Let \(0 \leqslant w \leqslant 1\). Show that the solution to the convex optimization problem

\[ \begin{gather*} \min _{p_{1}, \ldots, p_{n}} \sum_{i=1}^{n} p_{i}^{2} \tag{7.28}\\ \text { subject to: } \sum_{i=1}^{n-1} p_{i}=w \text { and } \sum_{i=1}^{n} p_{i}=1 \end{gather*} \]

is given by \(p_{i}=w /(n-1), i=1, \ldots, n-1\) and \(p_{n}=1-w\).