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In mathematical optimization, the Rosenbrock function is a non-convex function, introduced by Howard H. Rosenbrock in 1960, which is used as a performance test problem for optimization algorithms. [1] It is also known as Rosenbrock's valley or Rosenbrock's banana function. The global minimum is inside a long, narrow, parabolic-shaped flat ...
Here some test functions are presented with the aim of giving an idea about the different situations that optimization algorithms have to face when coping with these kinds of problems. In the first part, some objective functions for single-objective optimization cases are presented.
The idea of Rosenbrock search is also used to initialize some root-finding routines, such as fzero (based on Brent's method) in Matlab. Rosenbrock search is a form of derivative-free search but may perform better on functions with sharp ridges. [6] The method often identifies such a ridge which, in many applications, leads to a solution. [7]
Figure 1: MCS algorithm (without local search) applied to the two-dimensional Rosenbrock function. The global minimum = is located at (,) = (,). MCS identifies a position with within 21 function evaluations. After additional 21 evaluations the optimal value is not improved and the algorithm terminates.
Test functions for optimization: Rosenbrock function — two-dimensional function with a banana-shaped valley; Himmelblau's function — two-dimensional with four local minima, defined by (,) = (+) + (+) Rastrigin function — two-dimensional function with many local minima
More generally, if the objective function is not a quadratic function, then many optimization methods use other methods to ensure that some subsequence of iterations converges to an optimal solution. The first and still popular method for ensuring convergence relies on line searches , which optimize a function along one dimension.
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Pages in category "Test functions for optimization" ... Rosenbrock function; S. Shekel function This page was last edited on 29 December 2023, at 15:34 ...