Numeric.GSL.Minimization

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:

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.

Documentation

minimize

Arguments

:: 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

minimizeV

Arguments

:: 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

Minimization without derivatives (vector version)

data MinimizeMethod

Constructors

NMSimplex

NMSimplex2

Instances

Bounded MinimizeMethod

Methods

minBound :: MinimizeMethod

maxBound :: MinimizeMethod

Enum MinimizeMethod

Methods

succ :: MinimizeMethod -> MinimizeMethod

pred :: MinimizeMethod -> MinimizeMethod

toEnum :: Int -> MinimizeMethod

fromEnum :: MinimizeMethod -> Int

enumFrom :: MinimizeMethod -> [MinimizeMethod]

enumFromThen :: MinimizeMethod -> MinimizeMethod -> [MinimizeMethod]

enumFromTo :: MinimizeMethod -> MinimizeMethod -> [MinimizeMethod]

enumFromThenTo :: MinimizeMethod -> MinimizeMethod -> MinimizeMethod -> [MinimizeMethod]

Eq MinimizeMethod

Methods

(==) :: MinimizeMethod -> MinimizeMethod -> Bool

(/=) :: MinimizeMethod -> MinimizeMethod -> Bool

Show MinimizeMethod

Methods

showsPrec :: Int -> MinimizeMethod -> ShowS

show :: MinimizeMethod -> String

showList :: [MinimizeMethod] -> ShowS

minimizeD

Arguments

:: 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.

minimizeVD

Arguments

:: 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

Minimization with derivatives (vector version)

data MinimizeMethodD

Constructors

ConjugateFR

ConjugatePR

VectorBFGS

VectorBFGS2

SteepestDescent

Instances

Bounded MinimizeMethodD

Methods

minBound :: MinimizeMethodD

maxBound :: MinimizeMethodD

Enum MinimizeMethodD

Methods

succ :: MinimizeMethodD -> MinimizeMethodD

pred :: MinimizeMethodD -> MinimizeMethodD

toEnum :: Int -> MinimizeMethodD

fromEnum :: MinimizeMethodD -> Int

enumFrom :: MinimizeMethodD -> [MinimizeMethodD]

enumFromThen :: MinimizeMethodD -> MinimizeMethodD -> [MinimizeMethodD]

enumFromTo :: MinimizeMethodD -> MinimizeMethodD -> [MinimizeMethodD]

enumFromThenTo :: MinimizeMethodD -> MinimizeMethodD -> MinimizeMethodD -> [MinimizeMethodD]

Eq MinimizeMethodD

Methods

(==) :: MinimizeMethodD -> MinimizeMethodD -> Bool

(/=) :: MinimizeMethodD -> MinimizeMethodD -> Bool

Show MinimizeMethodD

Methods

showsPrec :: Int -> MinimizeMethodD -> ShowS

show :: MinimizeMethodD -> String

showList :: [MinimizeMethodD] -> ShowS

uniMinimize

Arguments

:: UniMinimizeMethod

The method used.

-> Double

desired precision of the solution

-> Int

maximum number of iterations allowed

-> (Double -> Double)

function to minimize

-> Double

guess for the location of the minimum

-> Double

lower bound of search interval

-> Double

upper bound of search interval

-> (Double, Matrix Double)

solution and optimization path

Onedimensional minimization.

data UniMinimizeMethod

Constructors

GoldenSection

BrentMini

QuadGolden

Instances

Bounded UniMinimizeMethod

Methods

minBound :: UniMinimizeMethod

maxBound :: UniMinimizeMethod

Enum UniMinimizeMethod

Methods

succ :: UniMinimizeMethod -> UniMinimizeMethod

pred :: UniMinimizeMethod -> UniMinimizeMethod

toEnum :: Int -> UniMinimizeMethod

fromEnum :: UniMinimizeMethod -> Int

enumFrom :: UniMinimizeMethod -> [UniMinimizeMethod]

enumFromThen :: UniMinimizeMethod -> UniMinimizeMethod -> [UniMinimizeMethod]

enumFromTo :: UniMinimizeMethod -> UniMinimizeMethod -> [UniMinimizeMethod]

enumFromThenTo :: UniMinimizeMethod -> UniMinimizeMethod -> UniMinimizeMethod -> [UniMinimizeMethod]

Eq UniMinimizeMethod

Methods

(==) :: UniMinimizeMethod -> UniMinimizeMethod -> Bool

(/=) :: UniMinimizeMethod -> UniMinimizeMethod -> Bool

Show UniMinimizeMethod

Methods

showsPrec :: Int -> UniMinimizeMethod -> ShowS

show :: UniMinimizeMethod -> String

showList :: [UniMinimizeMethod] -> ShowS

minimizeNMSimplex :: ([Double] -> Double) -> [Double] -> [Double] -> Double -> Int -> ([Double], Matrix Double)

Deprecated: use minimize NMSimplex2 eps maxit sizes f xi

minimizeConjugateGradient :: Double -> Double -> Double -> Int -> ([Double] -> Double) -> ([Double] -> [Double]) -> [Double] -> ([Double], Matrix Double)

Deprecated: use minimizeD ConjugateFR eps maxit step tol f g xi

minimizeVectorBFGS2 :: Double -> Double -> Double -> Int -> ([Double] -> Double) -> ([Double] -> [Double]) -> [Double] -> ([Double], Matrix Double)

Deprecated: use minimizeD VectorBFGS2 eps maxit step tol f g xi