{-# LANGUAGE Rank2Types, FlexibleContexts #-} module Numeric.AD.Lagrangian.Internal where import Numeric.Optimization.Algorithms.HagerZhang05 import qualified Data.Vector.Unboxed as U import qualified Data.Vector.Storable as S import Numeric.AD import GHC.IO (unsafePerformIO) import Numeric.AD.Types import Numeric.AD.Internal.Classes import Numeric.LinearAlgebra.Algorithms import qualified Data.Packed.Vector as V import qualified Data.Packed.Matrix as M import Numeric.AD.Internal.Tower infixr 1 <=> -- | This is just a little bit of sugar for (,) to make constraints look like -- equals (<=>) :: ([a] -> a) -> a -> Constraint a g <=> c = (g,c) -- | The type for the contraints. -- Given a constraint @g(x, y, ...) = c@, we would represent it as @(g, c)@. -- or with sugar @g@ '<=>' @c@ type Constraint a = ([a] -> a, a) type AD2 s r a = AD s (AD r Double) -- | This is the lagrangian multiplier solver. It is assumed that the -- objective function and all of the constraints take in the -- same amount of arguments. minimize :: Double -> (forall s r. (Mode s, Mode r) => [AD2 s r Double] -> AD2 s r Double) -- ^ The function to minimize -> (forall s r. (Mode s, Mode r) => [Constraint (AD2 s r Double)] ) -- ^ The constraints as pairs @g \<=\> c@ which represent equations -- of the form @g(x, y, ...) = c@ -> Int -- ^ The arity of the objective function which should equal the arity of -- the constraints. -> Either (Result, Statistics) (S.Vector Double, S.Vector Double) -- ^ Either an explanation of why the gradient descent failed or a pair -- containing the arguments at the minimum and the lagrange multipliers minimize tolerance toMin constraints argCount = result where -- The function to minimize for the langrangian is the squared gradient obj argsAndLams = squaredGrad (lagrangian toMin constraints argCount) argsAndLams -- The mode does matter but I need to add annotation for the type checker constraintCount = length (constraints :: [Constraint (AD Tower (AD Tower Double))]) -- perhaps this should be exposed guess = U.replicate (argCount + constraintCount) (1.0 :: Double) result = case unsafePerformIO $ optimize (defaultParameters { printFinal = False }) tolerance guess (VFunction (lowerFU obj . U.toList)) (VGradient (U.fromList . grad obj . U.toList)) Nothing of (vs, ToleranceStatisfied, _) -> Right (S.take argCount vs, S.drop argCount vs) (_, x, y) -> Left (x, y) -- | Finding the maximum is the same as the minimum with the objective function inverted maximize :: Double -> (forall s r. (Mode s, Mode r) => [AD2 s r Double] -> AD2 s r Double) -- ^ The function to maximize -> (forall s r. (Mode s, Mode r) => [Constraint (AD2 s r Double)] ) -- ^ The constraints as pairs @g \<=\> c@ which represent equations -- of the form @g(x, y, ...) = c@ -> Int -- ^ The arity of the objective function which should equal the arity of -- the constraints. -> Either (Result, Statistics) (S.Vector Double, S.Vector Double) -- ^ Either an explanation of why the gradient descent failed or a pair -- containing the arguments at the minimum and the lagrange multipliers maximize tolerance toMax constraints argCount = minimize tolerance (negate1 . toMax) constraints argCount lagrangian :: Num a => ([a] -> a) -> [Constraint a] -> Int -> [a] -> a lagrangian f constraints argCount argsAndLams = result where args = take argCount argsAndLams lams = drop argCount argsAndLams -- g(x, y, ...) = c <=> g(x, y, ...) - c = 0 appliedConstraints = fmap (\(f, c) -> f args - c) constraints -- L(x, y, ..., lam0, ...) = f(x, y, ...) + lam0 * (g0 - c0) ... result = f args + (sum . zipWith (*) lams $ appliedConstraints) squaredGrad :: Num a => (forall s. Mode s => [AD s a] -> AD s a) -> [a] -> a squaredGrad f vs = sum . fmap (\x -> x*x) . grad f $ vs -- | WARNING. Experimental. -- This is not a true feasibility test for the function. I am not sure -- exactly how to implement that. This just checks the feasiblility at a point. -- If this ever returns false, 'solve' can fail. feasible :: (forall s r. (Mode s, Mode r) => [AD2 s r Double] -> AD2 s r Double) -> (forall s r. (Mode s, Mode r) => [Constraint (AD2 s r Double)] ) -> [Double] -> Bool feasible toMin constraints points = result where obj argsAndLams = squaredGrad (lagrangian toMin constraints $ length points) argsAndLams hessianMatrix = M.fromLists . hessian obj $ points -- make sure all of the eigenvalues are positive result = all (>0) . V.toList . eigenvaluesSH $ hessianMatrix