 # Restricted basis entry rule

This is a restriction on which variables can enter the basis in a simplex method. A common rule arises in separable programming, which uses specially ordered sets: a group of non-negative variables must sum to 1 such that at most two variables are positive, and if two are positive, they must be adjacent. For example, suppose the variables are $LaTeX: (x_1,x_2,x_3).$ Then, it is feasible to have $LaTeX: (.5,.5,0)$ and $LaTeX: (0,.2,.8),$ but it is not feasible to have $LaTeX: (.5,0,.5)$ or $LaTeX: (.2,.2,.6).$ In this case the rule is not to permit a variable to enter the basis unless it can do so without violating the adjacency requirement. For example, if $LaTeX: x_1$ is currently basic, $LaTeX: x_3$ would not be considered for entry.
Another restricted entry rule pertains to the delta form of separable programming (plus other applications): Do not admit a variable into the basis unless its predecessor variables are at their upper bound. This means there is an ordered set of bounded variables, say $LaTeX: (x_1,x_2,\dots,x_n)$ such that $LaTeX: \textstyle 0 \le x \le (U_1,\dots,U_n).$ Then, $LaTeX: x_k$ is not considered for basis entry unless $LaTeX: x_j=U_j$ for all $LaTeX: j < k.$