Can ADMM be applied to optimisation with matrix variables and inequality constraint? If so, how may I do it? My optimisation problem looks something like this:
\begin{align} \min_{X \in \mathbb{R}^{n \times n}} \quad & \|Xv \|_2 + \|X\|_F^2 \\ \quad &\text{Tr}(AX) \geq \text{Tr}(B) \end{align} where $v \in \mathbb{R}^n$, $A,B \in \mathbb{R}^{n \times n}$ are known constant vector and matrices.
Thanks a lot!
Honestly, I do not see big issues here. Assume $n=2$ to clarify the ideas, and set $$ v = \begin{pmatrix} v_1 \\ v_2 \end{pmatrix}, \ A= \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}, \ B= \begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix}, \ X= \begin{pmatrix} x_{11} & x_{12} \\ x_{21} & x_{22} \end{pmatrix}. $$Then $$ \|Xv\|_2 + \|X\|_F ^2 = \sqrt{ (v_1 x_{11} + v_2 x_{12} )^2 + (v_1 x_{21} + v_2 x_{22})^2 }+ x_{11} ^2 + x_{12} ^2 + x_{21} ^2 + x_{22} ^2, $$ $$\text{Tr}(AX) = a_{11} x_{11} + a_{12} x_{21} + x_{22} a_{22} + a_{21} x_{12} $$ and $$\text{Tr}(B)=b_{11} + b_{22}. $$
Then you want to minimize a nonlinear objective function (differentiable almost everywhere, but maybe you wanted $\|Xv\|_2 ^2$ instead?) under linear inequality constraints. This paper briefly explains how ADMM can deal with linear inequality constraints.