Safe Haskell | None |
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Numerically solve convex lagrange multiplier problems with conjugate gradient descent.

Here is an example from the Wikipedia page on Lagrange multipliers. Maximize f(x, y) = x + y, subject to the constraint x^2 + y^2 = 1

`>>>`

Right ([0.707,0.707], [-0.707])`maximize 0.00001 (\[x, y] -> x + y) [(\[x, y] -> x^2 + y^2) <=> 1] 2`

The first elements of the result pair are the arguments for the objective function at the minimum. The second elements are the lagrange multipliers.

- type AD2 s r a = AD s (AD r a)
- newtype FU a = FU {}
- (<=>) :: (forall s r. (Mode s, Mode r) => [AD2 s r a] -> AD2 s r a) -> a -> Constraint a
- type Constraint a = (FU a, a)
- maximize :: Double -> (forall s r. (Mode s, Mode r) => [AD2 s r Double] -> AD2 s r Double) -> [Constraint Double] -> Int -> Either (Result, Statistics) (Vector Double, Vector Double)
- minimize :: Double -> (forall s r. (Mode s, Mode r) => [AD2 s r Double] -> AD2 s r Double) -> [Constraint Double] -> Int -> Either (Result, Statistics) (Vector Double, Vector Double)
- feasible :: (forall s r. (Mode s, Mode r) => [AD2 s r Double] -> AD2 s r Double) -> [Constraint Double] -> [Double] -> Bool

### Helper types

A newtype wrapper for working with the rank 2 types constraint functions.

(<=>) :: (forall s r. (Mode s, Mode r) => [AD2 s r a] -> AD2 s r a) -> a -> Constraint aSource

Build a `Constraint`

from a function and a constant

type Constraint a = (FU a, a)Source

A constraint of the form `g(x, y, ...) = c`

. See `<=>`

for constructing a `Constraint`

.

## Solver

:: Double | |

-> (forall s r. (Mode s, Mode r) => [AD2 s r Double] -> AD2 s r Double) | The function to maximize |

-> [Constraint Double] | The constraints as pairs |

-> Int | The arity of the objective function which should equal the arity of the constraints. |

-> Either (Result, Statistics) (Vector Double, Vector Double) | Either an explanation of why the gradient descent failed or a pair containing the arguments at the minimum and the lagrange multipliers |

Finding the maximum is the same as the minimum with the objective function inverted

:: Double | |

-> (forall s r. (Mode s, Mode r) => [AD2 s r Double] -> AD2 s r Double) | The function to minimize |

-> [Constraint Double] | The constraints as pairs |

-> Int | The arity of the objective function which should equal the arity of the constraints. |

-> Either (Result, Statistics) (Vector Double, Vector Double) | Either an explanation of why the gradient descent failed or a pair containing the arguments at the minimum and the lagrange multipliers |

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.

### Experimental features

feasible :: (forall s r. (Mode s, Mode r) => [AD2 s r Double] -> AD2 s r Double) -> [Constraint Double] -> [Double] -> BoolSource

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.