Generalized inequality

A generalized inequality is defined in terms of a proper cone $$K$$ and is symbolized with the notation $$\preceq_K$$. The generalized inequality $$\preceq_K$$ is defined as

$$x \preceq_K y \implies y-x \in K$$.

A strict generalized inequality $$\prec_K$$ is defined as

$$x \prec_K y \implies y-x \in \textbf{int}\, K$$,

where $$\textbf{int}\, K$$ is the interior of the set $$K$$.

Properties of generalized inequalities
A generalized inequality has the following interesting properties
 * 1) $$\preceq_K$$ is preserved under addition: If $$x \preceq_K y$$ and $$u \preceq_K y$$, then $$x+u \preceq_K y+v$$
 * 2) $$\preceq_K$$ is transitive: If $$x \preceq_K y$$ and $$y \preceq_K z$$, then $$x \preceq_K z$$.
 * 3) $$\preceq_K$$ is preserved under nonnegative scaling: If $$x\preceq_K y$$ and $$\alpha > 0$$, then $$\alpha  x \preceq_K \alpha y$$.
 * 4) $$\preceq_K$$ is reflexive: If $$x \preceq_K x$$.
 * 5) $$\preceq_K$$ is antisymmetric: If $$x \preceq_K y$$ and $$y \preceq_K x$$, then $$x = y$$.
 * 6) $$\preceq_K$$ is preserved under limits: If $$x_i \preceq_K y_i$$ for $$i=1,2,\dots$$ and $$x_i \to x$$ and $$y_i \to i$$, then $$x \preceq_K y$$.

Minimum and minimal elements
Generalized inequalities allow us to generalize the concept of a minimum element to the space of vectors.

An essential difference between a regular inequality and a generalized one is that not all points are comparable. That is, one of the two inequalities $$x \le y$$ or $$x \le y$$ has to hold. This is not the case for generalized inequality.

Example. Consider the proper cone $$K=\mathbf R^n_+$$, and points $$x=(3,3)$$, $$y=(5,5)$$ and $$z=(4,2)$$. Clearly, $$x$$ and $$y$$ are comparable and $$x\preceq_K y$$. Similarly, $$y$$ and $$z$$ are comparable and $$z \preceq_K y$$. However, $$x$$ and $$z$$ are not comparable.

Also, a natural concept to define is that of the minimal point. The formal definitions of minimum and minimal points are as below.

Minimum element. We say that $$x \in S$$ is the minimum element of $$S$$ (w.r.t. $$\preceq_K$$) if for every $$y \in S$$ we have $$x \preceq_K y$$, which happens if and only if $$S \subseteq x+K$$ where $$x+K$$ is the set of all the points that are (i) comparable to $$x$$ and (ii) greater than or equal to $$x$$ (confer Fig. 2.17 of or Fig. 43 of ).

There can be at most one minimum point. Minimal element. First of all, a minimum point is also a minimal point. But a minimal point can exist even if there is no minimum. There can be more than one minimal points.

We say that $$x\in S$$ is the minimal point of $$S$$ (w.r.t. $$\preceq_K$$) if for any $$y\in S$$, $$y\preceq_K$$ holds only if $$y=x$$. Or, equivalently,

$$ (x-K) \cap S = \{x\} $$

where $$x-K$$ denotes the set of all points that are comparable to $$x$$ and are less then or equal to $$x$$ w.r.t. $$\preceq_K$$.

Confer Fig. 2.17 of or Fig. 43 of. Note that in Fig. 43b it's impossible to draw the cone $$K$$ (centered on any of the minimal points) that would contain the entire $$\mathcal C_2$$, therefore $$\mathcal{C}_2$$ has no minimum.