Finding solution to $Xa=b$ where $X$ has to be symmetric positive-definite

Question

I want to reconstruct a $K\times K$ matrix $X$, knowing that that:

1. $X1=b$ ($1$ and $b$ are $K\times 1$)
2. $X$ is symmetric positive definite.

I understand this is a kind of under-determined system, but I am wondering whether and how I can find a solution $X$, such that $X$ is symmetric and positive-definite? Or do I need to add more information? Can a solution be unique, or is there anyway an infinity of those?

Thanks!

Answer 1

Just solve the following semidefinite program (SDP)

$$\begin{array}{ll} \text{minimize} & 0\\ \text{subject to} & \mathrm X \mathbb 1 = \mathrm b\\ & \mathrm X \succeq \mathrm O\end{array}$$

The solution set is the intersection of the positive semidefinite cone with a hyperplane.