what is meant by minimum element ? what's the difference between minimum and minimal element?

Question

We say that $x ∈ S$ is the minimum element of $S$ (with respect to the generalized inequality $≼_K$) if, for every $y ∈ S$, we have $x ≼_K y$.

We say that $x ∈ S$ is a minimal element of $S$ (with respect to the generalized inequality $≼_K$) if, for every $y ∈ S, y ≼K x$ we have $y = x$.

We can describe minimum and minimal elements using simple set notation. A point $x ∈ S$ is the minimum element of S if and only if $S ⊆ x + K$. Here $x + K$ denotes all the points that are comparable to $x$ and greater than or equal to $x$ (according to $≼_K$ ). A point $x ∈ S$ is a minimal element if and only if $(x − K) ∩ S = {x}$.

Answer 1

Your definitions are incorrect.

Something is minimal if nothing is smaller. Formally we write this as "$y$ is minimal if $\forall x, y\leq_K x$."

Something is the minimum if it is the unique minimal element. Formally we write this as "$y$ is the minimum if $\forall x(x\leq_K y\Rightarrow x=y)$. Alternatively, we can write "$y$ is minimum if $\forall x(x=y\lor y<_K x)$," where $a<_Kb\iff a\leq_Kb\land a\neq_Kb$. Notice that the equality in both cases is actual equality, not equality in the partial ordering induced by $K$

Let's look at an example. Consider $\mathbb{C}$ equipped with the partial order induced by $|\cdot|$. The minimal points of $\{z:3\leq |z|\leq 5\}$ are the set $\{z:|z|=3\}$, but the function $f(z)=|z|$ has a minimum value of $3$ on that set. When talking about the output of the function, there is a unique smallest output, and so it's the minimum. When talking about the input there are multiple points that give rise to the smallest output, so those are minimal points.

A minimum point is always minimal, but the converse statement is not true in general. However, it is true for total orders.

Answer 2

You have almost correctly quoted the formal definitions (see @StellaBiderman 's answer), so I assume you're asking for the idea, expressed more in words.

"Minimum" means "smallest". In the usual ordering of the natural numbers $\{1, 2, \ldots \}$, the number $1$ is the minimum.

"Minimal" means "nothing is smaller". If you order the natural numbers starting with $2$ by divisibility, so $a "\le" b$ when $a$ divides $b$ then the minimal numbers are the prime numbers. $7$ isn't minimum - it's not smaller than everything else - but it is minimal - nothing is smaller than it.

Answer 3

I don't think you copy the wrong labels or having incorrect definition, at least I didn't find this to be an errata of the book.

In Section 2.4.2 of Boyd (Convex Optimization), the difference is detailed, minimum if exists, must be unique; however, several minimals may be found for $S$ given an generalized inequality $\le_K$ induced by a proper cone $K$.

To see why minimum is defined in this way, let $x, y \in S$ and each is a minimum. It follows that $x \ge_K y$ and $y \ge_K x$. But $x-y\in K$ and $y-x\in K$ implies that $d\in K$ and $-d \in K$. Assuming that $d \ne 0$, $td+t'(-d) = (t-t')d \in K$ for any nonnegative scalars $t, t'$. So, $K$ contains the line defined by $0$ and $d$ which is impossible because $K$ must be proper. Thus, minimum must be unique where $d=0 \implies x=y$.

Intuitively, for any $y \in S$, $y$ must be comparable with $x$. Hence, the minimum can be found because $x$ is smaller than or equal to all $y$.

To see the difference between minimum and minimal, for any $y \in S$ and $y \le_K x$ where $x$ is a minimal of $S$ with respect to $\le_K$, $y=x$. However, $S$ may have several minimals because there may exist an $x' \in S$ which is not comparable with $x$. If $\le_K$ is fortunately a total/linear order on $S$, you can talk about the minimum. Otherwise, you can only talk about minimals. (See https://en.wikipedia.org/wiki/Total_order) Figure 2.17 in the book clearly illustrates these concepts.