How to solve the constrained convex optimization problem with nuclear norm?

Question

Let $f(x,y)=\sum_{i=1}^n\sum_{j=1}^{n}(x_{ij}+\sqrt{-1}y_{ij})B_{ij}(x,y)$

Now I want to solve the following problem: \begin{equation} \begin{split} &\arg\min_{C} \iint_{[0,1]^2}|(f_x+\sqrt{-1}f_y)-g(f_x-\sqrt{-1}f_y)|^2\mathrm{d}x\mathrm{d}y+\omega\|C\|_{*}\\ s.t. \quad &C_{i1},C_{in},C_{1j},C_{nj}\quad (i=1,\cdots,n;j=1,\cdots,n)\quad \text{are given}. \end{split} \end{equation} Where $C\in\mathbb{C}^{n\times n}$ is a complex matrix with the following form: $$ \begin{equation} C=\left( \begin{array}{cccc} x_{11}+\sqrt{-1}y_{11} & x_{12}+\sqrt{-1}y_{12} & \cdots &x_{1n}+\sqrt{-1}y_{1n} \\ x_{21}+\sqrt{-1}y_{21} & x_{22}+\sqrt{-1}y_{22} & \cdots &x_{2n}+\sqrt{-1}y_{2n} \\ \vdots & \vdots & \ddots& \vdots \\ x_{n1}+\sqrt{-1}y_{n1} & x_{n2}+\sqrt{-1}y_{n2} & \cdots &x_{nn}+\sqrt{-1}y_{nn} \end{array} \right) =X+\sqrt{-1}Y \end{equation} $$

$g=sin(xy)+\sqrt{-1}cos(x+y)$, and $\|\cdot\|_{*}$ is the nuclear norm of a matrix.

How to solve this convex problem?

I convert this problrm into the following form: \begin{equation} \begin{split} &\arg\min_{C} \frac{1}{2}c^TAc+\omega\|C\|_{*}\\ s.t. \quad &x_{i1},y_{i1},x_{in},y_{in},x_{1j},y_{1j},x_{nj},y_{nj}\quad (i=1,\cdots,n;j=1,\cdots,n)\quad \text{are given}. \end{split} \end{equation} where $A$ is a real positive-definite matrix, $c$ is a combination of the vectorization of $X$ and $Y$ which is defined as: $c=(\text{vec}(X),\text{vec}(Y))$.

But I don't know how to solve this.

Answer 1

Easiest approach is to simply use a modelling tool (YALMIP, CVX etc) with any semidefinite programming solver (Mosek, Sedumi,SDPT3 etc). It is a fairly standard problem.

Here is the corresponding YALMIP code (MATLAB modelling toolbox)

X = sdpvar(n,n,'full');
Y = sdpvar(n,n,'full');
C = X + sqrt(-1)*Y
c = [X(:);Y(:)];
Model = [X(:,1) == values,Y(:,1) == someothervalues,...];
optimize(Model,c'*A*c + norm(C,'nuclear');