# Quipu, a simple Nelder Mead solver in F#

15 Apr 2023Some time back, I wrote a small post digging into the mechanics behind the Nelder Mead solver. As it turns out, I had a use for it recently, and after copy-pasting my own code a few times, I figured it would make my life easier to turn that into a NuGet package, Quipu.

So what does it do, and why might you care?

A code example might be the quickest explanation here. Suppose that, for
whatever reason, you were interested in the function `f(x) = x ^ 2`

, and wanted
to know for what value of `x`

this function reaches its minimum.

That is easy to solve with the Quipu Nelder-Mead solver:

```
open Quipu
open Quipu.NelderMead
let f x = x ** 2.0
let solution =
NelderMead.solve
Configuration.defaultValue
(Objective.from f) [ 100.0 ]
printfn $"{solution}"
```

… which produces the following result:

```
Optimal (0.0001556843433, [|0.01247735322|])
```

The function `f`

reaches a minimum of `0.0001`

, for `x = 0.0124`

.

`NelderMead.solve`

expects 3 arguments:

- The
`Configuration`

describes how the solver should behave, - The
`Objective`

is the function we are trying to minimize, - The starting value,
`[ 100.0 ]`

, is our initial guess.

Now the mathematically inclined reader might point out that surely, this is
not correct. `f`

reaches a minimum of `0.0`

, for `x = 0.0`

. The
Nelder-Mead algorithm is a numerical method which will produce an
approximation for the answer.

If the accuracy is insufficient, you can set up a tighter tolerance:

```
let config = {
Configuration.defaultValue with
Termination = {
Tolerance = 0.000_0001
MaximumIterations = None
}
}
let closerSolution =
NelderMead.solve config (Objective.from f) [ 100.0 ]
printfn $"{closerSolution}"
```

```
Optimal (6.663562871e-08, [|0.000258138778|])
```

This is closer, and the minimum value, `6.663e-8`

, is within `0.000_0001`

, or
`1e-07`

, of the correct answer, `0.0`

.

While we are discussing caveats, Nelder-Mead is not guaranteed to find the global minimum. It might give you a local minimum only.

So what would happen if we gave the solver a function that does not have a
minimum, like `f(x) = x`

?

```
let f x = x
let solution =
NelderMead.solve
Configuration.defaultValue
(Objective.from f) [ 100.0 ]
printfn $"{solution}"
```

```
Unbounded
```

The solver returns `Unbounded`

as a solution, that is, the problem has no
minimum.

In circumstances where abnormal situations are encountered (for instance, `nan`

value during the search), the solver will return `Abnormal`

, with the values
that caused the error.

What if you had a more complicated function, say, `g(x, y) = sin x * cos y`

?

```
let g (x, y) = sin x * cos y
let solution =
NelderMead.solve
Configuration.defaultValue
(Objective.from g) [ 0.0; 0.0 ]
printfn $"{solution}"
```

```
Optimal (-0.9995738601, [|-1.59440106; -0.01718172454|])
```

Note how the starting value is now `[ 0.0; 0.0 ]`

. Because `g`

expects 2
arguments, we need to provide an initial value for both.

Functions of 3 arguments follow the same pattern. After 4, you are on your own,
and will need to do a little manual wrapping, converting the function into a
form `Objective.from`

can handle: `(int: dimension, f: float [] -> float)`

,
like so:

```
// convert f(a,b,c,d) = sin a + cos b + (c * d) ^ 2
// into a function that takes an array of floats:
let h (args: float[]) =
sin args.[0] + cos args.[1] + (args.[2] * args.[3]) ** 2.0
// call Objective.from (4, h), where 4 is the dimension,
// that is, the number of arguments we expect in the array:
let solution =
NelderMead.solve
Configuration.defaultValue
(Objective.from (4, h)) [ 0.0; 0.0; 0.0; 0.0 ]
printfn $"{solution}"
```

```
Optimal
(-1.99962865, [|-1.559102568; 3.117602262; 0.7363555181; 0.005298690767|])
```

And… that’s what I got at the moment! It is version `0.1.0`

for a reason: it
works on my machine, for the problem I needed it for. There is obviously quite
a bit that can be improved around usability, too. So your mileage may vary, but
it was useful to me, so I figured I would share!

If you have comments or questions, hit me up on Mastodon!

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