# Variables are not values: types and expressions in mathematical optimization

This week, I came across Richard Oberdieck’s post, “Why ‘evaluate’ is the feature I am missing the most from commercial MIP solvers”. It would indeed be practical to have for the reasons listed by the author, but some barriers stand to have it as it is expressed in the snippets presented.

# Initial problem statement

The author first tests the optimization of a non-linear function through scipy as such:

func = lambda x: np.cos(14.5 * x - 0.3) + (x + 0.2) * x
func(5) # 25.001603108415402

So far so good, we are defining a scalar function, passing it a scalar value at which it evaluates and returns the value, which is what it is supposed to do.

Now the real gripe comes when moving on to developing against a black box solver (often commercial, closed-source), commonly used for linear, mixed-integer problems:

import xpress as xp

# Define the model and variables
Model = xp.problem()
x = xp.var(lb=0, ub=10)

# Define the objective and solve
test_objective = 5*x
Model.setObjective(test_objective)
Model.solve()
# test_objective(5) does not work

One first problem to notice here is that test_objective is at best an expression, not a function, meaning it does not depend on an input argument but on decision variables declared globally. That is one point why it cannot be called.

Now, the rest of this article will be some thoughts on how optimization problems could be structured and represented in a programming language.

One hack that could be used is being able to set the values of x, but this needs to be done at the global level:

x = xp.var(lb=0, ub=10)

# Define the objective
test_objective = 5*x

x.set(5)
# evaluates test_objective with the set value of x
xp.evaluale(test_objective)

Having to use the global scope, with an action on one object (the variable x) modifying another (the test_objective expression) is called a side-effect and quickly makes things confusing as your program grows in complexity. You have to contain the state in some way and keep track. Keeping track of value changes is more or less fine, but the hardest part is keeping track of value definitions. Consider the following example:

x = xp.var(lb=0, ub=10)
y = xp.var(lb=0, ub=10)

# Define the objective and solve
test_objective = 5*x + 2*y
xp.evaluale(test_objective) # no variable set, what should this return?

x.set(5)
xp.evaluale(test_objective) # y is not set, what should this return?

# A terminology problem

We are touching a more fundamental problem here, variables are not values and cannot be considered as such. Merging the term “variable” for variables of your Python/Julia/other program with the decision variables from an optimization problem creates a great confusion. Just like variables, the term function is confusing here: most optimization techniques exploit the problem structure, think linear, disciplined convex, semi-definite; anything beyond non-linear differentiable or black-box optimization will use the specific structure in a specialized algorithm. If standard functions from your programming language are used, no structure can be leveraged by the solver, which only sees a function pointer it can pass values to. So working with mathematical optimization forces you to re-think what you call “variables” and what you call “functions”.

There is something we can do for the function part, which is defining arithmetic rules over variables and expressions, which is for instance what the JuMP modelling framework does:

using JuMP
m = Model()
@variable(m, x1 >= 0)
@variable(m, x2 >= 0)

# random affine function
f(a, b) = π + 3a + 2b

f(x1, x2) # returns a JuMP.GenericAffExpr{Float64,VariableRef}

@variable(m, y)
f(x1, x2) + y  # also builds a JuMP.GenericAffExpr{Float64,VariableRef}

This works especially well with affine functions because composing affine expressions builds other affine expressions but gets more complex any time other types of constraints are added. For some great resource on types and functions for mathematical optimization, watch Prof. Madeleine Udell’s talk at JuliaCon17 (the Julia syntax is from a pre-1.0 version, it may look funny).

# Encoding possibilities as sum-types

Getting back to evaluation, to make this work, you need to know what values variables hold. What if the model hasn’t been optimized yet? You could take: 1. A numerical approach and return NaN (floating point value for Not-A-Number) 2. An imperative approach and throw an error when we evaluate an expression without values set or the model optimized 3. A typed functional approach and describe the possibility of presence/absence of a value through types

The first approach was JuMP 0.18 and prior, the second is JuMP 0.19 and onward, the third is the one of interest to us, if we want to describe what is happening through types.

If you show these three options to a developer used to statically-typed functional programming, they would tell you that the first option coming to mind is an option, a type which can be either some value or nothing. In the case of an optimization model, it would be some numerical value if we have a value to return (that is, we optimized the model and found a solution). The problem is, there are many reasons for which you may have or not a value. What you could do in that case is get more advanced information from your model. This is the approach JuMP is taking with a bunch of model attributes you can query at any time, see the documentation for things you can query at any time.

The problem is that querying information on the status of the problem (solved, unsolved, impossible to solve…) and getting values attached to variables can be unrelated.

m = Model()
@variable(m, x >= 0)
@variable(m, y >= 0)

# getting status: nothing because not optimized
termination_status(m)
# OPTIMIZE_NOT_CALLED::TerminationStatusCode = 0

primal_status(m) # NO_SOLUTION::ResultStatusCode = 0

JuMP.value(x) # ERROR: NoOptimizer()
# woops, we forgot that we hadn't optimized yet

This is indeed because x does not exist by itself, there is a “magic bridge” between the variable x and the model m. The computer science term for this “magic bridge” is a side-effect, the same kind as mentioned earlier when we set the value of a variable at the global scope. Again, they are fine at a small scale but are often the parts making a program confusing. Every time I’m reviewing some code by researchers starting out, the first thing I encourage them to do is to create self-contained bits of code within functions and remove mutable global state.

# A typed solution for describing mathematical problems

We stated that the variables and model are bound together. In that case, let us not split them but describe them as one thing and since this one thing accepts different possible states, we will use tagged unions, which you can think of as C enumerations with associated values. Other synonyms for this construct are sum types (as in OCaml and Haskell).

We can think of the solution process of an optimization problem at a high level as a function:

solve(Model(Variables, Constraints, Objective)) -> OptimizationResult


Where OptimizationResult is a sum type:

OptimizationResult = Infeasible(info) | Unbounded(info) | Optimal(info) | NearOptimal(info) ...


In this case, everything can stay immutable, expressions including objective and constraints are only used to build the model in input, they can be evaluated at any points and just describe some expressions of variables. The value of the variables resulting from the optimization are on available in cases where it makes sense. If the results are stored in the solution info structure, we can query values where it makes sense only, here in the Optimal and NearOptimal cases, with a syntax like:

match OptimizationResult {
Optimal(info) -> value(info, x) # or info.value(x)
Infeasible(info) -> ...
Unbounded(info)  -> ...
}


Internally, info would keep an association from variables to corresponding values. No more confusion on what binding of your computer program represents what symbolic variable of your problem.

So why would we keep using these bindings associated with variables, if they have never been independent from the problem in the first place? The obvious reason that comes to mind is practical syntax, we can write expressions in a quasi-mathematical way (here in JuMP):

@expression(m, 3x + 2x^2 <= 4y)

While if variables were attached to the model, the required syntax would be in the flavour of:

@expression(m, 3m[:x] + 2m[:x]^2 <= 4m[:y])

Which quickly becomes hard to read. Can we do better?

# Stealing a solution elsewhere

I stumbled upon an interesting solution to such problem while reading the documentation for various probabilistic programming languages built on top of Julia. Here is one example from Turing.jl

@model gdemo(x, y) = begin
s ~ InverseGamma(2,3)
m ~ Normal(0,sqrt(s))
x ~ Normal(m, sqrt(s))
y ~ Normal(m, sqrt(s))
end

# sample from the model using an algorithm
chn = sample(gdemo(1.5, 2), HMC(1000, 0.1, 5))

It’s just one step away from imagining the same for optimization:

@optim_model linmodel(a, b) = begin
x[1:10] >= 0
5 <= y <= 10
z ∈ 𝔹
cons1: y - 5 <= 5z
cons2: x + y >= 3
Min x
end

result = optimize(linmodel)

Naming the constraints would be necessary to retrieve associated dual values. Retrieving values associated with variables could be done in an associative structure (think a dictionary/hash map). This structure removes any confusion as to what belongs where in an optimization model. The variables x, y, z are indeed defined within a given model and explicitly belong to it.

Why are interfaces not built this way? Warning, speculative opinions below:

One reason is the ubiquity of C & C++ in optimization. The vast majority of commonly used solvers is built in either of these, supporting limited programming constructs and based on passing pointers around to change the values pointed to. Because the solvers are built like this, interfaces follow the same constructions. Once a dominant number of interfaces are identical, building something widely different is a disadvantage with a steeper learning curve.

Another more nuanced reason is that declarative software is hard to get right. One often has to build everything upfront, here in the @optim_model block. Getting meaningful errors is much harder, and debugging optimization models is already a tricky business.

Lastly, lots of algorithms are based on incremental modifications of models (think column and row generation), or combinations with other bricks. This requires some “hackability” of the model. If one looks at Algebraic Modelling Languages, everything seems to fall apart once you try to implement decompositions. Usually it involves a completely different syntax for the decomposition scheme (the imperative part) and for the model declaration (the declarative part).

So overall, even though side-effects are a central part of the barrier to the expression of mathematical optimization in a mathematical, type-based declarative way, they are needed because of the legacy of solvers and some algorithms which become hairy to express without it.

# Further resources

As pointed above, Prof. Madeleine Udell’s talk gives some great perspectives on leveraging types for expressive optimization modelling. For the brave and avid readers, this PhD thesis tackles the semantics of a formal language for optimization problems. If you have further resources on the subject, please reach out.

Thanks Richard for the initial post and the following discussion which led to this post. For shorter and nicely written posts on optimization, go read his blog.

Note: I try never to use the terms “mathematical programming” and “mathematical program” which are respectively synonyms for “mathematical optimization” and “mathematical optimization problem” respectively. We can see why in this post: this kind of context where the term “program” could refer to a computer program or a mathematical problem becomes very confusing. We are in 2019 and the term “program” is now universally understood as a computer program. Moreover, “mathematical programming” merely refers to a problem specification, it is very confusing to say that “linear/semi-definite/convex programming” is merely meant as putting together a bunch of equations, not at all about how to tackle these.

##### Mathieu Besançon
###### PhD candidate in mathematical optimization

My research interests include bilevel optimization, demand response and pricing.