I list my codes, could you please help me figure out the problems? Thanks

Dear experts, I have some problems when I use CVX. I list relevant codes in the following. Could you please help me figure out the problems? Thanks for your time.

  1. Code 1

    cvx_begin
    variables x y z r;
    rdr=-rel_entr(1-2z,1-2z+ytheta1);
    rbd=z
    theta2;
    rdc=-rel_entr(z,z+xtheta2);
    maximize (w1
    r+w2*rdr);
    subject to
    x+y<=z;
    r<=rbd;
    r<=rdc;
    x>0;
    y>0;
    z>0;
    z<0.5;
    cvx_end

In this code, x,y,z and r are variables, and theta1, theta2 and theta3 are constants. I can run this code successfully.

  1. Code 2

     cvx_begin
     variables x y m n p q R;  
     R10=-rel_entr(y,y+x*rho10);
     R12=-rel_entr(y,y+x*rho12);
     R20=-rel_entr(m,m+p*rho2);
     R2=-rel_entr(n,n+q*rho2);
     R=min(R10+R20,R12);
     maximize (w1*R+w2*R2);
     subject to
     x+y+m+n<=1;
     p+q<=x;
     R<=R12;
     R<=R10+R20;
     x>=0;
     y>=0;
     m>=0;
     n>=0;
     p>=0;
     q>=0;
     cvx_end
    

In this code, x, y, m, n, p, q and R are variables, and rho10, rho12 and rho2 are constants. When I run this code, the notice is listed as follows:
cvxprob/newcnstr (line 192)
Disciplined convex programming error:
Invalid constraint: {concave} <= {concave}

<= (line 21)
b = newcnstr( evalin( ‘caller’, ‘cvx_problem’, ‘[]’ ), x, y, ‘<=’ );

bishe (line 39)
R<=R12;

In summary, I can confirm these two mathematical problems are both convex optimization problems. Moreover, they are same in essence. Could youo please help me correct the second code?

R<=R12; and R<=R10+R20; are both {concave} <= {concave} .

That is because R10, R12, R20 are all concave. R is the min of concave arguments, so is also concave.

The constraint R<=R12; must always hold, and therefore can be eliminated from the problem formulation. The same with R<=R10+R20;. In both cases, these constraints are of the form: min of {something} and {something else} <= {something}., so of course are always satisfied.

So you’re right. Code 2 is a convex optimization problem, or in any event is equivalent to the convex optimization problem obtained by eliminating the two vacuous constraints.

As to what problem(s) you actually want to solve, I will not hazard a guess.

Thanks for your kind reply. Yes, this problem is a convex optimization problem. Is there any method to make this code (Code 2) successfully perform?

As I wrote, if those two constrains are eliminated, the problem will be accepted by CVX, and is mathematically equivalent to your code. II believe it will, however, be unbounded. So likely not a useful code without some modifications.

Yes. Thanks for your time.