Let $x_i,y_i\in\mathbb{R}$, $a_i\in[0,1]$, with $y_1>y_2$ and $y_3>y_4$. We also have $(y_1-y_3)^2 + (y_2 -y_4)^2 <\varepsilon $. I need to solve the following constrained optimization problem
$$\min_{x_1,x_2,x_3,x_4}\sum_{i = 1}^4 a_i(x_i-y_i)^2/2$$ $$\text{subject to } x_2\geq x_1,$$ $$ \quad \quad \quad \; x_4 \geq x_3$$ $$ \quad \quad \quad \;\quad \quad \quad \;\quad \quad \quad \; (x_1-x_3)^2 + (x_2-x_4)^2\leq \varepsilon$$
Attempted solution:
$L(\lambda,x) = \sum_{i = 1}^4 a_i(x_i-y_i)^2/2 + \lambda_1(x_1-x_2)+ \lambda_2(x_3-x_4) + \lambda_3( (x_1-x_3)^2 + (x_2-x_4)^2-\varepsilon)$
Differentiating and assuming binding constraints ($x_1 = x_2$, $x_3 = x_4$, $\varepsilon = (x_1-x_3)^2 + (x_2-x_4)^2)$ we get
$$x_k = \frac{\sum_i{a_i y_i}\mp \sqrt{\varepsilon/2}}{\sum_i a_i }, \text{ for } k = 1,2$$ and $$x_k = \frac{\sum_i{a_i y_i}\pm \sqrt{\varepsilon/2}}{\sum_i a_i }, \text{ for } k = 3,4$$
Is the procedure correct? Thanks!
