Determining step size using golden ratio

**Keep** · December 3rd 2011, 09:24 PM

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!

**AlexisM** · December 12th 2011, 06:46 AM

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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