Properties of the lexicographic maximum of the set of solutions of a system of linear inequalities (Q2730527)

The paper deals with the system of linear inequalities \(\sum\limits_{j=1}^{n}a_{ij}x_{j}\leq b_{i},\;i=1,2,3\) under the conditions \(0\leq x_{j}\leq \beta_{j},\;j=1,2,\ldots,n\). Denote \(P^{k}=\{x\in \mathbb{R}^{k}\mid 0\leq x_{j}\leq\beta_{j}, j=1,2,\ldots,k\}\); \(X_{k}^{i}=\{x\in P^{k}\mid \sum\limits_{j=1}^{n}a_{ij}x_{j} \leq b_{i}\},\;i=1,2,3\); \(Y_{k}^1=X_{k}^{1}\cap X_{k}^{2}\), \(Y_{k}^2=X_{k}^{1}\cap X_{k}^{3}\), \(Y_{k}^3=X_{k}^{2}\cap X_{k}^{3}\), \(Y_{k}=X_{k}^{1}\cap X_{k}^{2}\cap X_{k}^{3}\); \(y^{*}=\max\limits^{L} Y_{k}\), \(1\leq k\leq n\). The main result is the following.NEWLINENEWLINENEWLINEIf \(Y_{n}\neq\emptyset\), \(y^1=\max\limits^{L} Y_{n}^1\), \(y^2=\max\limits^{L} Y_{n}^2\), \(y^3=\max\limits^{L} Y_{n}^3\), and \(y^1\mathop{\leq}^{L} y^2\mathop{\leq}^{L} y^3\), then the lexicographic maximal point of the set of solutions of the considered system \(y^{*}\) is equal to \(y^1\).