# 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.

## Fixed Point

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)))

## Good-Enough

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))

An `ideal`

is justified to be a global binding (a constant, of course! or wait...)

## Newton's Method

*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))

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