0

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!

Rodrigo de Azevedo
  • 1
Matifou
  • 173

1 Answers1

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.

Rodrigo de Azevedo
  • 1
  • great, thanks! Can you give a quick intuition/explanation why minimizing 0? And you don't mention the symmetry in the constraints, is that automatically satisfied? Finally, if I may ask, any specific method/algorithms to recommend, ideally in R? Thanks!!! – Matifou Oct 29 '19 at 22:25
  • Minimizing $0$ because it's a feasibility problem, not an optimization problem. One only cares about getting a point in the feasible region. There's no cost to minimize. Positive semidefiniteness usually implies symmetry. Since the skew-symmetric part of a matrix contributes nothing to a quadratic form, it only makes sense discussing the definiteness of symmetric matrices. I never used R for optimization. If you use MATLAB, try using CVX. If you use Python, there's CVXPY. In R, there's Rdsdp. – Rodrigo de Azevedo Oct 29 '19 at 23:21
  • thanks a lot for the clarification, much appreciated! – Matifou Oct 30 '19 at 00:48
  • I made a new post about the specific implementation of this, in case you want to look at it would be great, thanks! https://math.stackexchange.com/questions/3416911/semidefinite-programming-writing-down-a-feasibility-program-in-dsdp-or-csdp – Matifou Oct 31 '19 at 19:35
  • @Matifou If you try to use Python + CVXPY, take a look at this. – Rodrigo de Azevedo Oct 31 '19 at 19:43
  • One more question... do you know how one would add the constraint that the diagonal elements are non-negative? Can that be cast into the sdp constraint? Thanks!! – Matifou Nov 11 '19 at 21:45
  • @Matifou From Sylvester's criterion, it follows that the positive semidefiniteness constraint on matrix $\rm X$ does encapsulate the constraints you mentioned. – Rodrigo de Azevedo Nov 13 '19 at 16:12