TITLE: Least sum squares given constraints on subcomponents
QUESTION [5 upvotes]: Hi all,
I recently encounter a difficult problem.
I wish to minimize in $ \mathbf{x} $ the sum  $\min \sum_{i=1..n} (\mathbf{x}^T \mathbf{A}_i \mathbf{x})^2$ given the constraints on the norms of all $\mathbf{x}$'s subcomponents (let's say three 3-by-1 vectors) $|\mathbf{x}_1| = 1, |\mathbf{x}_2| = 1, |\mathbf{x}_3| = 1$. $\mathbf{A}_i$ may not be positive-definite.
Yes, it's quartic expression that we want to minimize. I'm not sure if any one has worked on this or similar problem in the math community. I search the literature for sometimes but no use. My question may be similar but actually much more difficult than this Least square given constraint on subcomponents
The 4th-order and constraints on all subcomponents makes it really hard for me to handle.
Any idea to a numerical/analytical solution, is greatly appreciated. Thanks for reading.
p/s: $\mathbf{x} = [\mathbf{x}_1^T , \mathbf{x}_2^T, \mathbf{x}_3^T]^T$. By "subcomponents", I mean the subvectors, as shown in the equation.

REPLY [3 votes]: As a practical matter, a lot depends on $n$ and the dimension of $x$.  If the problem is small enough then you might not be in deep trouble.  If the problem is large (e.g. $x$ might have thousands of components), then it could be very hard.  
If the problem is quite small, then you might consider using an approach that exploits the polynomial structure of your optimization problem.  There are convex relaxations of such polynomial optimization problems that provide very tight lower bounds and software can often use these lower bounds to find a globally optimal solution.  See for example the Gloptipoly2 software:
http://homepages.laas.fr/henrion/software/gloptipoly2/