Using F# to minimise a function

Finding the minimum of functions is at the heart of optimisation. Mathematicians, engineers and programmers have come up with a large number of approaches to solving this problem, including differentiation, genetic algorithms and even exhaustive search.

Consider a quadratic function such that could be written in F# as

let f x = (x ** 2.0) - (2.0 * x) + 1.0

Finding the minimum means finding the input value for the function that returns to lowest value. If we plot the curve, then the minimum is the lowest point of the curve.

One way to find this is find the derivative of the function:

(2.0 * x) - 2.0

and solve the equation where the derivative is equal to 0, which is when x is 1. This probably the best way to solve this problem in practice.

However, not all functions have derivatives that are easy to find. Therefore, an alternative way to minimise a function is an exhaustive search of the inputs in some range. This is not an elegant or generally efficient solution, but it works.

In F#, we might proceed as follows.

We define the range that we want to search and the step between candidate inputs:

let min = 0.0
let max = 10.0
let step = 0.01

We know where to start searching- with the lowest candidate input:

let firstCandidate = f min, min

This creates a tuple of two floats, the first being the output and the second being the lowest candidate input.

We also want a sequence of tuples of two floats that are the remaining candidate solutions:

let remainingCandidates = seq { for c in min + step .. step .. max do yield f c, c }

Note that at this point the sequence has not been enumerated and the function has not been run with each of the candidate inputs. Because sequences are evaluated lazily, the function will not be run for each of the candidate inputs until it is asked for.

We are trying to minimise the function, therefore we need a function that can compare the first elements of two candidate solutions:

let findMin currentMinSln candidateSln =
    match fst currentMinSln  currentMinSln
    | false -> candidateSln

Running this code in fsi shows its type to be:

val findMin : 'a * 'b -> 'a * 'b -> 'a * 'b when 'a : comparison

The F# compiler can infer that the function takes in two tuples, each with two elements. The tuples must be of the same type and the first element of each tuple must be comparable. Way to go, type inference!

Everything has been set up at this point. We just need to run the code:

let minSln = Seq.fold findMin firstCandidate remainingCandidates

The fold function is to reduce a sequence to a single value, starting with a given accumulator. In this case, the first candidate solution that we created earlier is our starting point. The output of the function for each candidate input is compared to that accumulator.

Running the code finds the same answer (1) that we found with calculus.

This code runs pretty quickly, but this approach is generally slow. Our range is only 10 wide and the step is 0.01, so there are only 1000 candidate inputs. Increasing the size of the range or decreasing the step would increase the number of inputs. More worryingly, the size of the search space increases exponentially as the number of inputs to the function increases. So a function that took two inputs for the same range for each input would have a million candidate inputs, three inputs would take a billion and so on.

However, it is also quite straightforward to parallelise this approach. The sequence of candidate solutions could be split up into partitions and each partition could be sent off to a different computer. Each batch would find a local minimum for the range it was given. Finding the minimum for the whole of our range of inputs is simply a matter of find the minimum in the returned list.

The complete code for this can be found here:

https://github.com/robert-impey/CodingExperiments/blob/master/F%23/Loose/MinimiseQuadratic.fs

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