Solver not applicable (mosek does not support signomial constraints)

[Kali]

Hi, 

My objective function is as follows:

max  \sum_{k=1}^K     log(   1   +   2*real(a_k * x' * v_k)  -   (b_k)   -   (c_k * x' * v_k * v_k' * x)   )

The constraints can be reformulated as cone(x,1).

The dimensions are as follows: 

x     = Nx1 complex vector (unknown)

v_k = N x 1 vector (known)

a_k = scalar (known)

b_k = scalar (known)

c_k = scalar (known)

The constraints are second-order cone constraints and the objective is non-linear. So, I reformulated the objective as an exponential cone programming problem using "https://yalmip.github.io/tutorial/exponentialcone/". To be more clear, I used "logsumexp" by considering the objective as follows:

max  \sum_{k=1}^K     log(   e^0   +   e^log(2*real(a_k * x' * v_k))  -   e^log((b_k))   -  e^log( (c_k * x' * v_k * v_k' * x))   )

I am trying to use mosek as the solver, but it is throwing the following error: "Warning: Solver not applicable (mosek does not support signomial constraints)". 

Please advice!

Thanks in advance.

[Johan Löfberg]

you would have to supply a reproducible example

[Johan Löfberg]

and it a bit weird as you speak of maxmization, but then you say you use the logsumexp operator which is convex, i.e. that should not be possible at all using mosek since it isn't convex problem

tisdag 9 mars 2021 kl. 14:28:11 UTC+1 skrev Kali:

[Kali]

Attaching the working code and data required. 

[Johan Löfberg]

As you have written it, it is very far from being exponential-cone representable

The data does not match the desrciption above (A instead of v and c), but if it is the case that you have a sum of logs of concave quadratics Qi(x), i.e that c_k in your desciption is positive, you should write it as

maximize sum log t

t <= Qi(x)

that will be exponential-cone representable

[Kali]

Thank you. I will implement this. 

[Kali]

I tried implementing this, but some unknown error is popping up. I also tried the following:

max log(t)

where t <= product(all the quadratics)

This is also not working 

[Kali]

Can I use non-linear solvers like fmincon? or is it better to reformulate the objective appropriately and use mosek?

[Johan Löfberg]

No idea what you are trying to do, impossible to know without reproducible code

The objective is -sum(log(t))

with constraints t(i) <= Q(x) and everything else

[Johan Löfberg]

Of course (although you should not use the cone operator then, better to simply write the quadratic constraints since YALMIP otherwise has to do various ugly stuff to handle the non-smooth cone operator when interfacing a standard local nonlinear solver)

Whether a standard nonlinear solver or mosek with exponential cone performs best is impossible to know before testing

[Kali]

Thank you. This was very helpful. 

[Kali]

Hi,

Attaching the objective function and the reproducible code. I tried different approaches, but I am unable to understand where I'm going wrong. Please le me know if you need additional information.

Thank you in advance!

[Johan Löfberg]

As I said, you will have to model the stuff manually and replace all log(quadratic) with new variables log(t) and epigraph constraint t<=quadratic

also, you seem to have numerical issues as the term aa-bb which you probably think is real, isn't due to floating-point numerics

>> aa - bb

Quadratic scalar (complex, 16 variables)

Coefficients range: 1.0588e-22 to 0.99825

hence you will have to use real(aa-bb) when creating your epigraphs constraints

[Kali]

Thank you very much!! It is working now.

[Kali]

Hi, 

I have another question regarding this optimization problem. What happens when the same objective function is modified a little and the sqrt(log(quadratic)) is to be maximized? Can the same method be applied or will it make the problem non-linear?

Thanks in advance!

[Johan Löfberg]

sqrt is socp representable so    t<= q , s <= log(t)  and then maximize sqrt(s) should work

[Kali]

Thank you. It is working.