><td>60.5</td><td>41.25</td><td>62.5</td><td>44.25</td><td>66.5</td><td>49.25</td><td>72.5</td><td>56.25</td><td>80.5</td><td>65.25</td><td>69.17</td></tr><tr><td> ± 5 {\displaystyle \pm 5} </td><td>25</td><td>45.25</td><td>26</td><td>47.25</td><td>29</td><td>51.25</td><td>34</td><td>57.25</td><td>41</td><td>65.25</td><td>50</td><td>53.92</td></tr><tr><td> ± 5.12 {\displaystyle \pm 5.12} </td><td>28.92</td><td>49.17</td><td>29.92</td><td>51.17</td><td>32.92</td><td>55.17</td><td>37.92</td><td>61.17</td><td>44.92</td><td>69.17</td><td>53.92</td><td>57.85</td></tr></tbody></table> <p>The abundance of local minima underlines the necessity of a global optimization algorithm when needing to find the global minimum. Local optimization algorithms are likely to get stuck in a local minimum. </p>