Lexicographic maximum of convex polyhedral set (Q2759408)

This paper deals with conditions of the existence of lexicographic maximum of convex polyhedral set in \(\mathbb{R}^{n}\). Let \(X=\{x\in \mathbb{R}^{n}\mid Ax=b\); \(x_{j}\geq 0\), \(j\in J\); \(x_{j}\leq\nu_{j}\), \(j\in I\}\), where \(A\) is a \(m\times n\)-matrix; \(\text{ rank} A<n\); \(J,I\subset \{1,2,\ldots,n\}\). The authors prove, in particular, that a non-empty set \(X\) has no lexicographic maximum if and only if it contains two points \(x\) and \(y\) with coordinates satisfying inequalities \(x\mathop{<}^{L} y\), \(x_{j}\leq y_{j}\), \(j\in J\), \(x_{j}\geq y_{j}\), \(j\in I\), where symbol \(\mathop{<}^{L}\) means lexicographic less. The non-empty set \(X\) defined above has a lexicographic maximum if and only if zero point is a lexicographic maximum of set Let \(X_0=\{x\in \mathbb{R}^{n}\mid Ax=0\); \(x_{j}\geq 0\), \(j\in J\); \(x_{j}\leq 0\), \(j\in I\}\).