Chapter 2.4 Generalized Ineuqaltities

2.4.1 Proper Cones and Generalized Inequalities

1. A cone \(K\subseteq \mathbb{R}^n\) is called a proper cone if it satisfies the following:
   • \(K\) is convex;
   • \(K\) is closed;
   • \(K\) is solid, which means it has nonempty interior;
   • \(K\) is pointed, which means it contains not lines. (or equivalently, \(x\in K\), \(-x\in K\), \(\rightarrow x=0\) ).
2. A proper cone \(K\) can be used to define a strict and a non-strict generalized inequality. I.e.,
   • non-strict: \(x\succeq_K y \Leftrightarrow y-x\in K\);
   • strict: \(x\succ_K y \Leftrightarrow y-x\in {\textbf{int}} K\). (interior).
3. Three proper cones provided in the textbook
   • The non-negative orthant \(K=\mathbb{R}_{+}^{n}\), with interior \(K=\mathbb{R}_{++}^{n}\);
   • The positive semidefinite cone \({\mathbf{S}}_{+}^{n}\), with interior the positive definite cone \({\mathbf{S}}_{++}^{n}\).
   • The coefficients of polynomials non-negative on [0,1]: \(\begin{aligned} K=\{c\in\mathbb{R}^n\vert c_1+c_2t+\ldots+c_nt^{n-1}\geq 0,{\text{ for } t\in[0,1]}\} \end{aligned}\),
     with interior: \(\begin{aligned} K=\{c\in\mathbb{R}^n\vert c_1+c_2t+\ldots+c_nt^{n-1}> 0,{\text{ for } t\in[0,1]}\} \end{aligned}\).
4. Please check some properties of the generalized inequality in the textbook.

2.4.2 Minimum and Minial Points

1. Minimum Point:
   • Official concept: \(x\in S\) is the minimum element of \(S\) (with respect to the general inequality \(\succeq_K\)), if for every \(y\in S\), we have \(x\succeq_K y\);
   • minimum point is unique;
   • Linfang understanding: The concept of minimum point requires all points in \(S\) is comparable to \(x\).
2. Minimal Points:
   • We say that \(x\in S\) is a minimal element of \(S\) (with respect to the general inequality \(\succeq_K\)), if \(y\succeq_K x\) only if \(y=x\).
   • A set can have many minimal points;
   • Linfang understanding: The concept of minimal point DOES NOT require all points in \(S\) is comparable to \(x\). It requires that all the points \(y\) that are COMPARABLE to \(x\) have \(x\succeq_K y\).