# 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.

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.Note

## 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.

: All Python code is copyrighted by Gurobi Optimization, LLCDisclaimer

### 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.

### Adding Constraints

#### 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 =
m.addConstrs(
(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 =
Model.addConstrs m "capacity" arcs
(fun [i; j] -> (sum flow ["*"; i; j] <== capacity.[[i; j]]))
```

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.