A Necessary Condition for Continuity in Parametric Linear Programming

Abstract

In a parametric linear optimization problem

$$ \eqalign{ & {\text{subject to}} \cr & {a_j}\left( t \right)x\mathop = \limits^ < {b_j}\left( t \right)forj = 1, \ldots ,m, \cr} $$

((1))

where $p \in {\mathbb{R}^{\text{n}}},{\text{ }}{{\text{a}}_{\text{j}}}:{\text{T }} \to {\text{ }}{\mathbb{R}^{\text{n}}},{{\text{b}}_{\text{j}}}:{\text{T }} \to \mathbb{R}$ and T a locally compact metric space, the set Z(t) of feasable points is a set-valued function in t ∊ T:

$$Z(t) = \{ x \in {R^n}|{a_j}(t)x \leqq {b_j}(t)\;,\;j = \;1,...,m\} \;.$$

(2)