Basis dictionaries and reduced costs for revised simplex

Use the standard-form problem of Revised simplex standard-form linear program. Assume $A$ has full row rank $m$. An ordered basis is a list $B=(B_1,\dots ,B_m)$ of distinct column indices for which the square matrix $A_B=[a_{B_1}\cdots a_{B_m}]$ is invertible. Write $N=\{1,\dots ,n\}\setminus \{B_1,\dots ,B_m\}$ and $c_B=(c_{B_1},\dots ,c_{B_m}{)}^{\mathsf{T}}$. The basic vector $x(B)$ is defined by $x_B(B)=A_B^{-1}b$ and $x_j(B)=0$ for $j\in N$; the basis is primal feasible, and $x(B)$ is a basic feasible solution, exactly when $A_B^{-1}b\ge 0$.

The basis multiplier is $y(B)=A_B^{-\mathsf{T}}{c}_B$. For every column $j$, its reduced cost is ${\bar{c}}_j=c_j-a_j^{\mathsf{T}}y(B)$. In particular, ${\bar{c}}_{B_i}=0$ for each basic index. The basis is dual feasible when ${\bar{c}}_j\ge 0$ for every $j\in N$. When $m=0$, take the empty basis and the empty multiplier; these formulas remain meaningful.

A primal-feasible basis is degenerate if some basic coordinate is zero. A simplex pivot is degenerate if its ratio-test step length is zero.