Minimize log(1+1/x) where 0<x<inf

Hi

I am regenerating the results of a paper published in IEEE transactions in wireless communications where they used CVX to solve a convex optimization problem. They are minimizing the following function

Minimize

log_2 (1+\frac{\alpha}{b^H X b +1})+ \lambda. tr(X)
S.T.:
X is a postive semi-definite matrix (covariance matrix).

where \alpha and \lambda are constant scalars. and b is a constant vector.

The above problem is convex. Since X is positive semi-definite, therefore b^H X b > 0.

I tried many ways to formulate this problem using CVX. However, none of them succeeded. e.g.

cvx_begin
variable X(N,N) complex semidefinite
expression summation
summation=summation+log(1+alphainv_pos(b’Xb+1))/log(2)
minimize(summation+lmda
trace(X) )
cvx_end

But I obtain the following error

Error using cvx/log (line 64)
Disciplined convex programming error:
Illegal operation: log( {convex} ).

The authors mentioned that they used CVX to solve this problem. So, I am sure it can be implemented using CVX. Can someone help?

Thanks

Can you reformulate in terms of minimizing

-log(concave function)

for some concave function? Note that to follow CVX’s rules, the argument of log must be a concave function. Then the log will be a concave function.

Note that your title question is easily handled as
minimize(-log(x))

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Thank you Mark for your quick response.

I tried a lot to formulate it in the form of -log (concave fn.) but unfortunately I couldn’t.

The authors must have found some work around to make CVX accept the formulation and that’s what I am trying to figure out.

Sorry for the title. It should have been log (1+1/x). I have updated that.

May be you should try to minimize -log(x/(1+x))

@Marc , I don’t see you to get -log(x/(1+x)) accepted by CVX. Specifically, x/(1+x) is indeed concave for x > 0, but I don’t see how to get it accepted.

@mohamedmarzban ,log(1+1/x) is convex for x > 0, but I don’'t know how it can be entered in CVX, and suspect it can not. Perhaps you can email the article authors and inquire how they used CVX.

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Thanks again Mark for your reply.

Actually, I emailed the first author but he did not answer. Maybe I will email the second author.

Thanks.

You should reach out to the authors to find out how they formulated their problem in CVX.

Oops, sorry, I didn’t scroll down enough. I see you’ve already reached out to them.

Frankly, while I greatly appreciate all of the citations for CVX in academic papers, I find that this kind of situation happens quite often—authors do not publish the models they use, so there’s no way to reproduce their work.

Sorry for the late answer. May be you can rewrite your problem in the following form

minimize y
such that log(1 + 1/x) \leq y

which is equivalent to

minimize y
such that exp(-y) + exp(-y-log(x)) \leq 1

The following code is working at least

cvx_begin
variables x y
minimize( y )
exp(-y) + exp(-y - log(x) ) <= 1
cvx_end

This should work :slight_smile:
log(1+1/x) = log((1+x)/x) = -log(x/(1+x)) = -log(1 - 1/(1+x)) = -log(1 - inv_pos(1+x))

1 Like

Indeed, @Dexin_Wang’s solution works. I then reformulated to use rel_entr , per 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 so as to invoke CVXQUAD, and received the following error, in CVX 2.1, which seems like a bug to me

cvx_begin
variable x
rel_entr(1,1 - inv_pos(1+x))
=====================================
Using Pade approximation for exponential
cone with parameters m=3, k=3
=====================================
Error using cvxprob/newcnstr (line 192)
Disciplined convex programming error:
   Invalid constraint: {concave} == {real affine}

Error in cvxprob/newcnstr (line 72)
            newcnstr( prob, x{k}, y{k}, op );

Error in  ==  (line 3)
b = newcnstr( evalin( 'caller', 'cvx_problem', '[]' ), x, y, '==' );

Error in cvx/rel_entr (line 80)
                { -q, xt, yt } == exponential( sz ); %#ok

However, in Writing x*log(1+x/y) ,the wizard of conic reformulation,@Michal_Adamaszek
produced the result
x*log(1+x/y) = rel_entr(x+y,y) + rel_entr(y,x+y)

Setting x = 1, then Interchanging the roles of x and y so as to match the problem in this thread, results in
log(1 + 1/x) = rel_entr(x,x+1) + rel_entr(x+1,x)
The RHS is accepted by CVX, and as a bonus, is already in a form needed by CVXQUAD. It executes without error.

And you can see how to formulate the matrix analog using CVXQUAD’s quantum_rel_entr at Perspective of log det function, and CVX formulations of related log det problems using Quantum Relative Entropy from CVXQUAD .

1 Like

Hello Dexin,

When I use your formulation, I get the following error:

Error using cvxprob/newobj (line 57)
Disciplined convex programming error:
Cannot maximize a(n) convex expression.

Error in maximize (line 21)
newobj( prob, ‘maximize’, x );

I am trying to maximize
log(1+1/x) = log((1+x)/x) = -log(x/(1+x)) = -log(1 - 1/(1+x)) = -log(1 - inv_pos(1+x))
but I get the above error.

log(1+1/x) is convex. Therefore, using one of the reformulations shown in the preceding posts, it can be minimized.in CVX. But it can’t be maximized in CVX, because that would be a non-convex optimization problem.