I am trying to solve question 4.8, part (e) from Convex Optimization by Boyd.
The problem is to find an explicit solution for the minimization problem:
Minimize $\textbf{c}^T \textbf{x}$
subject to
$\textbf{1}^T \textbf{x} = \alpha, \textbf{0} \preceq \textbf{x} \preceq \textbf{1}$
I attempted the problem by trying to decompose the vector $c$ as a vector orthogonal to $\textbf{1}$ and $\lambda \textbf{1}$, i.e. $c = \lambda \textbf{1} + \hat c$ with $\hat c^T \textbf{1} = 0$
I ended up an optimal value of $\lambda \alpha$. I've read the given solution to this problem, and the authors take a different route to solve the problem (but use this similar vector decomposition to solve previous parts of this problem i.e. parts (a) and (b)). I'm trying to understand why my approach fails to work or is insufficient, and if I have to take the approach that Boyd does in the solutions, or if there is a workable solution to the problem utilizing my approach (or something similar!)