How can I write this kind of constraint in cvx


(Dixiao Wu) #1

In my optimal problem I have two constraints as following
image
Q, t_k, l are all my variables. It’s a convex constraint but it can’t just solve for CVX.
(t3(1))sigma0/h(1)(power(2,beta*l(1)/B/(t3(1)))-1)+xi(1)*C(1)pow_pos(l(1),3)/(t2(1))^2 <= Talpha(1)real(trace(QH)). In this formula, l(1),t2(1),t3(1) and Q four variables. I want to know how to write this kind of constraint in cvx. Help needed. Thanks a lot.


(Mark L. Stone) #2

Re-write 2^x as exp(log(2)*x)

Then use the exponential cone construct:
y*exp(y/x) <= z, with y > 0
can be written in CVX as
{ x, y, z } == exponential(1)

or alternatively,
x + rel_entr(y,z) <= 0

Declare z as a CVX variable, replace y*exp(y/x) by z, and add one of the above constraints.

Either way, CVXQUAD can be used without modification to your program, and is recommended for your problem 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 .


How to express "t*e^(x/t)-t" in CVX?
(Dixiao Wu) #3

Thanks a lot, Mark~:grinning:
But I want to ask about the if the formula with variables l_k and t_k like
image can be written in the CVX?


(Mark L. Stone) #4

That doesn’t look convex to me. Why isn’t CVX accepting my model? READ THIS FIRST! You claim that constraint is convex, so show us the proof.


(Dixiao Wu) #5

Hi, Mark~
Can we made this formula as a perspective function as following
image


(Mark L. Stone) #6

You didn’t tell us before that l > 0. The determinant of the Hessian = 0, therefore at least one of the two eigenvalues is always zero. Examination of the trace of the Hessian shows that the function is convex for l >= 0 and certain combinations of t and l < 0; and is concave everywhere else.The Hessian is never indefinite.

Given that the function is not convex over its entire natural domain, and is convex only for certain combinations of t and l < 0, I am not optimistic you can get it into CVX for the entirety of its convex region. Perhaps you or another forum reader can determine whether there is a formulation which can be entered into CVX with the constraint that l >= 0 .


(Michal Adamaszek) #7

Surely something like (n+1)-power-over-n-power should be representable by iterating quad-over-lins. It corresponds to a power cone. Homogenizing an example from https://docs.mosek.com/modeling-cookbook/cqo.html#simple-sets-involving-power-functions we have that xy^2\geq z^3 with x,y,z\geq 0 is equivalent to sy\geq z^2,\ xz\geq s^2.


(Dixiao Wu) #8

I’m so sorry that I just saw the answer. Thanks a lot.:grinning:


(Dixiao Wu) #9

:grinning:Thanks very much for your answer and also apologize for that I miss the answer before…


#10

Thanks Dixiao.
It is true that x^3/t^2<=z can be equivalently turned into yt>=x^2, zx>=y^2 according to the topic you linked, and we can represent x^3/t^2 with z by introducing another two variables z and y and adding the two constraints. However, the two constraints yt>=x^2, zx>=y^2 are not convex and can not be used in CVX. By doing that, I have the following error:
%%%%%%%%%%%%%%%%%%%%%%%
Error using .* (line 262)
Disciplined convex programming error:
Invalid quadratic form(s): not a square.

Error in * (line 36)
z = feval( oper, x, y );

Error in Sample_Trajectory_CVX (line 45)
x2^2-y2*t2<=0;
%%%%%%%%%%%%%%%%%%%%%%

Do you know how to solve this error?


(Michal Adamaszek) #11

You have to use lorentz_cone, or whatever it is called in cvx, rather than write the non-convex version of these constraints.


#12

Thanks so much. I think the CVX function ‘rotated_lorentz(n)’ works for this case.