# Why is the problem infeasible?

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


• it solves for me to optimlaity, i.e., finds a feasible solution, using several different random number draws and all of Mosek, SeDuMi, and SDPT3. Can you poist reproducible code, copied and pasted, not an image, and show us a complete MATLAB session exhibiting infeasibility? Display the values of Vt and lambda. – Mark L. Stone Feb 14 at 20:43
• I copied and pasted, and ran it multiple times (with different random mumbers) for all 3 solvers, and it solved every time. – Mark L. Stone Feb 15 at 2:45
• Thanks a lot. Unfortunately, I could not get the correct result with CVX. Could you show me your results? – fengbiqian Feb 15 at 2:48
• All real and imaginary elements of V-Vt are less than 1r-9, 1e-10, 1e-15 in magnitude respectively for SeDuMi, SDPT3, Mosek. The exact output varies when random numbers are changed. Try running your program starting from a fresh MATLAB session. – Mark L. Stone Feb 15 at 3:12
• Your code and logic are fine. Something is screwed up in your MATLAB, CVX, or solver installation or session. – Mark L. Stone Feb 15 at 4:15