The unconstrained least-squares problem
$$ \min_x \| A x - b \|_2^2 $$
should be a convex objective with a global minimum having the pseudo inverse analytical solution
$$ x_{\min} = \left(A^T A \right)^{-1} A^T b $$
based on section 1.2.1 of Boyd & Vandenberghe's Convex Optimization. What confuses me is that the example below which has the above form, gives me local minima (which is not a the global minima) using the Excel solver.
$$ \min_x \| A x - b \|_2^2 $$
where
$$ A = \begin{bmatrix} p_1 & p_2 & 0 \\ 0 & p_2 & p_2 \end{bmatrix}, \qquad b = \begin{bmatrix} b_1 \\ b_2 \end{bmatrix} $$
where all $p_i, b_i > 0$. I thought this was quite strange since the least-squares objective seems convex regardless of what $A$ is? Here's the problem setup using the Excel solver. Not really sure what I might be missing. Any pointers would be much appreciated.
It could just be a careless mistake.
From your excel working, the intended matrix should be
$$\begin{bmatrix} p_1 & p_{\color{red}1} & 0 \\ 0 & p_2 & p_2\end{bmatrix}$$
rather than
$$\begin{bmatrix} p_1 & p_2 & 0 \\ 0 & p_2 & p_2\end{bmatrix}$$
The rank of $A$ is $2$ and hence it must be consistent. A possible solution is to set $x_2=0$, then $x_1 = \frac{b_1}{p_1}$ and $x_2=\frac{b_2}{p_2}$.