Why in the content of cvx, subs() does not work? - matlab

Question

TLDR
I wrote a code to solve a SDP problem. I used subs to change syms into numerical value before cvx content and I did see the numerical value after I printed out the variables. But whos() did show that the variable is still syms type which I think might be the key point where I went wrong. The error message shows like this:
Unable to convert 'cvx' to 'sym'.
It went wrong at the line trace(rhot*X_ele(:,:,i)) == value(i); where rhot is symbolic instead of numeric.
Details
I'm trying to write a SDP problem as follows:
function [res] = NH_SDP(rhot,p,value,t)
dim = size(rhot,1);
for i = 1:p
drhot(:,:,i) = diff(rhot,t(i));
end
drhot = subs(drhot,t,value);
rhot = subs(rhot,t,value);
rhott = kron(eye(p),rhot);
% cvx process
cvx_begin sdp
% cvx part
variable L_ele(dim,dim,(1+p)*p/2) hermitian;
variable X_ele(dim,dim,p) hermitian;
cnt = 1;
for i = 1:p
for j = i:p
if i == j
L((i-1)*dim+1:i*dim,(j-1)*dim+1:j*dim) = L_ele(:,:,cnt);
cnt = cnt + 1;
else
L((i-1)*dim+1:i*dim,(j-1)*dim+1:j*dim) = L_ele(:,:,cnt);
L((j-1)*dim+1:j*dim,(i-1)*dim+1:i*dim) = L_ele(:,:,cnt);
cnt = cnt + 1;
end
end
end
for i = 1:p
X((i-1)*dim+1:i*dim,1:dim) = X_ele(:,:,i);
end
for i = 1:p
X2(1:dim,(i-1)*dim+1:i*dim) = X_ele(:,:,i);
end
% rhott = [[0, 0, 0, 0, 0, 0, 0, 0]
% [0, 1/2, 7/20, 0, 0, 0, 0, 0]
% [0, 7/20, 1/2, 0, 0, 0, 0, 0]
% [0, 0, 0, 0, 0, 0, 0, 0]
% [0, 0, 0, 0, 0, 0, 0, 0]
% [0, 0, 0, 0, 0, 1/2, 7/20, 0]
% [0, 0, 0, 0, 0, 7/20, 1/2, 0]
% [0, 0, 0, 0, 0, 0, 0, 0]];
rhott = double(rhott);
rhot = double(rhot);
drhot = double(drhot);
minimize(trace(rhott*L));
subject to
% rhot = [0 0 0 0;0 1/2 7/20 0;0 7/20 1/2 0;0 0 0 0];
% drhot
for i =1:p
trace(rhot*X_ele(:,:,i)) == value(i);
end
for i = 1:p
for j = 1:p
if i == j
trace(drhot(:,:,i)*X_ele(:,:,j)) == 1;
else
trace(drhot(:,:,i)*X_ele(:,:,j)) == 0;
end
end
end
% semidefinit constraint.
[L X; X2 eye(dim)] >= 0;
cvx_end
res = cvx_optval;
X_ele
end
There are three line in the above code:
rhott = double(rhott);
rhot = double(rhot);
drhot = double(drhot);
If I delete those three lines, there will be error Unable to convert 'cvx' to 'sym'. while my rhot have already changed into numerical value by subs function and I print it out to see numerical numbers even though whos function show that the type of rhot is still syms.

Answer 1

To cite the documentation on subs():
snew = subs(s,old,new) returns a copy of s, replacing all occurrences
of old with new, and then evaluates s. Here, s is an expression of
symbolic scalar variables or a symbolic function, and old specifies
the symbolic scalar variables or symbolic function to be substituted.
emphasis mine. In other words: subs() changes your symbolic expression to a new symbolic expression.
double() on the other hand, converts whatever you feed it to a type double, i.e. a numeric value. rhott thus initially contains a symbolic value, even if that is a number. double(rhott) then evaluates it to a numeric number, rather than a symbolic one.
I presume the problem is here
drhot = subs(drhot,t,value);
rhot = subs(rhot,t,value);
rhott = kron(eye(p),rhot);
You're substituting all occurrences of t within drhot with value, and then again on rhot. if drhot is already symbolic, i.e. if the inputs rhot, value, t are symbolic, this makes sense and indeed a double call will make your system numeric, rather than symbolic. If, on the other hand, all inputs are numeric, subs() unnecessarily converts everything to symbolic variables, slowing execution speed.