Solution to a non-convex problem — LP with unit norm constraint

The answer to this problem is ill-posed, the solution can be arbitrarily small and close to zero. Consider $A$ be the subspace formed by $\{a_i\}_{i=1}^{n}$. Then the solution for the problem, we call it $a^*$, can be written as $a^* = \bar{a}^* + a^{\bot}$ where $\bar{a}^* \in A$ and $a^{\bot}$ is in the null-space of $A$, i.e. $A^{\bot}$ .

Because there is the constraint $\langle a, a_i\rangle>0$, therefore the trivial solution $a = a^{\bot}, \lVert a^{\bot}\rVert = 1$ is not valid, however we can have a solution of the form $a = \epsilon_1 \frac{a_1}{\left\Vert a_1\right\Vert} +\epsilon_2 \frac{a_2}{\left\Vert a_2\right\Vert} + \cdots + \epsilon_n \frac{a_n}{\left\Vert a_n\right\Vert} + \sqrt{1-\epsilon_1^2-\epsilon_2^2-\cdots-\epsilon_n^2}a',a'\in A^{\bot}$, where $\forall i: \epsilon_i >0$ can be arbitrarily small.