Pivot element tolerance

Default value: 1e-6

Minimum value: 0
Maximum value: 1

rosepy parameter name:tol_element

See the linked article for information on How to change and view the solver parameters.

Recommendations

A larger tolerance value is generally better for avoiding degeneracy and ensuring numerical stability, which is especially important if  the problem is improperly scaled, meaning the values in the tableau have a large range of magnitude. But larger tolerance values may lead to slower convergence and potentially less accurate solutions.

That said, for most applications, the default settings provided by Rose strike a good balance between these factors, so there is generally no need to adjust this parameter.

Technical details

Pivot element tolerance refers to a threshold used to determine whether a candidate pivot element (the element in the matrix that is chosen to enter or leave the basis) is sufficiently non-zero to proceed with a pivot operation in the Simplex method.

When performing the pivot operation in the Simplex method, the algorithm involves selecting a non-zero element from the tableau (the matrix representation of the linear program) to serve as a pivot. However, due to numerical precision and rounding errors, an element that should theoretically be non-zero might be very close to zero in the computed tableau. If this near-zero element is used as a pivot, it can lead to numerical instability or incorrect results.

Mathematically, if a candidate pivot element \(p\) in the tableau satisfies:

$$|p| \gt \epsilon$$

then \(p\) is considered a valid pivot element. If \(|p| \le \epsilon\), the element is treated as effectively zero, and the algorithm may rejects it as a pivot to prevent numerical issues.

In practical terms, this tolerance helps ensure the stability and accuracy of the Simplex algorithm by preventing the selection of elements that are too small to be reliable pivot choices, thus reducing the risk of introducing significant numerical errors into the solution process.