Adjusting weights of a weighted sum so that the sum has a lower bound.

Question

Suppose I have this weighted sum (there are many $x_i$ sequences, so, I actually have many such sums):

$$ S = \sum_{i=0}^n{x_i \times w_i} $$

With:

$$ \sum_{i=0}^n{w_i} = 1 $$

Now, let's say I want the sum to have one of the $x_i$s as a lower bound, e.g., say I want:

$$ S \geq x_7 $$

Can the weights $w_i$ be adjusted such that $S$ will have such a lower bound for any sequence of $x_i$s? If yes, then how?

PS: I don't want $w_7=1$ and $w_i=0~~\forall i \neq 7$. All $w_i$'s must be non-zero.

Answer 1

What prevents $w_7=1$ and $w_i=0, \forall i \neq 7$, for example?

With the additional constraint added to the problem, there is no solution that works with any set of $x_i$. If you apply any non-zero weight to any value $x_i, i\neq 7$, that $x_i$ could be arbitrary low (say $-10^{1000}$) and make your sum $S$ lower than $x_7$.

Answer 2

Here's an answer to the question with the added PS.

No.

If $w_7 < 1$ then there is some other $w_i > 0$. Then it's easy to find values $x_i$ and $x_7$ such that $w_i x_i + w_7x_7 < x_7$.