Given $\mathbf V_t=\mathbf v_t\mathbf v_t^H$ where $\mathbf v_t=\left(e^{j\theta_{1}},e^{j\theta_{2}}\right)^H$: \begin{equation*} \begin{array}{ll} \underset{\mathbf V}{\operatorname{minimize}} & 1 \\ \text { subject to } & \operatorname{diag}(\mathbf V)=1\\ &\|\mathbf V\|_*-\operatorname{trace}\left[\boldsymbol\lambda\boldsymbol\lambda^H\left(\mathbf V-\mathbf V_t\right)\right]-\|\mathbf V_t\|_2\leq 0 \end{array} \end{equation*} where $\|\bullet\|_*$ and $\|\bullet\|_2$ is the nuclear norm and 2-norm. $\boldsymbol\lambda$ is the leading eigenvector of $\mathbf V_t$.
Since $\mathbf V=\mathbf V_t$ is a feasible solution, the optimization problem is feasible. Why does CVX show the problem is infeasible?
clear;clc;close all;
Vt = rand(2,1);
Vt = (exp(1i*Vt)*exp(1i*Vt)');
[lambda,~] = eigs(Vt,1,'largestabs');
cvx_begin sdp
variable V(2,2) hermitian semidefinite
minimize(1)
subject to
diag(V) == 1
norm_nuc(V) - real(trace(lambda*lambda'*(V-Vt))) - norm(Vt,2) <= 0
cvx_end
