Suppose I have a matrix valued function 

$$
F:\mathbb{R}\rightarrow\mathbb{R}^{m\times n},\qquad F(x)=\tilde Q\tilde R+xu_1v_1^T+xu_2v_2^T
$$
where $\tilde Q\in\mathbb{R}^{m\times m}$ is orthogonal, $\tilde R\in\mathbb{R}^{m\times n}$ upper triangular, $u_1,u_2\in\mathbb{R}^m$ and $v_1,v_2\in\mathbb{R}^n$. 

Is there anything that can be said about the QR decomposition of $F(x)=Q(x)R(x)$ depending on $x$? 

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To give a bit more background: I would like to minimize
$$
g(x)=||Q^T_2(x)z||^2_2
$$
for some vector $z$ and 
$$
F(x)=Q(x)R(x)=\begin{bmatrix}Q_1(x)& Q_2(x)\end{bmatrix}\begin{bmatrix}R_1(x)\\0\end{bmatrix}=Q_1(x)R_1(x).
$$

I plotted the function $g$ for a few different cases, and it always looks similar to this:
![enter image description here][1]

I have been wrapping my mind around the following two questions:
 
- Is it just coincidence that I see exactly one local minimum and one local maximum, or might that be proven?
- Might it even be possible to give a direct algorithm that finds the minimum of this function?

It is not to difficult to employ a nonlinear optimizer to find the minimum, however, in that case I would like the guarantee that I in fact only have one local minimum and that my optimizer in case it does not diverge is ensured to find the global optimum.

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What I have tried: There are algorithms for updating a QR decomposition with rank 1 matrices, e.g. by [Daniel, Gragg, Kaufman and Stewart][2]. I tried to follow those steps symbolically but using a series of Givens Rotations to ensure triangularity of a the matrix $R(x)$ quickly leads to terms that I found not to be good to handle. However, maybe I am just missing a good idea for a clear notation or do not see the system behind.

Any help (even if it is just a pointer to a paper that does something similar) is greatly appreciated. 


  [1]: https://i.sstatic.net/azJr7.jpg
  [2]: http://www.ams.org/journals/mcom/1976-30-136/S0025-5718-1976-0431641-8/S0025-5718-1976-0431641-8.pdf