steps Input: A, b and $\stackrel{˜}{c}$ Output: either V as the optimal point & VTb as optimal value or declare the LP problem as infeasible. 0 $V={V}_{0}=\left(0,0,\cdots ,0\right)$ % the coordinate cone; ${r}_{1}=\left(1,0,\cdots ,0\right),{r}_{2}=\left(0,1,\cdots ,0\right),\cdots ,{r}_{m}=\left(0,0,\cdots ,1\right)$ ; 1 while (true) 2 % For all facets not in C & reject V, find one that rejects V the least. $\overline{\Delta }=\left\{1:\left(m+n\right)\right\}\\Delta$ ; % $\overline{\Delta }$ are facets not in C ${j}^{*}=\mathrm{arg}{\mathrm{min}}_{j}\left\{{e}_{j}|{e}_{j}=\left({V}^{\text{T}}{\tau }_{\overline{\Delta }\left(j\right)}-{c}_{\overline{\Delta }\left(j\right)}\right),j\in \overline{\Delta }\right\}$ 3 if ${e}_{{j}^{*}}\ge 0$ return (V & VTb) as optimal vertex & optimal value else ${J}^{*}=\overline{\Delta }\left({j}^{*}\right)$ ; % j* is the facet index to enter 4 ${t}_{i}=\frac{{c}_{{J}^{*}}{V}^{\text{T}}{\tau }_{{J}^{*}}}{{r}_{i}^{\text{T}}{\tau }_{{J}^{*}}};i=1,\cdots ,m$ ; $i=1,\cdots ,m$ }; ${q}_{i}=V+{t}_{i}{r}_{i}$ ; 5 if $\left({t}_{i}<0\text{\hspace{0.17em}}\forall i=1,\cdots ,m\right)$ return(“LP is infeasible”) else % For all the real cuts qi,, find the qI* that is closest to V ${I}^{*}=\mathrm{arg}{\mathrm{min}}_{i}\left\{{q}_{i}^{\text{T}}b|{t}_{i}>0\right\}$ % I* is the facet index to leave 6 % Form new cone Ck+1 by updating Vk+1; edge vectors and facets ${V}_{k+1}={q}_{{I}^{*}}$ ; ${r}_{i}=\left\{\begin{array}{ll}{r}_{i}\hfill & i={I}^{*}\hfill \\ sign\left({t}_{i}\right)\left[{q}_{i}-{V}_{k+1}\right]\hfill & i\ne {I}^{*}\hfill \end{array}$ $\text{Δ}\left({I}^{*}\right)={J}^{*}$ ; 7 end