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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 ...
The artificial landscapes presented herein for single-objective optimization problems are taken from Bäck, [1] Haupt et al. [2] and from Rody Oldenhuis software. [3] Given the number of problems (55 in total), just a few are presented here. The test functions used to evaluate the algorithms for MOP were taken from Deb, [4] Binh et al. [5] and ...
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]
The derivatives provide detailed information for such optimizers, but are even harder to calculate, e.g. approximating the gradient takes at least N+1 function evaluations. For approximations of the 2nd derivatives (collected in the Hessian matrix), the number of function evaluations is in the order of N².
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Three Markov chains running on the 3D Rosenbrock function using the Metropolis–Hastings algorithm. The chains converge and mix in the region where the function is high. The approximate position of the maximum has been illuminated. The red points are the ones that remain after the burn-in process. The earlier ones have been discarded.
These formulas are equivalent for a quadratic function, but for nonlinear optimization the preferred formula is a matter of heuristics or taste. A popular choice is β = max { 0 , β P R } {\displaystyle \displaystyle \beta =\max\{0,\beta ^{PR}\}} , which provides a direction reset automatically.
The NAG Library contains several routines [10] for minimizing or maximizing a function [11] which use quasi-Newton algorithms. In MATLAB's Optimization Toolbox, the fminunc function uses (among other methods) the BFGS quasi-Newton method. [12] Many of the constrained methods of the Optimization toolbox use BFGS and the variant L-BFGS. [13]