Reduction To A Local Optimum
An optimum is a value such that any change will produce a worse result. Think of a tip of mountain...
This is also the definition of perfection - Perfection is obtained not when there is nothing more to add, but when there is nothing more to remove.
This is also the notion of a balance (a balanced system), an equilibrium. Homeostasis is the proper balance of (and between) all major subsystems of an organism.
The notion of a global optimum depends on the domain. The tip of the Everest is a global optimum in the domain of 8000-meter summits. The notion of a local optimum is that there are a lot of high mountains in the world.
The notion of a Fixed Point is a Global Optimum of a convex function.
Fixed Point of a function is a value such that a function applied to it produces the same value.
This means that the function converges to its optimum.
For less abstract purposes the function encapsulates a test - usually a predicate.
(define (fixed-point f start) (define (iterate old new) (if (good-enough? old new) new (iterate new (f new)))) (iterate start (f start)))
The processes of Nature are vastly complex, so we usually could not hope for finding (or even defining!) a global optimum. We, mere mortals, could define a predicate - how small is the difference between an ideal and what we really have (are we close enough to the unattainable ideal?).
(define (good-enough? value) (< (abs (- ideal value)) tolerance))
ideal is justified to be a global binding (a constant, of course! or wait...)
better and better successive approximations
Unless there is nothing to improve - improve''
(define (sqrt x) (define tolerance 0.00001) (define (good-enough? y) (< (abs (- (* y y) x)) tolerance)) (define (improve y) (average (/ x y) y)) (define (try y) (if (good-enough? y) y (try (improve y)))) (try 1))