Create a strictly increasing sequence following criterias Problem
Let y be a sequence of real numbers (of length $n$) bounded in the range [0,1]. I am trying to calculate the sequence x that respects the following criteria:


*

*x must be of length $n$ too

*x must also be bounded in the range [0,1]

*No two values of x must be closer to each other than $\epsilon$. That is $ \epsilon \le x_i - x_j  \space \forall i \neq j$


So far, there are an infinite number of possible solutions. There is therefore a statistic to optimize


*

*The sum of square differences between x and y must be minimized. That is $\sum_{i=1}^n (x_i - y_i)^2$ must be minimized.



Case specific solution 1
$y = [0,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{3}{4}]$
$x = [0,\frac{1}{2}-\epsilon,\frac{1}{2},\frac{1}{2}+\epsilon,\frac{3}{4}]$
Case specific solution 2
$y = [0,0,0]$
$x = [0,\epsilon,2\cdot \epsilon]$
Of course, there are cases where no solution exist. Typically, in the second example if $2\cdot \epsilon > 1$, then there are no solution.  
 A: Partial answer: So the problem maybe formulated as
$$ \begin{align} 
\text{minimize}& & f(\mathbf{x}) &= \sum_{i=1}^n (x_i - y_i)^2 \\
\text{subject to}& & x_{j+1} - x_j &\geq \varepsilon, j = 1, 2, \ldots, n-1 \\
& & x_1 &\geq 0 \\
& & x_n & \leq 1 \\
\end{align}$$
As you have mentioned, for the feasibility of the problem, obviously one has to restrict that 
$$ \varepsilon \leq \frac {1} {n - 1}$$
Following https://en.wikipedia.org/wiki/Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions
Note the following problem share the same optimal solution $\mathbf{x}^*$:
$$ \begin{align} 
\text{maximize}& & -f(\mathbf{x}) &= -\sum_{i=1}^n (x_i - y_i)^2 \\
\text{subject to}& & \varepsilon - x_{j+1} + x_j &\leq 0, j = 1, 2, \ldots, n-1 \\
& & -x_1 &\leq 0 \\
& & x_n - 1 &\leq 0 \\
\end{align}$$
So now we have $n+1$ inequality constraints. Consider the necessary conditions in Karush–Kuhn–Tucker conditions:
$$ \begin{align}
\frac {\partial f} {\partial x_i} = 2(x_i - y_i) &= 
\begin{cases} 
-\mu_{i-1}+\mu_i, i = 2, 3, \ldots, n-1 \\
\mu_1 - \mu_n, i = 1 \\
-\mu_{n-1} +\mu_{n+1}, i = n
\end{cases} \\
\mu_j &\geq 0, j = 1, 2, \ldots, n+1 \\
\mu_j(\varepsilon - x_{j+1} + x_j) &= 0, j = 1, 2, \ldots, n-1 \\
-\mu_n x_1 &= 0 \\
\mu_{n+1}(x_n - 1) & = 0
\end{align} $$
Sorry it is just the formulation. To solve this you need to check whether the inequalities constraints are active, and proceed to solve the system. See if I / someone have time to do this later.
