Solving a vector of quadratic equations

Question

I have a system of $n \times 1$ equations $$ 0 = A\,vec(xx^t) + B x + C $$ where

• $x$ is a $n \times 1$ vector of unknowns

• $x^t$ means transpose

• $vec$ means $xx^t$ has been vectorized so has dimension $n^2 \times 1$

• $A$ is a known matrix with dimensions $n \times n^2$

• $B$ is a known matrix with dimensions $n \times n$

• $C$ is a known vector of dimension $n \times 1$

I can solve these problems using a nonlinear solver. However, I am trying to find out if there are any theoretical results on how to solve this class of problems. I can find a lot of work on solving matrix quadratic equations. However, I can't find anything that has specifically this form, and I cannot figure out if I can rewrite this system in a way that is equivalent to other matrix quadratic problems I come across.

Answer 1

Shameless advertisement to a paper of mine: http://www.sciencedirect.com/science/article/pii/S0024379511004484 Quadratic vector equations, in Linear Algebra and its Applications, volume 438, 2013. Arxiv version on https://arxiv.org/abs/1004.1500.

I studied this exact problem in the case in which $A,B,C$ all contain nonnegative entries apart from the diagonal of $B$ which is nonpositive ($-B$ has to be an M-matrix, more precisely).

There are a few matrix equations that can be reduced to this form (for instance, some nonsymmetric Riccati equations or matrix quadratic polynomial equations); in this paper I tried to study them all in a unified fashion and work with the most general hypotheses. I dealt with the existence of a minimal nonnegative solution, and with the convergence of several numerical methods.

In future works, we have developed other algorithms that are tailored to the case in which there are two very close solutions.