Two-step segmentation rule

input: The number of decision variables (n), the number of constraints (m), the coefficient matrix of the problem (A), the vector of the right-hand side coefficients (b), the vector of the objective coefficients (c).

Output:

(Initialization) Compute:

$\cos {\theta }_i$ , $i=1,2,\cdots ,m$ , using (4)

$Q_{1c}$ , the first quartile of the values of $\cos {\theta }_i$ , $i=1,2,\cdots ,m$

Step 1:

Remove the constraints that: $\cos {\theta }_i\le Q_{1c}$ , $i=1,2,\cdots ,m$

Step 2: for the new problem, compute

${\lambda }_i$ , $i=1,2,\cdots ,k-m$ , using (3)

$1/{\lambda }_i^2$ , for $i=1,2,\cdots ,k-m$

$\cos {{\theta }^{\prime }}_i$ , $i=1,2,\cdots ,m$ , using (4)

${Q^{\prime }}_{1c}$ , the first quartile of the values of $\cos {{\theta }^{\prime }}_i$ , $i=1,2,\cdots ,k-m$

${Q^{\prime }}_{1\lambda }$ , the first quartile of the values of $1/{\lambda }_i^2$ , $i=1,2,\cdots ,k-m$

Remove the constraints that: $\cos {{\theta }^{\prime }}_i\le {Q^{\prime }}_{1c}$ & $1/{\lambda }_i^2\le {Q^{\prime }}_{1\lambda }$ , $i=1,2,\cdots ,k-m$