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Piecewise

A Piecewise is a collection of Pieces, which each define a valid range for a variable \(v\)

\[ \label{eq:active_piece} \beta_\text{k} = \lambda_\text{0, k} + \lambda_\text{1, k} \]
\[ \label{eq:piece} v_\text{k} = \lambda_\text{0, k} * \text{v}_{\text{start,k}} + \lambda_\text{1,k} * \text{v}_{\text{end,k}} \]
\[ \label{eq:piecewise_in_pieces} \sum_{k=1}^k \beta_{k} = 1 \]

With:

  • \(v\): The variable to be defined by the Piecewise
  • \(\text{v}_{\text{start,k}}\): the start point of the piece for variable \(v\)
  • \(\text{v}_{\text{end,k}}\): the end point of the piece for variable \(v\)
  • \(\beta_\text{k} \in \{0, 1\}\): defining wether the Piece \(k\) is active
  • \(\lambda_\text{0,k} \in [0, 1]\): A variable defining the fraction of \(\text{v}_{\text{start,k}}\) that is active
  • \(\lambda_\text{1,k} \in [0, 1]\): A variable defining the fraction of \(\text{v}_{\text{end,k}}\) that is active

Which can also be described as \(v \in 0 \cup [\text{v}_\text{start}, \text{v}_\text{end}]\).

Instead of \eqref{eq:piecewise_in_pieces}, the following constraint is used to also allow all variables to be zero:

\[ \label{eq:piecewise_in_pieces_zero} \sum_{k=1}^k \beta_{k} = \beta_\text{zero} \]

With:

  • \(\beta_\text{zero} \in \{0, 1\}\).

Which can also be described as \(v \in \{0\} \cup [\text{v}_{\text{start_k}}, \text{v}_{\text{end_k}}]\)

Combining multiple Piecewises

Piecewise allows representing non-linear relationships. This is a powerful technique in linear optimization to model non-linear behaviors while maintaining the problem's linearity.

Therefore, each Piecewise must have the same number of Pieces \(k\).

The variables described in Piecewise are created for each Piece, but nor for each Piecewise. Rather, \eqref{eq:piece} is the only constraint that is created for each Piecewise, using the start and endpoints \(\text{v}_{\text{start,k}}\) and \(\text{v}_{\text{end,k}}\) of each Piece for the corresponding variable \(v\)