I'm attempting to solve the following problem in $x \in \mathbb{R}^n$ and $y \in \mathbb{R}^n$
$$\begin{array}{ll} \text{minimize} & (x-x^0)^\intercal \Lambda (x-x^0) + (y-y^0)^\intercal \Lambda (y-y_0)\\ \text{subject to} & 1-\varepsilon \le x_i^2 + y_i^2 \le 1+\varepsilon \qquad \forall i \in \{1,2,\ldots,n\}\end{array}$$
where $x^0, y^0$ and $\varepsilon$ are all known parameters and square matrix $\Lambda$ is symmetric and positive definite.
Clearly, if
$$\quad 1-\varepsilon \le (x^0_i)^2 + (y^0_i)^2 \le 1+\varepsilon$$
for $i = 1,2,\ldots,n$, then the optimal solution is $x = x^0, y=y^0$. But what if $x^0$ and $y^0$ do not satisfy the constraint? Is there still an analytical solution (explicit form) to this optimization problem?
Thanks!