# Linear Objective Minimization on Intersection of Two Ellipsoid Surfaces

Let $$D=\text{diag}(d_1,d_2,\dots,d_n)$$ be a positive definite $$n\times n$$ matrix, $$0\ne c\in \mathbb R^n,$$ and $$\alpha$$ be a positive real number such that $$\alpha \ne d_i$$ for $$i=1,2,\dots,n$$. Consider the following nonconvex minimization problem: $$\begin{equation}\tag{P} \label{pr: p} \min_{y\in \mathbb{R}^n} \ c^Ty \qquad \mbox{ subject to } \quad y^TDy=\alpha, \quad \text{ and } \quad y^Ty=1. \end{equation}$$

To cut to the chase, assume that the KKT conditions exist and the Lagrangian multipliers are nonzero for both equality constraints. It turns out these multipliers must satisfy a nonlinear system with two equations (that I cannot explicitly find any of its solutions). Is it possible to find any (local or global) closed-form solution of (\ref{pr: p}) for the described scenario?

Note that one can eliminate two variables from the constraints and convert this problem into an unconstrained one, then apply any decent numerical method. But that is not what I am looking for rather I am more interested in finding closed-form solutions if possible. Any ideas would be helpful.