### Rosenbrock Function And The Steepest Descent

It's interesting how the steepest descent method sweats a lot if not correctly set up inside the "right" parameters' interval of the Rosenbrock Function:

It's a well known testbed optimization function and evolutionary algorithms can find its optimum with no problems at all -- surely, as long as you try a traditional parameters' setting that any introductory Evolutionary Computation book gives you.

Using the steepest descent and setting it up as:

x = 2

y = 2

Step Size = 0.001

Stop Criterion = 10^{-6}

I got the following results (see the image below):

Pay attention to the path the steepest descent takes until finding the optimum at x = 1 and y = 1. It is said that an evolution strategy would follow a similar path if its population could be infinite. Since, so far, there are no real computers with infinite memory, such assumption cannot be verified in the real world.

During a second run I set the parameters x = 5 and y = 5. The optimization path run very far away from the previous one! And from the optimum too!

Labels: Evolution Strategy, Evolutionary Algorithm, Evolutionary Computation, Genetic Algorithm, Gradient Method, Optimization, Optimization Path, Optimum Seeking, Rosenbrock Function, Steepest Descent

## 3 Comments:

"It is said that an evolution strategy would follow a similar path if its population could be infinite. "

This is only true if also (i) the step-size is 'set' to zero and (ii) it uses an isotropic sample distribution. For non-zero step-size, there are two possibilities with infinite population size: (i) finite parent number (truncation ratio zero), in which case an ES steps right into the optimum. (ii) the usual truncation ratio. Then, as with finite population size, the behavior depends essentially (i) on the step-size and (ii) on the shape of the mutation distribution. When using the distribution which is optimal in the finite case, infinite population will not change a lot.

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