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|
{-# OPTIONS_GHC -fglasgow-exts #-}
-----------------------------------------------------------------------------
{- |
Module : Numeric.GSL.Minimization
Copyright : (c) Alberto Ruiz 2006-9
License : GPL-style
Maintainer : Alberto Ruiz (aruiz at um dot es)
Stability : provisional
Portability : uses ffi
Minimization of a multidimensional function using some of the algorithms described in:
<http://www.gnu.org/software/gsl/manual/html_node/Multidimensional-Minimization.html>
-}
-----------------------------------------------------------------------------
module Numeric.GSL.Minimization (
minimizeConjugateGradient,
minimizeVectorBFGS2,
minimizeNMSimplex
) where
import Data.Packed.Internal
import Data.Packed.Matrix
import Foreign
import Foreign.C.Types(CInt)
-------------------------------------------------------------------------
{- | The method of Nelder and Mead, implemented by /gsl_multimin_fminimizer_nmsimplex/. The gradient of the function is not required. This is the example in the GSL manual:
@minimize f xi = minimizeNMSimplex f xi (replicate (length xi) 1) 1e-2 100
\ --
f [x,y] = 10*(x-1)^2 + 20*(y-2)^2 + 30
\ --
main = do
let (s,p) = minimize f [5,7]
print s
print p
\ --
\> main
[0.9920430849306285,1.9969168063253164]
0. 512.500 1.082 6.500 5.
1. 290.625 1.372 5.250 4.
2. 290.625 1.372 5.250 4.
3. 252.500 1.372 5.500 1.
4. 101.406 1.823 2.625 3.500
5. 101.406 1.823 2.625 3.500
6. 60. 1.823 0. 3.
7. 42.275 1.303 2.094 1.875
8. 42.275 1.303 2.094 1.875
9. 35.684 1.026 0.258 1.906
10. 35.664 0.804 0.588 2.445
11. 30.680 0.467 1.258 2.025
12. 30.680 0.356 1.258 2.025
13. 30.539 0.285 1.093 1.849
14. 30.137 0.168 0.883 2.004
15. 30.137 0.123 0.883 2.004
16. 30.090 0.100 0.958 2.060
17. 30.005 6.051e-2 1.022 2.004
18. 30.005 4.249e-2 1.022 2.004
19. 30.005 4.249e-2 1.022 2.004
20. 30.005 2.742e-2 1.022 2.004
21. 30.005 2.119e-2 1.022 2.004
22. 30.001 1.530e-2 0.992 1.997
23. 30.001 1.259e-2 0.992 1.997
24. 30.001 7.663e-3 0.992 1.997@
The path to the solution can be graphically shown by means of:
@'Graphics.Plot.mplot' $ drop 3 ('toColumns' p)@
-}
minimizeNMSimplex :: ([Double] -> Double) -- ^ function to minimize
-> [Double] -- ^ starting point
-> [Double] -- ^ sizes of the initial search box
-> Double -- ^ desired precision of the solution
-> Int -- ^ maximum number of iterations allowed
-> ([Double], Matrix Double)
-- ^ solution vector, and the optimization trajectory followed by the algorithm
minimizeNMSimplex f xi sz tol maxit = unsafePerformIO $ do
let xiv = fromList xi
szv = fromList sz
n = dim xiv
fp <- mkVecfun (iv (f.toList))
rawpath <- ww2 withVector xiv withVector szv $ \xiv' szv' ->
createMIO maxit (n+3)
(c_minimizeNMSimplex fp tol (fi maxit) // xiv' // szv')
"minimizeNMSimplex"
let it = round (rawpath @@> (maxit-1,0))
path = takeRows it rawpath
[sol] = toLists $ dropRows (it-1) path
freeHaskellFunPtr fp
return (drop 3 sol, path)
foreign import ccall "gsl-aux.h minimize"
c_minimizeNMSimplex:: FunPtr (CInt -> Ptr Double -> Double) -> Double -> CInt -> TVVM
----------------------------------------------------------------------------------
{- | The Fletcher-Reeves conjugate gradient algorithm /gsl_multimin_fminimizer_conjugate_fr/. This is the example in the GSL manual:
@minimize = minimizeConjugateGradient 1E-2 1E-4 1E-3 30
f [x,y] = 10*(x-1)^2 + 20*(y-2)^2 + 30
\ --
df [x,y] = [20*(x-1), 40*(y-2)]
\ --
main = do
let (s,p) = minimize f df [5,7]
print s
print p
\ --
\> main
[1.0,2.0]
0. 687.848 4.996 6.991
1. 683.555 4.989 6.972
2. 675.013 4.974 6.935
3. 658.108 4.944 6.861
4. 625.013 4.885 6.712
5. 561.684 4.766 6.415
6. 446.467 4.528 5.821
7. 261.794 4.053 4.632
8. 75.498 3.102 2.255
9. 67.037 2.852 1.630
10. 45.316 2.191 1.762
11. 30.186 0.869 2.026
12. 30. 1. 2.@
The path to the solution can be graphically shown by means of:
@'Graphics.Plot.mplot' $ drop 2 ('toColumns' p)@
-}
minimizeConjugateGradient ::
Double -- ^ initial step size
-> Double -- ^ minimization parameter
-> Double -- ^ desired precision of the solution (gradient test)
-> Int -- ^ maximum number of iterations allowed
-> ([Double] -> Double) -- ^ function to minimize
-> ([Double] -> [Double]) -- ^ gradient
-> [Double] -- ^ starting point
-> ([Double], Matrix Double) -- ^ solution vector, and the optimization trajectory followed by the algorithm
minimizeConjugateGradient = minimizeWithDeriv 0
{- | Taken from the GSL manual:
The vector Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm. This is a quasi-Newton method which builds up an approximation to the second derivatives of the function f using the difference between successive gradient vectors. By combining the first and second derivatives the algorithm is able to take Newton-type steps towards the function minimum, assuming quadratic behavior in that region.
The bfgs2 version of this minimizer is the most efficient version available, and is a faithful implementation of the line minimization scheme described in Fletcher's Practical Methods of Optimization, Algorithms 2.6.2 and 2.6.4. It supercedes the original bfgs routine and requires substantially fewer function and gradient evaluations. The user-supplied tolerance tol corresponds to the parameter \sigma used by Fletcher. A value of 0.1 is recommended for typical use (larger values correspond to less accurate line searches).
-}
minimizeVectorBFGS2 ::
Double -- ^ initial step size
-> Double -- ^ minimization parameter tol
-> Double -- ^ desired precision of the solution (gradient test)
-> Int -- ^ maximum number of iterations allowed
-> ([Double] -> Double) -- ^ function to minimize
-> ([Double] -> [Double]) -- ^ gradient
-> [Double] -- ^ starting point
-> ([Double], Matrix Double) -- ^ solution vector, and the optimization trajectory followed by the algorithm
minimizeVectorBFGS2 = minimizeWithDeriv 1
minimizeWithDeriv method istep minimpar tol maxit f df xi = unsafePerformIO $ do
let xiv = fromList xi
n = dim xiv
f' = f . toList
df' = (fromList . df . toList)
fp <- mkVecfun (iv f')
dfp <- mkVecVecfun (aux_vTov df')
rawpath <- withVector xiv $ \xiv' ->
createMIO maxit (n+2)
(c_minimizeWithDeriv method fp dfp istep minimpar tol (fi maxit) // xiv')
"minimizeDerivV"
let it = round (rawpath @@> (maxit-1,0))
path = takeRows it rawpath
sol = toList $ cdat $ dropColumns 2 $ dropRows (it-1) path
freeHaskellFunPtr fp
freeHaskellFunPtr dfp
return (sol,path)
foreign import ccall "gsl-aux.h minimizeWithDeriv"
c_minimizeWithDeriv :: CInt -> FunPtr (CInt -> Ptr Double -> Double)
-> FunPtr (CInt -> Ptr Double -> Ptr Double -> IO ())
-> Double -> Double -> Double -> CInt
-> TVM
---------------------------------------------------------------------
iv :: (Vector Double -> Double) -> (CInt -> Ptr Double -> Double)
iv f n p = f (createV (fromIntegral n) copy "iv") where
copy n' q = do
copyArray q p (fromIntegral n')
return 0
-- | conversion of Haskell functions into function pointers that can be used in the C side
foreign import ccall "wrapper"
mkVecfun :: (CInt -> Ptr Double -> Double)
-> IO( FunPtr (CInt -> Ptr Double -> Double))
-- | another required conversion
foreign import ccall "wrapper"
mkVecVecfun :: (CInt -> Ptr Double -> Ptr Double -> IO ())
-> IO (FunPtr (CInt -> Ptr Double -> Ptr Double->IO()))
aux_vTov :: (Vector Double -> Vector Double) -> (CInt -> Ptr Double -> Ptr Double -> IO())
aux_vTov f n p r = g where
V {fptr = pr} = f x
x = createV (fromIntegral n) copy "aux_vTov"
copy n' q = do
copyArray q p (fromIntegral n')
return 0
g = withForeignPtr pr $ \p' -> copyArray r p' (fromIntegral n)
--------------------------------------------------------------------
createV n fun msg = unsafePerformIO $ do
r <- createVector n
app1 fun vec r msg
return r
createMIO r c fun msg = do
res <- createMatrix RowMajor r c
app1 fun mat res msg
return res
|
