Griewank function

The Griewank <a href="/facts/Test_functions_for_optimization/lSafL6bc">test function</a> is a <a href="/facts/Smoothness/mffL4ch2">smooth</a> multidimensional mathematical function used in unconstrained optimization. It is commonly employed to evaluate the performance of <a href="/facts/Global_optimization/nksWlDqU">global optimization</a> algorithms.
The function is defined as:

f
        (
        x
        )
        =
        1
        +
        
          
            
              1
              4000
            
          
        
        
          ∑
          
            i
            =
            1
          
          
            n
          
        
        
          x
          
            i
          
          
            2
          
        
        −
        
          ∏
          
            i
            =
            1
          
          
            n
          
        
        
          P
          
            i
          
        
        (
        
          x
          
            i
          
        
        )
        
           with 
        
        
          P
          
            i
          
        
        (
        
          x
          
            i
          
        
        )
        =
        cos
        ⁡
        
          (
          
            
              
                
                  x
                  
                    i
                  
                
                
                  i
                
              
            
          
          )
        
      
    
    {\displaystyle f(x)=1+{\tfrac {1}{4000}}\sum _{i=1}^{n}x_{i}^{2}-\prod _{i=1}^{n}P_{i}(x_{i}){\text{ with }}P_{i}(x_{i})=\cos \left({\tfrac {x_{i}}{\sqrt {i}}}\right)}

where 
 
 
 
 x
 =
 (
 
 x
 
 1
 
 
 ,
 …
 ,
 
 x
 
 n
 
 
 )
 ∈
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle x=(x_{1},\dots ,x_{n})\in \mathbb {R} ^{n}}
 
 is a vector of real-valued variables. The nonlinear and nonconvex function is characterized by its unique multimodal structure, featuring multiple local minima and maxima around its single global minimum at 
 
 
 
 
 x
 
 ∗
 
 
 =
 0
 
 
 {\displaystyle x^{\ast }=0}
 
, where the function has minimal value 
 
 
 
 f
 (
 
 x
 
 ∗
 
 
 )
 =
 0
 
 
 {\displaystyle f(x^{\ast })=0}
 
. The number of stationary points increases as the dimensionality 
 
 
 
 n
 ∈
 
 N
 
 
 
 {\displaystyle n\in \mathbb {N} }
 
 grows.

For instance, the one-dimensional Griewank function 
 
 
 
 f
 (
 
 x
 
 1
 
 
 )
 :=
 1
 +
 (
 1
 
 /
 
 4000
 )
 ⋅
 
 x
 
 1
 
 
 2
 
 
 −
 cos
 ⁡
 (
 
 x
 
 1
 
 
 )
 
 
 {\displaystyle f(x_{1}):=1+(1/4000)\cdot x_{1}^{2}-\cos(x_{1})}

has 6365 <a href="/facts/Critical_point_(mathematics)/MrEk9PnY">critical points</a> in the interval [−10000,10000], where the first derivative vanishes. These points satisfy the equation 
 
 
 
 ∇
 f
 (
 
 x
 
 1
 
 
 )
 =
 
 
 
 1
 2000
 
 
 
 ⋅
 
 x
 
 1
 
 
 +
 sin
 ⁡
 (
 
 x
 
 1
 
 
 )
 =
 0
 
 
 {\displaystyle \nabla f(x_{1})={\tfrac {1}{2000}}\cdot x_{1}+\sin(x_{1})=0}
 
.

The multimodal structure of the Griewank function presents a challenge for many <a href="/facts/Deterministic_algorithm/v8x4wz35">deterministic</a> optimization algorithms, which may become trapped in one of the numerous local minima. This is particularly problematic for <a href="/facts/Gradient/TsR7uukj">gradient</a>-based methods. The Griewank function is commonly used to benchmark global optimization algorithms, such as <a href="/facts/Genetic_algorithms/WP2AFWuW">genetic algorithms</a> or <a href="/facts/Particle_swarm_optimization/QYo2YaP7">particle swarm optimization</a>. In addition to the original version, there are several variants of the Griewank function specifically designed to test algorithms in high-dimensional optimization scenarios.

Griewank function open-in-new

Griewank function