How do we solve the expression log(1/(1+x))?

1

Hi,all.
I want to write the expression log(1/(1+x)),but there is a error: ‘Disciplined convex programming error:Illegal operation: rel_entr( {convex}, {convex} ).’

My code is shown as follows,
function [a,Theta,theta]= opt_Theta(w,W,lambda,Pmax,Rmin,P1)
load(‘channel.mat’,‘K’,‘N’,‘M’,‘ite’,‘pd’,‘ps’,‘pde’,‘pse’,‘Hbe’,‘Hd’,‘Hd_w’,‘Hbe_w’,‘theta_init’,‘AP_angle’,‘IRS_angle’,…‘G_sig’,‘User_angle’,‘Eve_angle’,‘Hie_sig’,‘Hr_sig’,‘eb1’,‘eb2’,‘path_du’,‘path_iu’,‘path_de’,‘path_ie’);
ite=100;
sigma_e = db2pow(-200);%噪声功率
sigma_u = db2pow(-200);
N1=1/sigma_u;
N2=1/sigma_e;

      cvx_begin quiet
        %variable Theta(N,N) complex
        variable theta(1,N)            
            for t0=1:ite
                  Hd=pd*Hd_w(:,:,t0);
                  G=channel_G(AP_angle,IRS_angle,G_sig(:,:,t0),eb1,eb2,N,M);
                  Hr=ps*channel_Hr(User_angle,Hr_sig(:,:,t0),eb1,eb2,K,N);
                  Theta = diag(exp(1j*theta));
                  H=Hd+Hr*Theta*G; 
                       Rateu=0; 
                       for k0=1:K
                            P(k0)=norm(H(k0,:)*w);                           
                             sinr(k0)=norm(H(k0,:)*w)*N1;                                                                                                                                             
                             y=sinr(k0);                               
                             Rateu=Rateu+rel_entr(1+y,y)+rel_entr(y,1+y)-rel_entr(1,y);
                       end
                       
            end
           
                 for t0=1:ite
                  Hbe=pde*Hbe_w(:,:,t0);
                  G=channel_G(AP_angle,IRS_angle,G_sig(:,:,t0),eb1,eb2,N,M);
                  Hie=pse*channel_Hie(Eve_angle,Hie_sig(:,:,t0),eb1,eb2,1,N);
                  Theta = diag(exp(1j*theta));
                  H_e=Hbe+Hie*Theta*G;
                        for k0=1:1
                             P=norm(H_e(k0,:)*w);
                             sinr(k0)=P*N2;
                             x=sinr(k0); 
                             Ratee=rel_entr(1+x,x)+rel_entr(x,1+x)-rel_entr(1,x); 

                        end
                 end

          minimize (-(Rateu-Ratee-lambda*(P1+(w'*w))))
      subject to
          Rateu-Ratee>=Rmin;
          for n=1:1:N
              0<=theta(:,n)<2*pi;
          end
      cvx_end     
      Theta = diag(exp(1j*theta));

end
Who can explain the problem?Thank you.

3

Reformulate log(1/(1+x)) as -log(1+x), which is convex. So you need to use it is a convex way.

4

The error you are seeing is related to the use of the rel_entr function in your optimization problem. The rel_entr function computes the relative entropy of two vectors, which is only defined when both vectors are positive and their entries sum up to one. In your case, it seems that this assumption is not satisfied, which leads to the error you are seeing.

Based on your code, it seems that the rel_entr function is used to compute the rate of a communication system. If this is the case, you may want to consider using a different expression for the rate that does not involve the rel_entr function. One possibility is to use the Shannon capacity formula, which is a well-known expression for the maximum rate at which information can be transmitted over a communication channel. Alternatively, you could try to reformulate your problem so that the rel_entr function is not needed, or use a different optimization solver that can handle this function.

5

@ToneyWulff got placed in the moderator queue and was apparently written before my post above.

I am now a little confused as to what @Lenka wants to do. One or more of these formulas hopefully covers what you want to do:

log(1/(1+x)) = -log(1+x) = rel_entr(1,1+x)

log(1+1/x) = rel_entr(x,x+1) + rel_entr(x+1,x)