Ackley function

In <a href="/facts/Mathematical_optimization/oRn8Iv5I">mathematical optimization</a>, the Ackley 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 was proposed by David Ackley in his 1987 PhD dissertation. The function is commonly used as a minimization function with global minimum value 0 at 0,.., 0 in the form due to Thomas Bäck. While Ackley gives the function as an example of "fine-textured broadly unimodal space" his thesis does not actually use the function as a test.
For 
 
 
 
 d
 
 
 {\displaystyle d}
 
 dimensions, is defined as

f
        (
        x
        )
        =
        −
        a
        exp
        ⁡
        
          (
          
            −
            b
            
              
                
                  
                    1
                    d
                  
                
                
                  ∑
                  
                    i
                    =
                    1
                  
                  
                    d
                  
                
                
                  x
                  
                    i
                  
                  
                    2
                  
                
              
            
          
          )
        
        −
        exp
        ⁡
        
          (
          
            
              
                1
                d
              
            
            
              ∑
              
                i
                =
                1
              
              
                d
              
            
            cos
            ⁡
            (
            c
            
              x
              
                i
              
            
            )
          
          )
        
        +
        a
        +
        exp
        ⁡
        (
        1
        )
      
    
    {\displaystyle f(x)=-a\exp \left(-b{\sqrt {{\frac {1}{d}}\sum _{i=1}^{d}x_{i}^{2}}}\right)-\exp \left({\frac {1}{d}}\sum _{i=1}^{d}\cos(cx_{i})\right)+a+\exp(1)}

Recommended variable values are 
 
 
 
 a
 =
 20
 
 
 {\displaystyle a=20}
 
, 
 
 
 
 b
 =
 0.2
 
 
 {\displaystyle b=0.2}
 
, and 
 
 
 
 c
 =
 2
 π
 
 
 {\displaystyle c=2\pi }
 
.
The global minimum is 
 
 
 
 f
 (
 
 x
 
 ∗
 
 
 )
 =
 0
 
 
 {\displaystyle f(x^{*})=0}
 
 at 
 
 
 
 
 x
 
 ∗
 
 
 =
 0
 
 
 {\displaystyle x^{*}=0}
 
.

Ackley function open-in-new

Ackley function