Optimality

Abstract

In this chapter we consider the LP-problem(P- or D-directed; P- and D-regular; ρ < 1). In that case the theorems 5.2 and 5.3 imply that P-feasible solutions (x,y) and D-feasible solutions (u,v) such that x_Π ∉ 1 ⁿ₁ , y_Π ∉ 1 ^m₁ and u_ρ ∉ 1 ^m₁ , v_ρ ∉ 1 ⁿ₁ , cannot be optimal. Therefore, we may restrict ourselves to feasible solutions ((x,y),(u,v)) for which

$${\rm x}_\Pi \, \in \,1^{\rm n} _{1 + } ,\,y_\Pi \, \in \,1^{\rm m} _{1 + } ,\,{\rm u}_{\rm \rho } \, \in \,1^{\rm m} _{1 + } ,\,{\rm v}_{\rm \rho } \, \in \,1^{\rm n} _{1 + } .$$