I'm considering non-linear quadratic optimization problem, that may be represented as follows:
\begin{align} &\min_{x,y,z_i} f(x)+g(y) + h(z_i) \\ s.t. \quad &x_l \leq x \leq x_u \\ & y_l \leq y \leq y_u \\ & z^l_i\leq z_i \leq z^u_i \\ & x + \sum_{i \in J_k} z_{i} + C_k = y \quad \forall J_k \end{align}
$f(x)$ and $g(y)$ are quadratic, strictly convex functions and $h(z_i) = \sum_{i \in I} c_i z_i$ -linear combination of $z_i$. Subsets of indices $J_k \subset I$ are given beforehand and constitute a partition of $I$.
This is a convex problem, so local and global minima coincide, but there might be no uniqueness. What I'm trying to understand is whether there's uniqueness w.r.t. $x$ and $y$. It intuitively seems to be true, but I wasn't able to prove it.
For a trivial example, $c_i = c \, \forall i$, then we can write \begin{equation} h(z_i) = c \sum_i z_i = c \Big[ \sum_{k} \sum_{j \in J_k} z_j \Big] = c \sum_k \big[ y- x - C_k \big], \end{equation}
Eliminating $z_i$ in the objective function, thus we can have a unique solution w.r.t. $x$ and $y$. But is there any consideration for the general case?