I was studying a problem in a paper in which the author tries to use the semidefinite relaxation (SDR) technique in order to solve it. After changing the original problem using the SDR technique, a rank-1 constraint is added to the problem. The author states that this constraint is non-convex, and it should be relaxed. I do not know why the rank-1 constraint is a non-convex constraint. Could someone help me out?
2026-03-25 11:22:44.1774437764
In an optimization problem, why is a rank-1 constraint non-convex?
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The matrices $A = (-1)$ and $B=(1)$ are both rank one. However, the matrix $\lambda A + (1-\lambda)B$ with $\lambda=0.5$ does not have rank one. The set of rank one matrices is therefore not convex.