What are non-zeros?

Non-zeros refer to the elements in the coefficient matrix of an optimization problem that are not equal to zero. In sparse matrices, most elements are typically zero, and only the non-zero elements carry meaningful information. The non-zeros field indicates the count of these non-zero values in the matrix, which is important for understanding the structure and complexity of the problem. This value is reported and can be used to calculate the density of the matrix which may affect the efficiency of solving the optimization problem.

As the density of the matrix increases, the solving efficiency often decreases. Sparse problems are usually easier to solve due to the reduced computational complexity, while dense problems require more resources, slowing down the solver. However, the correlation between density and efficiency can vary depending on the specific problem type.

In most cases, a problem is considered sparse if it has 1% to 5% non-zeros, though this can go up to around 10% in some contexts. You can calculate the percentage of non-zeros in the coefficient matrix by dividing the number of non-zeros by the product of the number of decision variables and the number of constraints.

For example, in the following problem snippet, there are 5 non-zero coefficients, 3 decision variables, and 2 constraints. The density of this problem would be:

$$\frac{5}{3 \times 2} \approx 0.83$$

Problem snippet:

$$3x + 5y + 6z \le 80$$

$$0x + 9y + 8x \le 95$$