clear;
fs = 100;
tt = 0:1/fs:1;
I = sin(2pi10tt);
Q = cos(2pi10tt);
A = [];
at = [];
for i = 1:length(I)
at = [I(i)*I(i) I(i)*Q(i) Q(i)*Q(i) I(i) Q(i) 1];
A = [A
at];
end
m=length(I);

cvx_begin
variable X(7,7)
X(1,1) == 1;
X(3,3)-4X(2,4) <= 0;
X == semidefinite(7);
r = AX(2:7,1)
minimize norm(r,1)
cvx_end

the result is
sorry, i can not paste the picture!
1.00000000000069 8.22675090326961e-17 -1.32600005447270e-27 8.22675090325424e-17 -3.70910063708033e-28 -3.75622517313046e-28 -8.22675090325573e-17
8.22675090326961e-17 26.3193626275072 4.95608144995849e-16 9.59592111769102 4.75200415146021e-16 4.75200415120391e-16 4.75200415129057e-16
-1.32600005447270e-27 4.95608144995849e-16 19.1918422352120 4.95608144996138e-16 6.76750690884502e-16 6.76750690850753e-16 6.76750690862163e-16
8.22675090325424e-17 9.59592111769102 4.95608144996138e-16 26.3193626275099 4.75200415146306e-16 4.75200415120676e-16 4.75200415129343e-16
-3.70910063708033e-28 4.75200415146021e-16 6.76750690884502e-16 4.75200415146306e-16 22.7889058762847 6.60446414547032e-16 6.60446414558269e-16
-3.75622517313046e-28 4.75200415120391e-16 6.76750690850753e-16 4.75200415120676e-16 6.60446414547032e-16 22.7889058760098 6.60446414525035e-16
-8.22675090325573e-17 4.75200415129057e-16 6.76750690862163e-16 4.75200415129343e-16 6.60446414558269e-16 6.60446414525035e-16 22.7889058761028

in this paper, y11 = y1y1, y12 = y1y2,
my code is:
clear;
fs = 100;
tt = 0:1/fs:1;
I = sin(2pi10tt);
Q = cos(2pi10tt);
A = [];
at = [];
for i = 1:length(I)
at = [I(i)*I(i) I(i)*Q(i) Q(i)*Q(i) I(i) Q(i) 1];
A = [A
at];
end
m=length(I);

cvx_begin
variable X(7,7)
% variable y(7)
% y(1) = 1;
% X = y*y’;
% X(1,3)-X(3,1) == 0;
% X(2,2) - X(1,2)*X(2,1) == 0;
% square(X(1,2)) - X(2,2) == 0; %this is some test for X(2,2) = X(1,2)X(1,2)
X(1,1) == 1;
X(3,3)-4X(2,4) <= 0;
X == semidefinite(7);

r = A*X(2:7,1)
minimize norm(r,1)
cvx_end

so the result is not show the y11 = y1y1, y12 = y1y2, and so on,

As @mcg pointed out to you in your previous thread on the same topic Help for SDP problem , your additional constraints y11 == y1y1, y12 == y1y2 are not convex.

Doesn’t the paper show that a relaxation of the original problem is a convex SDP? So not all constraints of the original problem are necessarily satisfied?

Maybe you need to contact the authors of the paper and seek their guidance.

Note: You really shouldn’t make this a new thread. If you have additional information or specific questions, add them to the previous thread.

ok i will contact the author of the paper, i create the new thread. because no new reply for long time, i think maybe something is wrong, i will mind next time.