Solving A Problem We Already Solved
Last post we upgraded our swarm so that it could successfully minimize problems with any number of variables. All that’s left now is to teach it how to maximize and how to aim for specific results.
By now you should understand that maximization is just minimization turned upside down and hitting a target is the same as minimizing the difference between the results you are getting and the results you want.
Thanks to this insight we were able to easily write a 2D maximizer and target seeker by just using our already written minimizer. Now it’s time to do the same thing for the general swarm.
The tricky part here has to do with arguments.
In the 2D swarm we knew that everything would have two variables, so we were able to use lambda to build new two variable functions on demand. Then we passed that new two variable function to swarm-minimize-2d, which gave it the two arguments it expected.
But this time things are different. The general swarm-minimize uses apply to pass an unpredictable number of variables to our function. Which means that the new lambda function we build has to be able to accept a flexible number of variables and then pass them all on the the original function.
Fortunately for us Lisp is all ready to handle this problem thanks to the keyword &rest. Put inside of a function declaration this basically means “A list of every extra variable in the function call”.
A quick demo:
(defun rest-test (first second &rest extra) (format t “This is the first argument: ~a~%” first) (format t “This is the second argument: ~a~%” second) (format t “There were ~a extra arguments:~%” (length extra)) (loop for argument in extra do (format t “~a~%” argument)))
[9]> (rest-test “one” “two” “three”)
This is the first argument: one
This is the second argument: two
There were 1 extra arguments:
three
NIL
[10]> (rest-test “one” “two” “three” “four” “five”)
This is the first argument: one
This is the second argument: two
There were 3 extra arguments:
three
four
five
NIL
This is exactly what we need in order to create a lambda function that can take a flexible number of variables and pass them all along to an inner function. Upgrading the 2D maximize and target functions is now simple:
(defun swarm-maximize (fn boundaries) (swarm-minimize (lambda (&rest arguments) (* -1 (apply fn arguments) boundaries)))) (defun swarm-target (fn target boundaries) (swarm-minimize-2d (lambda (&rest arguments) (abs (- target (apply fn arguments)))) boundaries))
Let’s take these new functions for a spin:
[16]> (defparameter *swarm-iterations* 100)
*SWARM-ITERATIONS*
[17]> (defparameter *swarm-output* (swarm-maximize #’inverse-double-parabola ‘((-10 10)(-10 10))))
*SWARM-OUTPUT*
[18]> (first *swarm-output*)
(5.114901E-5 4.665053E-7)
[19]> (defparameter *swarm-output* (swarm-target #’triple-parabola 7 ‘((-10 10)(-10 10)(-10 10))))
*SWARM-OUTPUT*
[20]> (first *swarm-output*)
(-1.5512012 0.78995675 -1.9924128)
[21]> (triple-parabola -1.5512012 0.78995675 -1.9924128)
6.9999657
Success on both counts!
Coincidentally, this also shows off that our general swarm optimizer can handle 2D problems just as well as the specific 2D swarm optimizer. This is nice because it let me test maximization on a function we already had and saved me the trouble of writing an inverse triple parabola.
Time To Give The Swarm A Real Run For It’s Money
Ok, so we were able to maximize and target a couple of parabolas using the general swarm. But that’s not really very impressive. Parabolas are easy to solve and these weren’t even high dimensional problems.
So for my next and final “Let’s Program A Swarm Intelligence” I’m going to pull out all the stops and try to find some real world multi-variable problems to test this thing on. I’m sure I have a few engineering and calculus textbooks laying around with interesting problems just waiting to be solved.