Let $\boldsymbol{a}=(a_1,a_2,...,a_n)\in \mathbb{R}^{n}$, $b,c\in \mathbb{R}$. I need to study a problem in the form
\begin{align} \min_{\boldsymbol{x}}& \sum_{i=1}^{n} x_i a_i\\ s.t.& \sum_{i=1}^{n} x_i=b,\\ &\sum_{i=1}^{n} x^2_i=c,\\ &\boldsymbol{x}\geq0. \end{align}
Which is non-convex due to the quadratic equality constraint $\sum_{i=1}^{n} x^2_i=c$. Any chance that it is possible to rewrite it as a convex linear program? Or that maybe one can find a closed-form solution for the global minimum? Any help with this would be much appreciated.