Newton's method in optimization

<p>In <a href="/facts/Calculus/MEmUTYoz">calculus</a>, <a href="/facts/Newton%27s_method/VlRhI2FL">Newton's method</a> (also called Newton–Raphson) is an <a href="/facts/Iterative_method/uKMAJhEh">iterative method</a> for finding the <a href="/facts/Zero_of_a_function/mlK3dhoT">roots</a> of a <a href="/facts/Differentiable_function/jQqTgLk1">differentiable function</a> 
  
    
      
        f
      
    
    {\displaystyle f}
  
, which are solutions to the <a href="/facts/Equation/pxFzLAVR">equation</a> 
  
    
      
        f
        (
        x
        )
        =
        0
      
    
    {\displaystyle f(x)=0}
  
. However, to optimize a twice-differentiable 
  
    
      
        f
      
    
    {\displaystyle f}
  
, our goal is to find the roots of 
  
    
      
        
          f
          ′
        
      
    
    {\displaystyle f'}
  
. We can therefore use Newton's method on its <a href="/facts/Derivative/7Ito70Dh">derivative</a> 
  
    
      
        
          f
          ′
        
      
    
    {\displaystyle f'}
  
 to find solutions to 
  
    
      
        
          f
          ′
        
        (
        x
        )
        =
        0
      
    
    {\displaystyle f'(x)=0}
  
, also known as the <a href="/facts/Critical_point_(mathematics)/MrEk9PnY">critical points</a> of 
  
    
      
        f
      
    
    {\displaystyle f}
  
. These solutions may be minima, maxima, or saddle points; see section <a href="/facts/Critical_point_(mathematics)/MrEk9PnY">"Several variables"</a> in <a href="/facts/Critical_point_(mathematics)/MrEk9PnY">Critical point (mathematics)</a> and also section "Geometric interpretation" in this article. This is relevant in <a href="/facts/Mathematical_optimization/oRn8Iv5I">optimization</a>, which aims to find (global) minima of the function 
  
    
      
        f
      
    
    {\displaystyle f}
  
.
</p>

Newton's method in optimization open-in-new

Newton's method in optimization