The linear program \begin{align} \min &\sum_{i=1}^nc_{i}x_{i}\\\ \mbox{s.t.:}&\sum_{i=1}^nx_{i}=1,\\\ &x_{i}\geq 0,&&\forall i=1,\dots,n \end{align} has a trivial optimal solution which is to simply: find an $i^*\in\arg\min_{i=1,...,n}\{c_{i}\}$ and then set $x_{i^*}=1$ - the rest of the $x$'s should be zero. However, the quadratic program \begin{align} \min &\sum_{i=1}^nc_{i}x_{i} + \sum_{i=1}^n \beta_{i}x_{i}^2\\\ \mbox{s.t.:}&\sum_{i=1}^nx_{i}=1,\\\ &x_{i}\geq 0,&&\forall i=1,\dots,n, \end{align} doesn't exhibit this nice "single assignment" property. Albeit, the structure is rather simple, and one could expect that there is a simple way to solve the program anyway. So the questions are
- Does the quadratic program have a closed form solution, that I just haven't thought of?
- If the answer to 1. is "No", what would be your approach to solving (to optimality) this problem fast, given the simple structure?
For my purpose, we can assume that both $c_{i}$ and $\beta_{i}$ are non-negative.