Griewank function

<h2 id="non-smooth-variant-of-the-griewank-function">Non-Smooth Variant of the Griewank Function</h2>
<p>A non-smooth version of the Griewank function has been developed<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> to emulate the characteristics of objective functions frequently encountered  in optimization problems from <a href="/facts/Machine_learning/e0w0XJTu">machine learning</a> (ML). These functions often exhibit <a href="/facts/Piecewise_function/nlQwptpe">piecewise</a> smooth or non-smooth behavior due to the presence of <a href="/facts/Regularization_(mathematics)/K601A3y2">regularization terms</a>, <a href="/facts/Activation_function/S4NImL6L">activation functions</a>, or constraints in learning models.
</p><p>A non-smooth variant of the Griewank function is the following:
</p>

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
                    
                  
                  
                    2
                    
                      
                        i
                      
                    
                  
                
              
              )
            
          
          |
        
        −
        
          |
          
            sin
            ⁡
            
              (
              
                
                  
                    x
                    
                      i
                    
                  
                  
                    2
                    
                      
                        i
                      
                    
                  
                
              
              )
            
          
          |
        
        .
      
    
    {\displaystyle f(x)=1+{\frac {1}{4000}}\sum _{i=1}^{n}x_{i}^{2}-\prod _{i=1}^{n}P_{i}(x_{i}){\text{ with }}P_{i}(x_{i})=\left|\cos \left({\frac {x_{i}}{2{\sqrt {i}}}}\right)\right|-\left|\sin \left({\frac {x_{i}}{2{\sqrt {i}}}}\right)\right|.}

<p>By incorporating non-smooth elements, such as absolute values in the cosine and sine terms, this function mimics the irregularities present in many ML loss landscapes. It offers a robust benchmark for evaluating optimization algorithms, especially those designed to handle the challenges of non-convex, non-smooth, or high-dimensional problems, including sub-gradient, hybrid, and evolutionary methods.
</p><p>The function's resemblance to practical ML objective functions makes it particularly valuable for testing the robustness and efficiency of algorithms in tasks such as <a href="/facts/Hyperparameter_tuning/bzcJJrxs">hyperparameter tuning</a>, neural network training, and constrained optimization.
</p>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Griewank, A. O. "Generalized Descent for Global Optimization." J. Opt. Th. Appl. 34, 11–39, 1981 <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Locatelli, M. "A Note on the Griewank Test Function." J. Global Opt. 25, 169–174, 2003 <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Bosse, Torsten F.; Bücker, H. Martin (2024-10-29). "A piecewise smooth version of the Griewank function". Optimization Methods and Software: 1–11. doi:10.1080/10556788.2024.2414186. ISSN 1055-6788. <a href="https://doi.org/10.1080%2F10556788.2024.2414186" target="_blank">https://doi.org/10.1080%2F10556788.2024.2414186</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
</ol>

Griewank function open-in-new

Griewank function