A Realization of the Simplex Method Based on Triangular Decompositions

Consider the following problem of linear programming $$Minimize\;{c_0} + {c_{ - m}}{x_{ - m}} + ... + {c_{ - 1}}{x_{ - 1}} + {c_1}{x_1} + ... + {c_n}{x_n}$$ (1.1.1a) subject to $${x_{ - 1}} + \sum\limits_{k = 1}^n {a{a_{ik}}{x_k} = {b_i},\;i = 1,2,...m} $$ (1.1.1b) $${x_i}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \geqslant } 0\;for\;i \in {I^ + },\;{x_i} = 0\;for\;i \in {I^0}$$ (1.1.1c) where I + , I 0, I ± are disjoint index sets with $${I^ + } \cup {I^0} = N: = \{ i| - m\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } i\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } - 1,\;1\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } i\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } n\} ]$$ .