Piecewise-linear Functions in Math Programs

How to handle a piecewise-linear function of one variable in an optimization model depends to a large extent on whether it represents an economy of scale, a diseconomy of scale, or neither. The concepts apply generally, but in this post I'll stick to cost functions (where smaller is better, unless you are the federal government) for examples; application to things like productivity measures (where more is better) requires some obvious and straightforward adjustments. Also, I'll stick to continuous functions today.

Let's start with a diseconomy of scale. On the cost side, this is something that grows progressively more expensive at the margin as the argument increases. Wage-based labor is a good example (where, say, time-and-a-half overtime kicks in beyond some point, and double-time beyond some other point). A general representation for such a function is \[f(x)=a_{i}+b_{i}x\mbox{ for }x\in[d_{i-1},d_{i}],\: i\in\{1,\ldots,N\}\] where $d_{0}=-\infty$ and $d_{N}=+\infty$ are allowed and where the following conditions must be met: \begin{eqnarray*}d_{i-1} & < & d_{i}\:\forall i\in\{1,\ldots,N\}\\b_{i-1} & < & b_{i}\:\forall i\in\{2,\ldots,N\}\\a_{i-1}+b_{i-1}d_{i-1} & = & a_{i}+b_{i}d_{i-1}\:\forall i\in\{2,\ldots,N\}.\end{eqnarray*}The first condition just enforces consistent indexing of the intervals, the second condition enforces the diseconomy of scale, and the last condition enforces continuity. The diseconomy of scale ensures that $f$ is a convex function, as illustrated in the following graph.

Letting $f_{i}(x)=a_{i}+b_{i}x$, we can see from the graph that $f$ is just the upper envelope of the set of functions $\{f_{1},\ldots,f_{N}\}$ (easily verified algebraically). To incorporate $f(x)$ in a mathematical program, we introduce a new variable $z$ to act as a surrogate for $f(x)$, and add the constraints\[z\ge a_{i}+b_{i}x\;\forall i\in\{1,\ldots,N\}\] to the model. Anywhere we would use $f(x)$, we use $z$ instead. Typically $f(x)$ is either directly minimized in the objective or indirectly minimized (due to its influence on other objective terms),
and so the optimal solution will contain the minimum possible value of $z$ consistent with the added constraints. In other words, one of the constraints defining $z$ will be binding. On rare occasions this will not hold (for instance, cost is relevant only in satisfying a budget, and the budget limit is quite liberal). In such circumstances, the optimal solution may have $z>f(x)$, but that will not cause an incorrect choice of $x$, and the value of $z$ can be manually adjusted
to $f(x)$ after the fact.

Now let's consider an economy of scale, such as might occur when $x$ is a quantity of materials purchases, $f(x)$ is the total purchase cost, and the vendor offers discounts for larger purchases. The marginal rate of change of $f$ is now nonincreasing as a function of $x$, and as a result $f$ is concave, as in the next graph.

Now $f$ is the lower envelope of $\{f_{1},\ldots,f_{N}\}$. We define the segments as before, save with the middle condition reversed to $b_{i-1}>b_{i}\:\forall i\in\{2,\dots,N\}$. We cannot simply add constraints of the form \[z\le a_{i}+b_{i}x\] if we are (directly or indirectly) minimizing $z$, since $z=-\infty$ would be optimal (or, if we asserted a finite lower bound for $z$, the lower bound would automatically be optimal). In the presence of an economy of scale, we are forced to add binary variables to the formulation, either directly or indirectly. The easiest way to do this is to use a type 2 specially ordered set (SOS2) constraint, provided that the solver (and any modeling language/system) understand SOS2, and provided that $d_{0}$ and $d_{N}$ are finite. Introduce new (continuous) variables $\lambda_{0},\ldots,\lambda_{N}\ge0$ along with $z$, and introduce the following  constraints:\begin{eqnarray*}\lambda_{0}+\cdots+\lambda_{N} & = & 1 \\ \lambda_{0}d_{0}+\cdots+\lambda_{N}d_{N} & = & x\\ \lambda_{0}f(d_{0})+\cdots+\lambda_{N}f(d_{N}) & = & z.\end{eqnarray*} Also communicate to the solver that $\{\lambda_{0},\ldots,\lambda_{N}\}$ is an SOS2 set. The SOS2 constraint says that at most two of the $\lambda$ variables can be nonzero, and that if two are nonzero they must be consecutive. Suppose that $\lambda_{i}$ and $\lambda_{i+1}$ are positive and all others are zero. Then the new constraints express $x$ as a convex combination of $d_{i}$ and $d_{i+1}$ (with weights $\lambda_{i}$ and $\lambda_{i+1}=1-\lambda_{i}$), and express $z=f(x)$ as a convex combination of $f(d_{i})$ and $f(d_{i+1}$) using the
same weights.

If SOS2 constraints are not an option, then binary variables have to be added explicitly. Let $\delta_{i}\in\{0,1\}$ designate whether $x$ belongs to the interval $[d_{i-1},d_{i}]$ ($\delta_{i}=1$) or not ($\delta_{i}=0$). Add the following constraints:\begin{eqnarray*}\delta_{1}+\cdots+\delta_{N} & = & 1\\ x & \le & d_{i}+M(1-\delta_{i})\:\forall i\in\{1,\ldots,N\}\\ x & \ge & d_{i-1}-M(1-\delta_{i})\:\forall i\in\{1,\ldots,N\}\\ z & \ge & a_{i}+b_{i}x-M(1-\delta_{i})\:\forall i\in\{1,\ldots,N\}\end{eqnarray*} where $M$ is a suitably large constant. (This formulation will again require that $d_{0}$ and $d_{N}$ be finite.) The first constraint says that $x$ must belong to exactly one interval. If $\delta_{j}=1$, the second and third constraints force $d_{j-1}\le x\le d_{j}$ and are vacuous for all $i\neq j$ (assuming $M$ is correctly chosen). The final constraint forces $z\ge f_{j}(x)$ and is again vacuous for all $i\neq j$. Since $z$ represents something to be minimized, as before we expect $z=f_{j}(x)$ in almost all cases, and can manually adjust it in the few instances were the optimal solution contains padding.