Rosenbrock function

In <a href="/facts/Mathematical_optimization/oRn8Iv5I">mathematical optimization</a>, the Rosenbrock function is a non-<a href="/facts/Convex_function/IbzG5SLF">convex</a> <a href="/facts/Function_(mathematics)/CdReEKCh">function</a>, introduced by <a href="/facts/Howard_Harry_Rosenbrock/blE4c3GI">Howard H. Rosenbrock</a> in 1960, which is used as a <a href="/facts/Test_functions_for_optimization/lSafL6bc">performance test problem</a> for optimization <a href="/facts/Algorithm/fnl5NmRt">algorithms</a>. It is also known as Rosenbrock's valley or Rosenbrock's banana function.
The <a href="/facts/Global_minimum/ADlgnyzV">global minimum</a> is inside a long, narrow, <a href="/facts/Parabola/Cweo3wgw">parabolic</a>-shaped flat valley. To find the valley is trivial. To converge to the global minimum, however, is difficult.
The function is defined by

f
        (
        x
        ,
        y
        )
        =
        (
        a
        −
        x
        
          )
          
            2
          
        
        +
        b
        (
        y
        −
        
          x
          
            2
          
        
        
          )
          
            2
          
        
      
    
    {\displaystyle f(x,y)=(a-x)^{2}+b(y-x^{2})^{2}}

It has a global minimum at 
 
 
 
 (
 x
 ,
 y
 )
 =
 (
 a
 ,
 
 a
 
 2
 
 
 )
 
 
 {\displaystyle (x,y)=(a,a^{2})}
 
, where 
 
 
 
 f
 (
 x
 ,
 y
 )
 =
 0
 
 
 {\displaystyle f(x,y)=0}
 
. Usually, these parameters are set such that 
 
 
 
 a
 =
 1
 
 
 {\displaystyle a=1}
 
 and 
 
 
 
 b
 =
 100
 
 
 {\displaystyle b=100}
 
. Only in the trivial case where 
 
 
 
 a
 =
 0
 
 
 {\displaystyle a=0}
 
 the function is symmetric and the minimum is at the origin.

Rosenbrock function open-in-new

Rosenbrock function