My objective function is quadratic in $\mathbf{a}$, which is vector version of matrix $\mathbf{A}$. But a rank-1 constrain i-e $\mathbf{A}=\mathbf{p}\mathbf{q}^{T}$ makes the problem nonconvex.
However if I fix $\mathbf{p}$, the remaining problem becomes an unconstrained minimization problem in $\mathbf{q}$. I solve for $\hat{\mathbf{q}}$, update $\mathbf{q}$, and then repeat to get $\hat{\mathbf{p}}$. This way I was able to split original problem into two unconstrained minimization problem wrt two vectors $\mathbf{p}$ and $\mathbf{q}$. I iterate till a stopping criterion is met.
Now I am looking into convergence issue of this approach.To be specific I want to be able to say that
- Regardless of how I initialize, my algorithm always returns $\hat{\mathbf{p}}$ and $\hat{\mathbf{q}}$.
- Let $\mathbf{p^*}$ and $\mathbf{q^*}$ denote the OPTIMAL solution. Under what conditions can we guarantee that $\hat{\mathbf{p}}=\mathbf{p^*}$ and $\hat{\mathbf{q}}=\mathbf{q^*}$.
Note that (1) holds true if (2) is true but (1) does not necessarily imply (2). Can someone please suggest some reference material or give some insight into how to approach such an analysis.