![]() ![]() Sometimes, even in a infinite and uniformly and non-atomic, continuous total order representation, nothing unique will come out. That is $F$ is a monotone strictly increasing function of some entity $x$ where $F(x)$ is the entity one wishes to order by $x$. Where in $(B)$ the numerical function, $F$ or representation is 'merely strong'. In contrast to $(B)$ a standard order embedding (strict monotone increasing function) (By cone, we mean that $\alpha K\equiv K$ for all $\alpha>0$ and by pointed, we mean that $K\cap-K=\ \, x=x_1 +x_2\, y=y_1+y_2 $$ Let $S$ be a vector space, and let $K\subset S$ be a closed, convex, and pointed cone with a non-empty interior. But given that the text deals with convex optimization, it was apparently considered helpful to refer to them as inequalities. And indeed, that's exactly what they are, and the book does refer to them that way as well. The book refers to these relations as generalized inequalities, but as Code-Guru rightly points out, they have been in use for some time to represent partial orderings. It is only when the quantities on the left- and right-hand sides are vectors, matrices, or other multi-dimensional objects that this notation is called for. It should be noted that the book does not use $\succeq$, $\preceq$, $\succ$, and $\prec$ with scalar inequalities for these, good old-fashioned inequality symbols suffice. I hope that I'm not being inappropriate by combining and expanding upon them here. Both Chris Culter's and Code Guru's answers are good, and I've voted them both up.
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