# Version 0.2 of the Quipu Nelder-Mead solver

Back in April ‘23, I needed a simple solver for function minimization, and published a basic F# Nelder-Mead solver implementation on NuGet. I won’t go over the algorithm itself, if you are curious I wrote a post breaking down how the Nelder-Mead algorithm works a while back.

In a nutshell, the algorithm takes a function, and finds the set of inputs that produces the smallest output for that function. The algorithm is not foolproof, but it is very useful, and has the benefit of being fairly simple.

After dog-fooding my library for a bit, I found some rough spots, and decided it was time to make improvements. As a result, the API has changed a bit - hopefully for the better! In this post, I’ll go over some of these changes.

## Basic usage

Imagine that you are interested in the following function:

$f(x,y)=(x-10)^2+(y+5)^2$

Specifically, you would want to know what values of $(x,y)$ produce the smallest value for $f$.

This is how you would go about it with Quipu in an F# script:

#r "nuget: Quipu, 0.2.0"

let f (x, y) = (x - 10.0) ** 2.0 + (y + 5.0) ** 2.0


This produces the following output:

val it: Solution = Optimal (2.467079917e-07, [|9.999611886; -4.999690039|])


The solver has found an Optimal solution, for $x=9.999,y=-4.999$, which yields $f(x,y)=2.467 \times 10^{-7}$, very close to the correct answer, $f(10,-5)=0$.

## Improvements: search starting point

There were a few reasons I wanted to make some changes to the API.

First, I wanted better control over the search starting point.

In the basic usage example, the search will start with a simplex centered on 0, with vertices on a sphere of radius 1. The new version allows you to specify a starting point, which will generate a regular simplex on a sphere of arbitrary radius. Stated more simply, you can now specify in which region you want the search to begin. Pass it a starting point and a radius, and the search will begin with n vertices located on a sphere of the requested radius:

NelderMead.minimize f
|> NelderMead.startFrom (StartingPoint.fromValue ([1.0; 2.0], 0.1))


This is useful in situations where you want to start in a specific region. You can specify where to start, and how wide or tight the original search simplex should be.

This improves on version 0.1, where the search for a function of $n$ arguments was initiated using a set of $2n+1$ vertices, where $n+1$ are sufficient.

Under the hood, this will call the following function:

Simplex.create ([| 1.0; 2.0 |], 0.1)


… creating the following regular simplex, centered on $(1,2)$:

val it: Simplex =
Vectors
(2,
[|[|1.025881905; 1.903407417|]; [|0.9034074174; 2.025881905|];
[|1.070710678; 2.070710678|]|])
{dimension = 2;
size = 3;}


## Improvements: termination

Version 0.1 stopped the search when all values of $f$ were within precision bounds. I decided to tighten the criterion, requiring that every argument should also be within the same bounds.

The reason I made that change was that the original approach could result in an premature termination. In situations where the function being minimized is “flat” around the optimum, the search could stop early, with a wide simplex. Tightening the rules forces the simplex to contract around the minimum.

As an example, in the basic usage, the tolerance is set to $0.001$. We can relax this:

NelderMead.minimize f
{ Configuration.defaultValue with
Termination = {
Tolerance = 0.1
MaximumIterations = None
}
}


This produces the following result, where $(x,y)$ and $f$ are within $0.1$ of the correct value:

Optimal (0.001308370319, [|10.03105359; -5.01854845|])


Tighten up the tolerance like so:

NelderMead.minimize f
{ Configuration.defaultValue with
Termination = {
Tolerance = 0.000_001
MaximumIterations = None
}
}


… and the search produces much tighter results, within the new tolerance:

Optimal (1.78601232e-13, [|10.00000009; -4.999999587|])


## Parting words

There are still a few improvements I want to make, but I believe the code is now in a much better state that previously! If you are interested in perusing the code, you can find it here:

Code on GitHub

I think my next steps will be focused on

• Improving the detection of abnormal situations,
• Improving the performance of the core algorithm.

Besides that, one thing I’d like to try out is adding constraints to the problem, something along the lines of $argmin_{x,y}(f(x,y))$, subject to a list of inequality constraints like $3 \times x - y^2 \geq 10$.

I imagine that’s something I should be able to do with penalty / barrier functions. However, I haven’t had much experience with that approach, and I can also already see in my mind all sorts of complications arising!

At any rate, this is where I will stop for today. Hope you found something of interest in this post, and perhaps you’ll even have a use for this library!

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