1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
|
{-# 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>
The example in the GSL manual:
@
f [x,y] = 10*(x-1)^2 + 20*(y-2)^2 + 30
main = do
let (s,p) = minimize NMSimplex2 1E-2 30 [1,1] f [5,7]
print s
print p
\> main
[0.9920430849306288,1.9969168063253182]
0.000 512.500 1.130 6.500 5.000
1.000 290.625 1.409 5.250 4.000
2.000 290.625 1.409 5.250 4.000
3.000 252.500 1.409 5.500 1.000
...
22.000 30.001 0.013 0.992 1.997
23.000 30.001 0.008 0.992 1.997
@
The path to the solution can be graphically shown by means of:
@'Graphics.Plot.mplot' $ drop 3 ('toColumns' p)@
Taken from the GSL manual:
The vector Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm 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).
The nmsimplex2 version is a new O(N) implementation of the earlier O(N^2) nmsimplex minimiser. It calculates the size of simplex as the rms distance of each vertex from the center rather than the mean distance, which has the advantage of allowing a linear update.
-}
-----------------------------------------------------------------------------
module Numeric.GSL.Minimization (
minimize, minimizeV, MinimizeMethod(..),
minimizeD, minimizeVD, MinimizeMethodD(..),
minimizeNMSimplex,
minimizeConjugateGradient,
minimizeVectorBFGS2
) where
import Data.Packed.Internal
import Data.Packed.Matrix
import Foreign
import Foreign.C.Types(CInt)
import Numeric.GSL.Internal
------------------------------------------------------------------------
{-# DEPRECATED minimizeNMSimplex "use minimize NMSimplex2 eps maxit sizes f xi" #-}
minimizeNMSimplex f xi szs eps maxit = minimize NMSimplex eps maxit szs f xi
{-# DEPRECATED minimizeConjugateGradient "use minimizeD ConjugateFR eps maxit step tol f g xi" #-}
minimizeConjugateGradient step tol eps maxit f g xi = minimizeD ConjugateFR eps maxit step tol f g xi
{-# DEPRECATED minimizeVectorBFGS2 "use minimizeD VectorBFGS2 eps maxit step tol f g xi" #-}
minimizeVectorBFGS2 step tol eps maxit f g xi = minimizeD VectorBFGS2 eps maxit step tol f g xi
-------------------------------------------------------------------------
data MinimizeMethod = NMSimplex
| NMSimplex2
deriving (Enum,Eq,Show,Bounded)
-- | Minimization without derivatives
minimize :: MinimizeMethod
-> Double -- ^ desired precision of the solution (size test)
-> Int -- ^ maximum number of iterations allowed
-> [Double] -- ^ sizes of the initial search box
-> ([Double] -> Double) -- ^ function to minimize
-> [Double] -- ^ starting point
-> ([Double], Matrix Double) -- ^ solution vector and optimization path
-- | Minimization without derivatives (vector version)
minimizeV :: MinimizeMethod
-> Double -- ^ desired precision of the solution (size test)
-> Int -- ^ maximum number of iterations allowed
-> Vector Double -- ^ sizes of the initial search box
-> (Vector Double -> Double) -- ^ function to minimize
-> Vector Double -- ^ starting point
-> (Vector Double, Matrix Double) -- ^ solution vector and optimization path
minimize method eps maxit sz f xi = v2l $ minimizeV method eps maxit (fromList sz) (f.toList) (fromList xi)
where v2l (v,m) = (toList v, m)
ww2 w1 o1 w2 o2 f = w1 o1 $ \a1 -> w2 o2 $ \a2 -> f a1 a2
minimizeV method eps maxit szv f xiv = unsafePerformIO $ do
let n = dim xiv
fp <- mkVecfun (iv f)
rawpath <- ww2 vec xiv vec szv $ \xiv' szv' ->
createMIO maxit (n+3)
(c_minimize (fi (fromEnum method)) fp eps (fi maxit) // xiv' // szv')
"minimize"
let it = round (rawpath @@> (maxit-1,0))
path = takeRows it rawpath
sol = cdat $ dropColumns 3 $ dropRows (it-1) path
freeHaskellFunPtr fp
return (sol, path)
foreign import ccall "gsl-aux.h minimize"
c_minimize:: CInt -> FunPtr (CInt -> Ptr Double -> Double) -> Double -> CInt -> TVVM
----------------------------------------------------------------------------------
data MinimizeMethodD = ConjugateFR
| ConjugatePR
| VectorBFGS
| VectorBFGS2
| SteepestDescent
deriving (Enum,Eq,Show,Bounded)
-- | Minimization with derivatives.
minimizeD :: MinimizeMethodD
-> Double -- ^ desired precision of the solution (gradient test)
-> Int -- ^ maximum number of iterations allowed
-> Double -- ^ size of the first trial step
-> Double -- ^ tol (precise meaning depends on method)
-> ([Double] -> Double) -- ^ function to minimize
-> ([Double] -> [Double]) -- ^ gradient
-> [Double] -- ^ starting point
-> ([Double], Matrix Double) -- ^ solution vector and optimization path
-- | Minimization with derivatives (vector version)
minimizeVD :: MinimizeMethodD
-> Double -- ^ desired precision of the solution (gradient test)
-> Int -- ^ maximum number of iterations allowed
-> Double -- ^ size of the first trial step
-> Double -- ^ tol (precise meaning depends on method)
-> (Vector Double -> Double) -- ^ function to minimize
-> (Vector Double -> Vector Double) -- ^ gradient
-> Vector Double -- ^ starting point
-> (Vector Double, Matrix Double) -- ^ solution vector and optimization path
minimizeD method eps maxit istep tol f df xi = v2l $ minimizeVD
method eps maxit istep tol (f.toList) (fromList.df.toList) (fromList xi)
where v2l (v,m) = (toList v, m)
minimizeVD method eps maxit istep tol f df xiv = unsafePerformIO $ do
let n = dim xiv
f' = f
df' = (checkdim1 n . df)
fp <- mkVecfun (iv f')
dfp <- mkVecVecfun (aux_vTov df')
rawpath <- vec xiv $ \xiv' ->
createMIO maxit (n+2)
(c_minimizeD (fi (fromEnum method)) fp dfp istep tol eps (fi maxit) // xiv')
"minimizeD"
let it = round (rawpath @@> (maxit-1,0))
path = takeRows it rawpath
sol = cdat $ dropColumns 2 $ dropRows (it-1) path
freeHaskellFunPtr fp
freeHaskellFunPtr dfp
return (sol,path)
foreign import ccall "gsl-aux.h minimizeD"
c_minimizeD :: CInt
-> FunPtr (CInt -> Ptr Double -> Double)
-> FunPtr TVV
-> Double -> Double -> Double -> CInt
-> TVM
---------------------------------------------------------------------
checkdim1 n v
| dim v == n = v
| otherwise = error $ "Error: "++ show n
++ " components expected in the result of the gradient supplied to minimizeD"
|
