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Math Help - Determining step size using golden ratio

  1. #1
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    Determining step size using golden ratio

    I would like to calculate the minimum of the function  g(x,y) = (y-x^2)^2+(1.1 -x)^2 using the steepest descent method. I am, however, stuck at determining the step size. I want to calculate the step size using golden ratio, however, as I have only use the method in 1 dimension, I don't know how to use it here. Here is what I did so far:

    Step 1: Choose starting points. These were given already as (0.5, 0.5)
    Step 2: Calculate the gradient \nabla. I did this and got x (2-4y)+4x^3-2.2,  2 (y-x^2). Substituting for the starting points 0.5, 0.5, we get -\nabla = 1.7, -0.5
    Step 3: Calculate g(x+ \alpha -\nabla).

    I am stuck at step 3. I don't know how I can get  \alpha. If there is a possibility to use the golden ratio to determine alpha here, I would welcome. Any help in explaining it is highly appreciated. Thanks in advance!
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  2. #2
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    Re: Determining step size using golden ratio

    If you know what to do in 1-dimension, it's easy to apply the algorithm in 2-dimensions. Just considers the 1-d function :
    f(\alpha) = g(x - \alpha \nabla_x, y - \alpha \nabla_y)

    And apply the method on f(\alpha).
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