Usage:

1. create an optimization problem of type Config by one of minimize, minimizeIO etc.
2. run it.

Let's optimize the following function f(xs). xs is a list of Double and f has its minimum at xs !! i = sqrt(i).

>>> let f = sum . zipWith (\i x -> (x*abs x - i)**2) [0..] :: [Double] -> Double
>>> let initXs = replicate 10 0                            :: [Double]
>>> bestXs <- run $ minimize f initXs
>>> f bestXs < 1e-10
True

If your optimization is not working well, try:

• Set scaling in the Config to the appropriate search range of each parameter.
• Set tolFun in the Config to the appropriate scale of the function values.

An example for scaling the function value:

>>> let f2 xs = (/1e100) $ sum $ zipWith (\i x -> (x*abs x - i)**2) [0..] xs
>>> bestXs <- run $ (minimize f2 $ replicate 10 0) {tolFun = Just 1e-111}
>>> f2 bestXs < 1e-110
True

An example for scaling the input values:

>>> let f3 xs = sum $ zipWith (\i x -> (x*abs x - i)**2) [0,1e100..] xs
>>> let xs30 = replicate 10 0 :: [Double]
>>> let m3 = (minimize f3 xs30) {scaling = Just (repeat 1e50)}
>>> xs31 <- run $ m3
>>> f3 xs31 / f3 xs30 < 1e-10
True

Use minimizeT to optimize functions on traversable structures.

>>> import qualified Data.Vector as V
>>> let f4 = V.sum . V.imap (\i x -> (x*abs x - fromIntegral i)**2)

>>> :t f4
f4 :: Floating c => V.Vector c -> c
>>> bestVx <- run $ minimizeT f4 $ V.replicate 10 0
>>> f4 bestVx < 1e-10
True

Or use minimizeG to optimize functions of any type that is Data and that contains Doubles. Here is an example that deal with Triangles.

>>> :set -XDeriveDataTypeable
>>> import Data.Data
>>> data Pt = Pt Double Double deriving (Typeable,Data)
>>> let dist (Pt ax ay) (Pt bx by) = ((ax-bx)**2 + (ay-by)**2)**0.5
>>> data Triangle = Triangle Pt Pt Pt deriving (Typeable,Data)

Let us create a triangle ABC so that AB = 3, AC = 4, BC = 5.

>>> let f5 (Triangle a b c) = (dist a b - 3.0)**2 + (dist a c - 4.0)**2 + (dist b c - 5.0)**2
>>> let triangle0 = Triangle o o o where o = Pt 0 0
>>> :t f5
f5 :: Triangle -> Double
>>> bestTriangle <- run $ (minimizeG f5 triangle0){tolFun = Just 1e-20}
>>> f5 bestTriangle < 1e-10
True

Then the angle BAC should be orthogonal.

>>> let (Triangle (Pt ax ay) (Pt bx by) (Pt cx cy)) = bestTriangle
>>> abs ((bx-ax)*(cx-ax) + (by-ay)*(cy-ay)) < 1e-10
True

When optimizing noisy functions, set noiseHandling = True (and increase noiseReEvals) for better results.

>>> import System.Random
>>> let noise = randomRIO (0,1e-2)
>>> let f6Pure = sum . zipWith (\i x -> (x*abs x - i)**2) [0..]
>>> let f6 xs = fmap (f6Pure xs +) noise
>>> :t f6
f6 :: (Floating b, Enum b, Random b) => [b] -> IO b
>>> xs60 <- run $ (minimizeIO f6 $ replicate 10 0) {noiseHandling = False}
>>> xs61 <- run $ (minimizeIO f6 $ replicate 10 0) {noiseHandling = True,noiseReEvals=Just 10}
>>> -- assert $ f6Pure xs61 < f6Pure xs60

(note : the above assertion holds with probability of about 70%.)