On Parametric Linear Optimization

Abstract

Consider the following parametric linear optimization problem: (PLO): Minimize

$$p(x): = \sum\limits_{v = 1}^n {p_v x_v }$$

subject to

$$\sum\limits_{v = 1}^n {a_{\mu v} (t)x_v \leqq b_\mu (t)} ,\mu = 1,2,....,m$$

where p₁p₂,…,p_n are given constants and a_με (t), b_μ (t) are given real valued functions depending continuously on the parameter t in T, T a metric space. Then for each t ∈ T we consider the set of feasible points

$$Z_t : = \left\{ {x \in \mathbb{R}^n \left| {\mathop \forall \limits_{\mu = 1}^m \sum\limits_{v = 1}^n {a_{\mu v} \left( t \right)x_v \underline \leqslant b_\mu \left( t \right)} } \right.} \right\},$$

the set of optimal solutions

$$p_t : = \left\{ {x_o \in Z_t \left| {p(x_o )} \right. = \mathop {\inf }\limits_{x \in Z_t } p(x)} \right\}$$

and the minimum value

$$E_t : = \mathop {\inf }\limits_{x \in Z_t } p(x)$$

Obviously the sets Z_t and P_t and the real number E_t depend on the parameter t.