How can I linearise this nonlinear proportional relation constraint?

Question

My optimisation problem has a constraint in the form

\begin{equation} \begin{array}{*{35}{l}} \text{}\hspace{16.5mm}\text{ C4:} \hspace{2mm}\sum_{u=1}^U d_{u,1}L_{u}:\sum_{u=1}^U d_{u,2}L_{u}:\cdots:\sum_{u=1}^U d_{u,C}L_{u}=\psi_1:\psi_2:\cdots:\psi_C \end{array} \end{equation}

This nonlinear and makes the problem even more complex.

Here $L_{u}$ and $\psi_{c}$ are known parameters.

I will be satisfied with approximate proportional relation, e..g, C4 can be

$\text{ C4:} \hspace{2mm}\sum_{u=1}^U d_{u,1}L_{u}:\sum_{u=1}^U d_{u,2}L_{u}:\cdots:\sum_{u=1}^U d_{u,C}L_{u}\approx\psi_1:\psi_2:\cdots:\psi_C$

How can I deal with this nonlinear constraint?

Answer 1

Based on the comments (thanks @RobPratt), C4 looks like $$\frac{\sum_{u=1}^U d_{u,1}L_{u}}{\sum_{u=1}^U d_{u,2}L_{u}} = \frac{\psi_1}{\psi_2}$$ with similar constraints for other ratios.

Just multiply both sides by the denominator, which males it a linear constraint. $$\sum_{u=1}^U d_{u,1}L_{u} = \frac{\psi_1}{\psi_2}\sum_{u=1}^U d_{u,2}L_{u}$$ Similarly with the other constraints in C4.

This is then a MILP.

If you only require the proportions to be approximately satisfied, one way of doing that is to add a term $a_i$ (declared as an optimization variable) on the right-hand side of the ith of these constraints, and add the inequality constraints $-f \le a_i \le f$ where $f$ is the allowed fudge (tolerance) factor for constraint satisfaction. This is still a MILP.

You could instead make the tolerance a multiplicative factor of the ratios of the $\psi 's$. This introduces binary, continuous product terms, which can be linearized using standard techniques well-covered in this forum.

Or change the equality constraints to double-sided inequality constraints building in the allowable tolerance, rather than introducing new variables. This can be done with either the additive or multiplicative tolerance.