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I am reading a text on convex optimisation, and there is a line:

$f_i(\tilde{x})\leq0$ and $h_i(\tilde{x})=0$, and $\lambda \succeq 0$

and I was just wondering why for one term, $\leq$ is used and for the other, $\succeq$ is used.

I have a computer science background and for some reason we never were taught much formal mathematical notation.

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2 Answers 2

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$\succeq$ is used typically in the context of matrices and vectors.

  • If used in the context of vectors, it typically means that all elements of the vectors are non-negative, i.e., $\vec{\lambda} \succeq 0$, if $\lambda_i \geq 0$ for all $i$.

  • If used in the context of matrices, it typically means that the matrix is non-negative definite, i.e., $A \succeq 0$, if $x^TAx \geq 0$ for all $x \in \mathbb{R}^{n}$, where $A \in \mathbb{R}^{n \times n}$. However, on extremely rare occasions, this symbol could also mean that all entries in a matrix are non-negative, i.e., $A \succeq 0$, if $A(i,j) \geq 0$ for all $i,j$.

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  • $\begingroup$ that makes a lot of sense as i believe it is talking about a vector, thanks! $\endgroup$
    – Gus Kenny
    May 1, 2015 at 2:24
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In a different context, succeed might be used when there is "discrete" ordering, e.g. for the natural numbers, as opposed to "continuous" ordering, e.g. the reals. For example, $5$ succeeds $4$ (as there are no intermediate natural numbers between $4$ and $5$) it is also said that $5$ is the immediate successor of $4$. On the other hand there is not such thing as an immediate successor for real numbers, e.g. one could say that $4.6>4.2$ but we could always insert more numbers in between, $4.6>4.56>4.48374883748>4.2$. So here one would only say that $4.6$ is greater than $4.2$

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  • $\begingroup$ the context was a vector one, as mentioned above - but this is an interesting use of it also. thanks for expanding on the answer. $\endgroup$
    – Gus Kenny
    May 1, 2015 at 2:25
  • $\begingroup$ @guskenny83 Thank you. In yet another context one may use the same notation $\prec$ with a different word and different meaning, that is for (usually open) covers $\mathcal U$ and $\mathcal V$ of a topological space, one says that $\mathcal U$ refines $\mathcal V$, written $\mathcal U\prec \mathcal V$ if if for every $U\in\mathcal U$ there is some $V\in\mathcal V$ such that $U\subseteq V$. More generally $\prec$ is sometimes used for pre-orders: When $x\prec y$ and $y\prec x$ does not necessarily imply that $x=y$. $\endgroup$
    – Mirko
    May 1, 2015 at 2:31

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