A Realization of the Simplex Method Based on Triangular Decompositions

Abstract

Consider the following problem of linear programming

$${\text{Minimize }}{c_{\text{0}}} + {c_{ - m}} {x_{ - m}} + \cdots + {c_{ - 1}}{x_{ - 1}} + {c_1}{x_1} + \cdots + {c_n}{x_n}$$

((1.1.1a))

subject to

$${x_{ - i}} + \sum\limits_{k = 1}^n {a{a_{ik}}{x_k} = {b_i},{\text{ }}i{\text{ = 1,2,}} \ldots {\text{,}}m{\text{,}}} $$

((1.1.1b))

$$ {x_i} \geqq 0{\text{ for }}i \in {I^ + },{\text{ }}{x_i} = 0{\text{ for }}i \in {I^0}, $$

((1.1.1c))

where I ⁺, I ⁰, I ^± are disjoint index sets with

$${I^ + } \cup {I^0} \cup {I^ \pm } = N: = \{ i| - m \leqq i \leqq - 1,1 \leqq i \leqq n\} .$$

The variable x _i is called a nonnegative (zero, free) variable if i∈I ⁺ (i∈I ⁰, i∈I)^±.