I've been working on solving some linear equations arising from different optimization problems, but I keep getting stuck. Right now I have the problem below:
I am trying to solve the system of equations for $x$:
$$ Ax-\alpha \frac{Bx}{ x^tBx}=c$$ $$e^tx=1$$ where $e=(1,...,1)^t$.
where $x,c\in \mathbb{R}^n$, both $A,B \in \mathbb{R}^{n\times n}$ are positive definite and indeed even $$A-\frac{\alpha B }{x^t B x}$$ is positive definite, so we have nice invertibility properties.
Any help, references, or much better - a solution - is very much appreciated!
EDIT: Some further work below.
If we set $x=\sqrt{B}^{-1}z$ we get $$A\sqrt{B}^{-1}z-\alpha\frac{\sqrt{B}z}{z^tz}-c=0$$ or for $D=\sqrt{B}^{-1}A\sqrt{B}^{-1}/\alpha$ and $p=\sqrt{B}^{-1}c/\alpha$ $$0=Dz-\frac{z}{z^tz}-p\Leftrightarrow (D-\frac{pz^t}{z^tz})z=\frac{z}{z^tz}.$$ So it appears that $z$ is a multiple of an eigenvalue of a matrix that in turn depends on $z$. Is there any way I can extract analyical solutions!?
If we decompose the vector $x$ as a component along and orthogonal to $e$, (i.e., $x=e+\Delta:\Delta \cdot e = 0$) and then apply this to the simplified form provided by @TheoBendit, we get:
$$(e+\Delta)^t(A(e+\Delta)-c)=\alpha \implies$$ $$e^tAe+\Delta^tAe+e^tA\Delta+\Delta^tA\Delta-e^tc-\Delta^tc = \alpha \implies $$ $$ \langle\Delta,\mathbf{a}_{\cdot j}\rangle+ \langle\Delta,\mathbf{a}_{i\cdot}\rangle +\mathbf{Q}_{A}(\Delta)-\langle c,\Delta\rangle=\alpha-\sum a_{ij}+\sum c_i \equiv K$$
So, we've (reduced?!) this to a quadratic equation in $n$ variables. If we let $\delta_i$ be the $i$th component of $\Delta$ then:
$$ \left(2\sum a_{ij}-\sum c_i\right) \sum \delta_i+\sum\sum a_{ij}\delta_i\delta_j-K=0 $$
Subject to:
$$\sum \delta_i = 0$$
Let $\mathbf{v}:=(v_1,v_2,...,v_n)$ be a solution to this formula, then your $x$ is $x=v+e$. There will likely be several solutions to this formula, but since its a quadratic form, you can apply any number of multivariate root finding algorithms to solve it. Or you can search the trove of answer on MSE. Here enter link description here.
Also, the general problem of solving underdefined quadratic systems (like that here), has been studied. See this paper and here.