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).

>>> import Test.DocTest.Prop
>>> 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
>>> assert $ f bestXs < 1e-10

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}
>>> assert $ f2 bestXs < 1e-110

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
>>> assert $ f3 xs31 / f3 xs30 < 1e-10

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) :: V.Vector Double -> Double
>>> bestVx <- run $ minimizeT f4 $ V.replicate 10 0
>>> assert $ f4 bestVx < 1e-10

Or use minimizeG to optimize functions of almost any type. Let's create a triangle ABC so that AB = 3, AC = 4, BC = 5.

>>> let dist (ax,ay) (bx,by) = ((ax-bx)**2 + (ay-by)**2)**0.5
>>> let f5 [a,b,c] = (dist a b - 3.0)**2 + (dist a c - 4.0)**2 + (dist b c - 5.0)**2
>>> bestTriangle <- run $ (minimizeG f5 [(0,0),(0,0),(0,0)]){tolFun = Just 1e-20}
>>> assert $ f5 bestTriangle < 1e-10

Then the angle BAC should be orthogonal.

>>> let [(ax,ay),(bx,by),(cx,cy)] = bestTriangle
>>> assert $ abs ((bx-ax)*(cx-ax) + (by-ay)*(cy-ay)) < 1e-10

When optimizing noisy functions, set noiseHandling = True 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
>>> xs60 <- run $ (minimizeIO f6 $ replicate 10 0) {noiseHandling = False}
>>> xs61 <- run $ (minimizeIO f6 $ replicate 10 0) {noiseHandling = True}
>>> assert $ f6Pure xs61 < f6Pure xs60