When does one use 'succeeds' and when does one use 'greater than'?

Question

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.

Answer 1

$\succeq$ 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 $i,j$.

Answer 2

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 $4.2$