Hello, I am trying to solve the following problem by cvx:

`Cij=aij*(qi-log2(∑(l≠i)2^ql* gli))+ aij* log2((M+N-1)/N* gij)`

M, q are the optimization variable that M is real variable and q(1,4).

By defining these variables the second part cij in cvx implementation is " cvx real affine expression (scalar)" .the type of M variable what should be that the second part of Cij be " cvx log-convex expression "?

# Illegal operation: rel_entr( {positive constant}, {log-convex} )

**rana_sedghi**(Rana) #1

**Mark_L_Stone**(Mark L. Stone) #2

Can you please provide a complete reproducible example, complete with all code and input data?

Have you proven this is convex or concave?

help rel_entr

rel_entr Scalar relative entropy.

rel_entr(X,Y) returns an array of the same size as X+Y with the

relative entropy function applied to each element:

{ X.*LOG(X./Y) if X > 0 & Y > 0,

rel_entr(X,Y) = { 0 if X == 0 & Y >= 0,

{ +Inf otherwise.

X and Y must either be the same size, or one must be a scalar. If X and

Y are vectors, then SUM(rel_entr(X,Y)) returns their relative entropy.

If they are PDFs (that is, if X>=0, Y>=0, SUM(X)==1, SUM(Y)==1) then

this is equal to their Kullback-Liebler divergence SUM(KL_DIV(X,Y)).

-SUM(rel_entr(X,1)) returns the entropy of X.`Disciplined convex programming information: rel_entr(X,Y) is convex in both X and Y, nonmonotonic in X, and nonincreasing in Y. Thus when used in CVX expressions, X must be real and affine and Y must be concave. The use of rel_entr(X,Y) in an objective or constraint will effectively constrain both X and Y to be nonnegative, hence there is no need to add additional constraints X >= 0 or Y >= 0 to enforce this.`

So as you can see, the 2nd argument of `rel_entr`

must be concave (or affine, which is a special case of concave), not log-convex as per your error message.

**rana_sedghi**(Rana) #3

very thanks Mark,

the original function that i want simulate is:

where

where

and , 0<yij<1, xij={1 if user j is associated with BS i, 0 otherwise} , gij is positive number.

Due to the represents of the q variable in the numerator and the denominator of fraction , also the multiplication of the M variable is clear that it is not convex. In the appendix section mentioned,also:

.

finally my cvx code is:

cvx_begin

variable q(1,Kt)

variable M

r_ij=zeros(Kt,Kr);

for e=1:Kr

for d=1:Kt

int=setdiff(Basic,d);

temp=2.^q(int)*lossovernoise(int,e);
temp=temp+sigma2;
if d==1
di=N; ei=(M+N-1)/N;
else
di=1; ei=1;
end
Cij=a0_ij(d,e)*(q(d)-(-rel_entr(1,temp))/0.6931)+a0_ij(d,e)

*(-rel_entr(1,(ei*lossovernoise(d,e))))/0.6931+b0_ij(d,e);

r_ij(d,e)=di*Cij;

end

end

[Part_I,Part_II]=algorithm_one(PMax,UMin,UMax,Pa,epsilon,M,lambda,w,mu,r_ij,X_1,Y_1,Kr,Kt,q);

maximaize(Part_I-Part_II)

subject to

M<=Mmax

for c=1:Kt

2^q©<=Pi©

end

for tt=1:Kt

sum(X_1(tt,:).*Y_1(tt,:).*r_ij(tt,:))<=Ci_BH(tt)

end

cvx_end

Error using cvx/rel_entr (line 71)

Disciplined convex programming error:

Illegal operation: rel_entr( {positive constant}, {log-convex} ).

Error in mainpart2017 (line 115)

Cij=a0_ij(d,e)*(q(d)-(-rel_entr(1,temp))/0.6931)+a0_ij(d,e)*(-rel_entr(1,(ei*lossovernoise(d,e))))/0.6931+b0_ij(d,e);

i have two another question ,too:

1: rel_entr can be used for any logarithmic function?

2:The ln function is implemented in cvx same as the logarithm, with this difference that it multiplied in 2.303 value?

**Mark_L_Stone**(Mark L. Stone) #4

It appears that you have copied a problem out of the paper https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=7990198 without understanding it.

Apparently, you are trying to solve the paper’s problem (33), which is proven to be a convex optimization problem in the paper’s Appendix B. The proof of convexity in that appendix provides a road map for how to formulate the problem in CVX, with use of `log_sum_exp`

being the key.

You will need to read the paper more carefully in order to understand how the solution to problem (33) fits into a larger algorithm (algorithm 1) for solving a higher level non-convex problem of interest.

Your use of `rel_ent`

r is as a substitute for `log`

, as described in CVXQUAD: How to use CVXQUAD's Pade Approximant instead of CVX's unreliable Successive Approximation for GP mode, log, exp, entr, rel_entr, kl_div, log_det, det_rootn, exponential cone. CVXQUAD's Quantum (Matrix) Entropy & Matrix Log related functions . In your case, I don’t recommend distracting yourself with that until you get a formulation which is accepted by CVX. Also niote that instances of `log`

occurring in what will need to be handled via `log_sum_exp`

, should indeed be handled by l`og_sum_exp`

, without use of `rel_entr`

.

Also note that terms of the form `2^x`

will need to replaced by `exp(log(2)*x)`

when determining the argument for `log_sum_exp`

.

**rana_sedghi**(Rana) #5

exactly, i want simulate this paper as a rival for my proposed method. i apply your recommendation about log_sum_exp.thank you so much for helping me.

**rana_sedghi**(Rana) #6

Dear mark, the problem in the paper’s Appendix B is solved with log_sum_exp and output of the problem(51) was concave.Now my problem is that in problem(33) how to calculate the logarithm of a concave term while log_sum_exp and rel_enter can not be used?

**Mark_L_Stone**(Mark L. Stone) #7

`log_sum_exp`

is convex, so its negative is concave. The problem basically is `log((-log_sum_exp)`

, which is log of a concave argument, which is concave and is allowed by CVX. That’s why I wrote that Appendix B provides a road map for how to formulate the problem in CVX,

**rana_sedghi**(Rana) #8

I sincerely appreciate your guidance and time. I couldn’t handle this problem yet. How are constraints 2 and 3 possible({concave} <= {real constant})? And that how 2^q is written in cvx, is a particular function used?

**Mark_L_Stone**(Mark L. Stone) #9

What are constraints 2 and 3? Equations (2) and (3) in the paper are calculational formulas, not constraints.

`q`

should be declared a variable in CVX, and then you should be able to enter `2^q`

. For use in `log_sum_exp`

, you need to rewrite `2^q`

as `exp(log(2)*q`

), which puts it into the form of a standard exponential, which provides the argument you need for `log_sum_exp`

.

**Mark_L_Stone**(Mark L. Stone) #11

Constraint C6 can be entered " as is".

Constraint C7 can also be entered as is because it is affine because `SINR`

is treated as input (numerical) data for this problem. In the outer algorithm (outside of an individual CVX invocation), `SINR`

is adjusted based on the optimal values of `q`

and `M`

from the previous CVX invocation, and must be initialized before the first CVX invocation, as outlined in Algorithm 1 on the top right-hand side of p. 4725.

**rana_sedghi**(Rana) #12

So the exp(log(2)*x) is not necessary used? In C6 if 2^q entered as is, the output of my code is convex for this case while mentioned in the appendix that the second term of f˜1 (q, M) is also a concave function .

In the case of C7, the SINR of previous repetition do not used and calculated with the equation (32).are you agree with me?

**Mark_L_Stone**(Mark L. Stone) #13

`exp(log(2)*x)`

is needed to get the argument of `log_sum_exp`

, which is in the objective function.

For C7, I’ve now lost track of what the variable(s) are. What are x and y? If \tilde{r_{ij}} is calculated using initialized or updated SINR, then unless x or y are variables, what is? I will let you sort out the mess because I don’ have the energy to read the paper carefully enough.

**rana_sedghi**(Rana) #14

X and Y calculated via algorithm 2. According to algorithm 4, First, Algorithms 2 and 3 run then the algorithm 1. the optimization variables are M and q vector .

**Mark_L_Stone**(Mark L. Stone) #15

I leace you to sort through how the decision (optimization) variables`q`

and `M`

come into play in constraint C7. You need to determine an explicit formulation for the formulation of problem (33) in CVX.

**rana_sedghi**(Rana) #16

In the Appendix, it is also written that:

.

Was this what you wanted or not?

thank you so much.

**Mark_L_Stone**(Mark L. Stone) #17

I will leave you to sort through this and show an explicit formulation of a convex constraint for C7, which means that the left-hand side must be convex. If \tilde{r_{ij}} is concave, then the left-hand side would be concave, beccause x_{ij} and y_{ij} are \ge 0. So you need to ressolve that.