Constraints introduced by horizontal obstacle

Question

I have a problem in the form

$$ f(x) = x^TAx - b^Tx $$

where $A \in \mathbb{R}^{n\times n}$ and $b \in \mathbb{R}^n$ are known and $x_{\frac{n}{4}} = x_{\frac{n}{4}+1} = \cdots = x_{\frac{n}{2}}$ (string is resting on some obstacle, so its second quarter is not bent down).

I'd like to rewrite it to the form

$$ min \,f(x)\\ s.t. \quad h(x) = 0 $$

but I have no idea, how to get $h(x)$ function from the one equality condition above. Could you, please, help me?

---

Note

Function f(x) looks similar this:

Answer 1

Assuming that $n$ is a (positive) multiple of $4$,

$$m := \frac n 4$$

is an integer. The string resting on the horizontal obstacle introduces $m$ equality constraints

$$x_m = x_{m+1} = \cdots = x_{2m}$$

which can be rewritten as follows

$$\begin{array}{cl} x_{m+1} - x_m &= 0\\ x_{m+2} - x_{m+1} &= 0\\ \vdots \\ x_{2m} - x_{2m-1} &= 0\end{array}$$

which can be written in matrix form

$$\mathrm H \mathrm x = 0_m$$

where $\mathrm H \in \{-1,0,1\}^{m \times 4m}$.