Intuitive Rosenbrock's Valley/Banana Function.

Question

The following is called Rosenbrock's valley or Rosenbrock's Banana Function,

$$f(x,y) = (1-x)^2+A(y-x^2)^2, \qquad (x,y) \in \Bbb R^2$$

Can you explain intuitively why is this function so important in the study of Optimization?

What is so special about this function that makes it so interesting in the study of Optimization?

Have you seen https://en.wikipedia.org/wiki/Rosenbrock_function? — Martín-Blas Pérez Pinilla, Dec 22 '16 at 20:40
@Martín-BlasPérezPinilla, Yes. I also have at least 10 Optimization e-books in my PC's hard drive. So, what do you think is the point of asking this question? — user366312, Dec 22 '16 at 20:58
Asking for an "intuitive" explanation sounds a bit unreasonable in this case, unless you can somehow ask Rosenbrock himself. At the very least, he does say in his article that it helped him develop his own minimization method. — J. M. ain't a mathematician, Dec 29 '16 at 22:05

score 2 · Answer 1 · answered Dec 22 '16 at 20:40

The Wikipedia page has a nice picture and explanation:

The global minimum is inside a long, narrow, parabolic shaped flat valley. To find the valley is trivial. To converge to the global minimum, however, is difficult.

Solving for the minimizer analytically is easy: $f$ is minimized at $(1,1)$.

However, if $A$ is large, optimization algorithms might pay a lot of attention to the second term $(y-x^2)^2$ and find the "parabolic valley" $\{(x,y) : y=x^2\}$ easily, but will have trouble going along the valley (which is relatively flat) to find the minimizer $(1,1)$.

Intuitive Rosenbrock's Valley/Banana Function.

1 Answers1