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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 $(x − K) ∩ S = {x}$.

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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.

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    i took that definition from boyd convex optimization book :\ – 2017-02-13
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    2.4.2 minimum and minimal elements page no 45 :\ – 2017-02-13
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    @MORAMREDDYRAKESHREDDY double check that you copied the labels correctly. There are a couple of different ways to formalize these ideas, and looking at it again the idea is right but they're backwards. The first is minimal and the second is minimum, though awkwardly worded in the second case. – 2017-02-13
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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 minumal - nothing is smaller than it.