Help in constraint expression

[Abdulrahman Aldeek]

Hello Johan

I'm trying to write the following constraint

sqrt(4*((PS(1:NT,1).^2)+(QS(1:NT,1).^2))+(Isqr(1:NT,1)-diag(W(TB(1:NT,1),TB(1:NT,1)))).^2)<=(Isqr(1:NT,1)+diag(W(TB(1:NT,1),TB(1:NT,1))))

in a way that would allow the use of the option "dualize", how can I do that?

[Johan Löfberg]

you will have to make it less specific to get rid of the irrelevant details

it simply looks like norm(x)  <= y which is a simple conic constraint cone([y;x])

(although the model does not appear to make sense as it looks like a vector on the right-hand side (you might mean norm(x)<=y with y vector meaning norm(x)<=y(i) 

[Abdulrahman Aldeek]

I'm trying to write

Wjj*Iij>PSij^2+QSij^2 in a factorized way. Wjj is the diagonal value of the PSD matrix W where j is the "to bus" index. This constraint should be written for each line (that has "from bus i" and "to bus j"). I tried to write it as norm([2PS;2QS;I-W])<=I+W but it didn't work. Can I write this constraint is a more proper way that can be used upon invoking the command "dualize"?

[Johan Löfberg]

you already have an socp-representation of that in some other post

it will never be a cone which benefits from dualization since it isn't a pure cone with equalities. Sure if the complete model is a massively simple primal conic + a few of these dualization might be beneficial, like

X = sdpvar(100,100);

Model = [X >= 0, X(1)==1, norm([X(3);X(100)]) <= X(7878)]

here the massive simple primal sdp conic model will benefit from dualization, but the non-primal socp wil have to be rewritten first but that is so little cost compared to gains in the SDP cone, hence the model YALMIP derives before dualization is

Model = [X >= 0, X(1)==1, cone(z), z == [X(7878);X(3);X(100)]]

i.e. it adds three equalities to the intended primal before dualization, meaning it effectively trades O(100^2) dual variables for 3 dual variables

[Abdulrahman Aldeek]

I'm sorry if my questions are trivial but I'm very recent to SDP. When you say:

"since it isn't a pure cone with equalities"

Does that mean that I should replace <= by ==??

Also, I have been noticing in papers that deal with SDP that the computational efficiency of SDP is O(n^3) by I don't understand what is the exact meaning of this notation. SDP is convex and should be solved in polynomial time but this notation is exponential, Why is that?

[Johan Löfberg]

Intending to use the command without understanding the (few) situations where it is relevant, and the underlying complexity theory is just not going to work. I do presume you have read the associated paper and understood it

https://yalmip.github.io/tutorial/automaticdualization

O(n^3) is not exponential. It is n*n*n, cubic.

[Johan Löfberg]

and you cannot say complexity is O(n^3) since you haven't even defined what n is but more importantly it depends on more things than just one number. If the largest semidefinite constraint has dimension n, then a lower bound on complexity is O(n^3) since you regardless of everything else will have to do O(n^3) linear algebra along the solution process