# Maximum of vector quadratic function over sub-spheres

Let $x \in \mathbb{R}^n$ and $P$ a $n \times n$ positive definite symmetrix matrix. It is known that $\max_{x|~x^Tx\leq 1} x^TPx=\lambda_{\text{max}}(P)$, where $\lambda_{\text{max}}(P)$ is the largest eigenvalue of $P$.

Now split the vector $x$ as $x=\begin{bmatrix}x^1\\...\\x^m\end{bmatrix}$, where $x^i \in \mathbb{R}^{n_i}$ and $\sum_{i=1}^m n_i =n$. Consider the following optimization problem $$y=\text{arg}\max_{x|~{x^i}^Tx^i\leq 1, \forall i } x^TPx$$ that is, every sub-variable is required to be in a unitary sub-sphere. Can I express $y$ in terms of the eigenvalues of $P$? Can I find $y$ analytically at all?

• Related: mathoverflow.net/q/277854/91764 Sep 8 '17 at 11:22
• This appears to be a of a Quadratically Constrained Quadratic Program, where your constraint matrices are just partial identity matrices. This particular case is convex, so many convex optimization methods are available, but I doubt you can express $\vec y$ any more eloquently than you could a solution to a linear program. Sep 9 '17 at 5:59