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 $$u \preceq_K v$$, 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 $$x \preceq_K x$$ for all $$x$$.
 * 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 in a generalized inequality. 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.

What's the point of generalized inequalities?
Generalized inequalities allow us to tackle some complicated, multi-dimensional optimization problems in principled ways. For example, we can questions like "what's the minimum/minimal ellipsoid that contains a set of points". In this example, the generalized inequality is the matrix inequality, and the proper cone that it is induced from is the cone of positive-definite matrices (i.e., $$\mathbf S^n_{++}$$, see Example 2.18 ).

What's the point of minimal points?
The concept of minimal points is directly related to Pareto optimal points, which, put simply and coarsely, is the set of points that are, as far as the objective function of an optimization problem is concerned, equivalent (see page 71 and Section 4.7.4 ).