# F# for Optimization Modeling

I recently attended a training event hosted by Gurobi. For those who don’t know, Gurobi produces one of the best mathematical solvers in the industry. It was a great event and we were able to spend ample time with engineers and experts in the field.

Using a mathematical solver requires the ability to formulate models and at this time one of the easiest languages for doing that is Python. Python is a great language for many use cases. One is providing a quick and easy means of formulating models that can then be fed to a solver. I was able to spend some time with one of the engineers who implemented Gurobi’s Python library, gurobipy. He pointed to the formulation of the netflow problem as an example of how terse and concise Python could be for modeling.

Since I love F#, I naturally wanted to see if I could accomplish the same thing using F#. What started as a silly proof of concept is slowly turning into a more full fledged library for wrapping the Gurobi .NET library in a functional F# wrapper. Below I give an example of how the power of functions in F# allows us to nearly duplicate the functionality of Python. The library I am working on can be found here.

Note I am not saying one language is better than another. I merely like to challenge myself with formulating ideas in different languages. It forces me to translate across paradigms which I find a useful exercise for the mind.

## Netflow Example

The following shows an example of a network flow problem provided by Gurobi and modeled in Python. The full formulation can be found here. In this example I am just comparing and contrasting the Python and F# constraint formulation methods.

Disclaimer: All Python code is copyrighted by Gurobi Optimization, LLC

### Creating a Model

#### Python

In Python the creation of the model and decision variables is quite straightforward.

# Copyright 2018, Gurobi Optimization, LLC

# Create optimization model
m = Model('netflow')

# Create variables
flow = m.addVars(commodities, arcs, obj=cost, name="flow")


#### With Gurobi.Fsharp

In F# we have a similar syntax but instead of flow being a Dictionary of decision variables indexed by tuples, we produce a Map<string list, GRBDecVar> which is essentially the same for our purposes. I am using string list as the index instead of tuples because we need an indexer which has dynamic length. I could do it with tuples but it would be less straightforward.

// Create a new instance of the Gurobi Environment object
// to host models
// create: GRBEnv
let env = Environment.create

// Create a new model with the environment variable
// create: env:GRBEnv -> name:string -> GRBModel
let m = Model.create env "netflow"

// Create a Map of decision variables for the model
// addVarsForMap: model:GRBModel -> lowerBound:float -> upperBound:float -> varType:string -> indexMap:Map<'a,float>
let flow = Model.addVarsForMap m 0.0 INF CONTINUOUS costs


Instead of using the methods on the object, functions have been provided which operate on the values that are passed in. This is more idiomatic for F#. The Model module in the library hosts all of the functions for working with objects of type Model.

The Model.adddVarsForMap function takes a Map<string list, float> and produces a Map<string list, GRBDecVar> for the modeler to work with. This is similar to how the Python tuples are working in the gurobipy library. Instead of indexing into a Python dictionary with tuples, F# uses a string list as the index.

#### Python

The gurobipy library offers a succinct way of expressing a whole set of constraints by using generators. There is additional magic going on under the hood though that may not be obvious at first. The following method generates a set of constraints for each element in arcs but also creates a meaningful constraint name. The prefix for the constraint name is the last argument of the method ("capacity" in this instance).

# Arc capacity constraints
capacityConstraints =
(flow.sum('*',i,j) <= capacity[i,j] for i,j in arcs), "capacity")


There is also special sauce occuring in the flow.sum('*',i,j) syntax. flow is a dictionary which is indexed by a 3 element tuple. What this sum() method is doing is summing across all elements in the dictionary which fit the pattern. The * symbol is a wildcard and will match against any element. This is a powerful way to sum across dimensions of the optimization model.

#### With Gurobi.Fsharp

In F# we can do something similar but instead of having a generator we pass in a lambda to create the constraints. The sinature of this function for creating the constraint set is: model->string->string list->(Map<string list, Gurobi.GRBConstr)

// addConstrs: model:GRBModel -> setName:string -> setIndexes: string list list -> constraintFunc:(string list -> ConstraintTuple) -> Map<string list, GRBConstr>
let capacityConstraints =

The function Model.addConstrs takes a model object as its first argument (m in this case), the prefix for what the constraints are going to be named ("capacity" in this case), and the set of indices the constraints will be created over, arcs in this case. The key point is that the types of the indices must match the input type of the lambda.
The addConstrs function will iterate through each of the indices in the set, create a constraint from the lambda that was passed, and name the constraint appropriatly. If the first element of the arcs set was ["Detroit"; "Boston"] then the name of the first constraint would be capacity_Detroit_Boston. This helps the modeler by maintaining a consistent naming scheme for the constraints in the model.