Rosenbrock’s banana function

Rosenbrock’s banana function is a famous test case for optimization software. It’s called the banana function because of its curved contours.

The definition of the function is

f(x, y) = (1 - x)^2 + 100(y - x^2)^2

The function has a global minimum at (1, 1). If an optimization method starts at the point (-1.2, 1), it has to find its way to the other side of a flat, curved valley to find the optimal point.

Rosenbrock’s banana function is just one of the canonical test functions in the paper “Testing Unconstrained Optimization Software” by MorĂ©, Garbow, and Hillstrom in ACM Transactions on Mathematical Software Vol. 7, No. 1, March 1981, pp 17-41. The starting point (-1.2, 1) mentioned above comes from the starting point required in that paper.

I mentioned a while back that I was trying out Python alternatives for tasks I have done in Mathematica. I tried making contour plots with Python using matplotlib. This was challenging to figure out. The plots did not look very good, though I imagine I could have made satisfactory plots if I had explored the options. Instead, I fired up Mathematica and produced the plot above easily using the following code.

Banana[x_, y_] := (1 - x)^2 + 100 (y - x*x)^2
ContourPlot[Banana[x, y], {x, -1.5, 1.5}, {y, 0.7, 2.0}]

7 thoughts on “Rosenbrock’s banana function

  1. John, don’t give up on Python! In SAGE, that will be something like

    x,y = var(‘x,y’)
    banana(x,y) = (1 – x)^2 + 100*(y – x*x)^2
    contour_plot(banana(x,y),(x,-1.5, 1.5), (y, 0.7, 2.0),cmap=’pink’)

  2. And in matplotlib it’s something like


    from pylab import mgrid, contourf, cm, show
    f = lambda x,y: (1 - x)**2 + 100*(y - x**2)**2
    x,y = mgrid[-1.5:1.5:50j, 0.7:2:50j]
    contourf(x, y, f(x,y), cmap=cm.Purples_r)
    show()

  3. +1 for Sage, and it’s even easier than Sergey makes it look; since variables get automatically recognized in a function definition, you can skip the line

    x, y = var('x,y')

    and go straight to

    banana(x, y) = (1 - x)^2 + 100(y - x * x)^2

    The “cmap” option in contour_plot is optional as well.

  4. Sorry to post again, but Sage also handles the optimization question nicely; using the “minimize” command, which can use a couple of different algorithms to find the minimum of a function:

    sage: minimize(banana, [-1.2, 1], algorithm='powell')
    Optimization terminated successfully.
    Current function value: 0.000000
    Iterations: 23
    Function evaluations: 665
    (1.0, 1.0)

    If I had left the “algorithm” option blank, I think it would have used Broyden-Fletcher-Goldfarb-Shannon. This algorithm yielded a slightly different answer.

  5. Hi John.
    Jon Harrop (Flying Frog) wrote about gradient and used banana to illustrate; no plot though.
    http://fsharpnews.blogspot.com/2011/01/gradient-descent.html

    And the following defines the famous Rosenbrock “banana” function that is notoriously difficult to minimize due to its curvature around the minimum:

    > let rosenbrock (xs: vector) =
    let x, y = xs.[0], xs.[1]
    pown (1.0 – x) 2 + 100.0 * pown (y – pown x 2) 2;;
    val rosenbrock : vector -> floatThe minimum at (1, 1) may be found quickly and easily using the functions defined above as follows:

    > let xs =
    vector[0.0; 0.0]
    |> gradientDescent rosenbrock (grad rosenbrock);;
    val xs : Vector = vector [|0.9977180571; 0.99543389|]

  6. There is a very simple algorithm that is competitive with current genetic and ES algorithms.
    I have some example code here:
    http://www.mediafire.com/?8i1pjhesw6s4oj7
    Well it is an ES algorithm but it uses a multi-scale approach. It is similar to Continuous Gray Code optimization. A few other people have come up with basically the same idea in the past.
    I also have some code on the Walsh Hadamard transform you may peruse if you wish:
    http://www.mediafire.com/?q4o3clxu1r18ufs
    But I have to say that some ideas in that are only joyful playing.

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