I noticed that in CVX, I can write my function \tau_1 \ log \left( 1 + \frac{ \mathbf{A_i} \tau_0 }{ c_i \tau_1 + \mathbf {B_i} \tau_0 } \right) as
- rel_{-} \operatorname{entr} \left( \tau_1+ \frac{ \mathbf{B_i} }{c_i} \tau_0 , \tau_1+ \frac{ \mathbf {B_i} }{c_i} \tau_0 + \frac{ \mathbf{A_i}}{ c_i} \tau_0 \right) -rel_{-} \operatorname{entr}( \frac{ \mathbf {B_i} }{ \mathbf {A_i} } \tau_1 + \frac{ \mathbf {B_i}^2}{ \mathbf {A_i} c_i} \tau_0, \frac{ \mathbf {B_i}}{ \mathbf {A_i}} \tau_1 + \frac{ \mathbf {B_i}^2}{ \mathbf {A_i} c_i} \tau_0+\frac{ \mathbf {B_i}}{ c_i} \tau_0)
- rel_{-} \operatorname{entr}( \frac{ \mathbf {B_i}}{ \mathbf {A_i}} \tau_1 + \frac{ \mathbf {B_i}^2}{ \mathbf {A_i} c_i} \tau_0+ \frac{ \mathbf {B_i} }{ c_i}\tau_0, \frac{ \mathbf{B_i} }{ \mathbf{A_i}} \tau_1 + \frac{ \mathbf{B_i}^2}{ \mathbf{A_i} c_i} \tau_0) .
Then, I wrote the following code for my optimization problem.
for i=1:K
cvx_begin
variable tau0 nonnegative
tau1=1-tau0;
R1=-rel_entr(tau1+ ((B(i)./c(i)) .* tau0 ), tau1+((B(i)+A(i))./c(i)) .* tau0 );
R2=-rel_entr( (B(i) ./ A(i)).*tau1 + (B(i)^2 ./( A(i).*c(i) ) ).*tau0, ...
(B(i) ./ A(i)).*tau1 + (B(i)^2 ./( A(i).*c(i) ) ).*tau0+ (B(i) ./ c(i)).*tau0) ;
R3=-rel_entr((B(i) ./ A(i)).*tau1 + (B(i)^2 ./( A(i).*c(i) ) ).*tau0+ (B(i) ./ c(i)).*tau0,...
(B(i) ./ A(i)).*tau1 + (B(i)^2 ./( A(i).*c(i) ) ).*tau0) ;
maximize (tau1.*(R1+R2+R3));
subject to
tau0<=1;
cvx_end
tau0_opt(i)=tau0;
end
Expressions R1, R2, R3, and R4 in my code are concave expressions. Hence, the objective function is a product of an affine and concave expression. Thus, when running this code, I am getting an error which is as follows.
Error using .* (line 173)
Disciplined convex programming error:
Cannot perform the operation: {real affine} .* {concave}
How to modify my code to handle this issue in CVX? Help required. Thanks in advance.