A Realization of the Simplex Method Based on Triangular Decompositions

Abstract

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

$${I^ + } \cup {I^0} = N: = \{ i| - m\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } i\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } - 1,\;1\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } i\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } n\} ]$$

.