What is a rank-1 constraint and why is it non-convex?

Question

When an optimal power flow problem is formulated using Semi-definite programming method, the equivalent OPF problem contains one constraint X or W = vv^H where v is a vector of bus voltages and H is the complex conjugate transpose operator (v = [v1 v2 .... vn]^T where (1,2,...n) are buses indexes) In various publications, authors say that this particular constraint is a rank-1 constraint and it is a non-convex. I want to know how is this constraint a rank-1 constraint (What is the actual meaning of rank-1 constraint)? Further, how is this constraint non-convex?

Answer 1

It is called a rank-1 constraint because it means that the matrix $X=xx^H$ has rank 1. The constraint is not convex because the set of matrices of rank 1 (and say symmetric, if you wish) is not convex.