If there anyone know how to represent this expression in CVX？Only B_k is variable . Thanks advance！
-rel_entr(Bk,Bk+p*hk/N0)/log(2) >= Rmin
x*log(1+y/x) can be formulated as
x is affine and
y is concave (note that a numerical constant is concave).
Thanks for your help. I am very sorry that there is another problem :is that can achieve in CVX? Only B_k is variable too. And constraint of B_k is :0<=B_k <=B_max(a constant)
You can apply
inv_pos, because it accepts a concave argument.
minimize(Lk*inv_pos(-rel_entr(Bk,Bk+p*hk/N0)/log(2))) 0 <= Bk <= B_max
inv_pos Reciprocal of a positive quantity.
inv_pos(X) returns 1./X if X is positive, and +Inf otherwise.
X must be real.
For matrices and N-D arrays, the function is applied to each element. Disciplined convex programming information: inv_pos is convex and nonincreasing; therefore, when used in CVX specifications, its argument must be concave (or affine).
You may benefit from a careful reading of the CVX Users’ Guide http://cvxr.com/cvx/doc/ , as well as the Mosek Modeling Cookbook https://docs.mosek.com/modeling-cookbook/index.html
Hello @Mark_L_Stone , is this approach still applicable when you have A+N_0B_k in the log function instead of N_0B_k? A is constant. Thanks in advance.
Presuming A>= 0, the same basic idea can be used, with some extra complication: The expression after the lattermost
= can be entered in CVX
x*log(1+y/(x+A)) = (x+A)*log(1+y/(x+A)) - A*log(1+y/(x+A)) = -rel_entr(x+A,x+A+y) - A*(rel_entr((x+A)/y,(x+A)/y+1) + rel_entr((x+A)/y+1,(x+A)/y))
If x and y are both variables, the decomposition in the first line does not result in a formulation which can be entered in CVX, although it is still mathematically correct. That is because
A*log(1+y/(x+A) is neither convex nor concave jointly in x; and y, and therefore can;'t be reformulated for CVX, even though
x*log(1+y/(x+A)) is jointly concave in x and y.
And of course you can apply
inv_pos to this, if you want the reciprocal of this expression. And that would be convex.