Abstract Versus Concrete Models

A mathematical model can be defined using symbols that represent data values. For example, the following equations represent a linear program (LP) to find optimal values for the vector \(x\) with parameters \(n\) and \(b\), and parameter vectors \(a\) and \(c\):

\begin{array}{lll} \min & \sum_{j=1}^n c_j x_j &\\ \mathrm{s.t.} & \sum_{j=1}^n a_{ij} x_j \geq b_i & \forall i = 1 \ldots m\\ & x_j \geq 0 & \forall j = 1 \ldots n \end{array}


As a convenience, we use the symbol \(\forall\) to mean “for all” or “for each.”

We call this an abstract or symbolic mathematical model since it relies on unspecified parameter values. Data values can be used to specify a model instance. The AbstractModel class provides a context for defining and initializing abstract optimization models in Pyomo when the data values will be supplied at the time a solution is to be obtained.

In many contexts, a mathematical model can and should be directly defined with the data values supplied at the time of the model definition. We call these concrete mathematical models. For example, the following LP model is a concrete instance of the previous abstract model:

\begin{array}{ll} \min & 2 x_1 + 3 x_2\\ \mathrm{s.t.} & 3 x_1 + 4 x_2 \geq 1\\ & x_1, x_2 \geq 0 \end{array}

The ConcreteModel class is used to define concrete optimization models in Pyomo.


Python programmers will probably prefer to write concrete models, while users of some other algebraic modeling languages may tend to prefer to write abstract models. The choice is largely a matter of taste; some applications may be a little more straightforward using one or the other.