Introductory Programming
Fall 2006

Optimization
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Optimization is the general name for finding the maximum (or minimum)
point in a design space.

For example, imagine that you have a function called "baseball" that
takes two input variables, launch velocity and launch angle, and
that computes the flight distance of a baseball hit with the given velocity and
angle.

Now imagine that you would like to compute the optimal angle for
a given velocity (the angle that maximizes the flight distance).

How would you do it?

Option 1:

Choose two values that you are pretty sure bracket the root.

       (in this case, 0 and 90 degrees would do it!)

Use linspace to enumerate a number of values in this range.

       (help linspace for details)

Evaluate the function for each value in the range.

Choose the value of x that maximizes f(x)

    in this case the angle that maximizes baseball(velocity, angle)

Maximum error is the space between values in the range.

Option 2:

Same as option 1, but after the first iteration, run it again
with the newly discovered two values that bracket the root.

Option 3:

Golden section search (see the handout from Numerical Recipes in C)

Option 4:

Brent's method (use a parabolic interpolation)

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If you can evaluate the derivative of the function as well as the
function itself, then there are two more option!

Option 5:

Use a root-finding algorithm (like fzero) on the derivative.
    (and check whether you found a minimum or a maximum!)

Option 6:

Amend Options 3 or 4 by using the derivative to decide which
subinterval to search in.