YALMIP and nullspace

[kirkrudolph]

Hi Johan,

Back again with a different optimization problem. I believe this one should be easier. Here I have 31 decision variables. The trouble I'm having is the cost function involves the null space of the decision variables and I'm getting an error trying to use the null function on the sdpvar types.

I didn't see null.m in the YALMIP/@sdpvar folder so I'm assuming it simply doesn't have an overload?

Problem Pseudo Code

% Decision Variables
r = sdpvar(31,1);

% Constant Weights
J = rand(21,1);

% Cost Function
Q = foo( r );                                 % Sparse [19 21] matrix
Objective = null( Q )'*diag( J )*null( Q );   % [2 21] x [21 21] x [21 2]

Troubleshooting

I verified Q is what I'd expect it to be:

>> Q
Linear matrix variable 19x21 (full, real, 31 variables)
Coeffiecient range: 1 to 1

Error Message

Check for missing argument or incorrect argument data type in call to function 'isfinite'.

Error in rat (line 57)
    tol = 1.e-6*norm(X(isfinite(X)),1);

Error in rref (line 26)
[num, den] = rat(A);

Error in null (line 50)
    [R,pivcol] = rref(A);

Error in YALMIP_Optimize (line 17)
null( Q, "rational" );

I tried using some other functions too

>> null(Q)
Error using svd
First input must be single or double.

Error in null (line 65)
    [~,S,V] = svd(A,0);

Thanks,
Kirk Rudolph

[johanlofberg]

Not sure what you expect to be returned from a null operator acting on a matrix involving decision variables

Just randomly, if

Q = [r1 r2 0;r2 r1 0]

then for any point in the solver iteration process where r1=r2 the result would be a 3x2 vector, and when r1 and r2 are different it is a 3x1 vector. There is no way to work with such an operator in a standard optimization framework or represent in YALMIP, as everything has defined sizes when the model is set up

[johanlofberg]

The only way to be able to attack it would be a black-box operator in a general nonlinear solver

https://yalmip.github.io/command/blackbox/

Leads to very slow and inefficient code (as YALMIP is not built for this) and probably poor performance in the solver as no derivatives are supplied.

I also believe you are missing a trace

objective = blackbox(r,@(r)trace((null(foo(r))'*diag(J)*null(foo(r)))))

[kirkrudolph]

Hmm, yeah I never thought about how certain selections of the decision variables could change the size of the null space. For my problem, the result of the null operator should always be a fixed size of [21 2].

[johanlofberg]

What you have to do then is to figure out if you can write that fixed null-space as a function of r, from the basis of Q (getbase(Q))

[kirkrudolph]

That makes sense. I don't believe it'll be too hard to make a function that computes the [21 2] matrix as a function of r.

The null operator is the generic way that works well for any "structure" of Q with predetermined values of r but this optimization flips everything around from work I've done in the past. It now requires a fixed "structure" of Q and varying values of r. I can see how using null would be problematic when the values are changing.

Thanks again for your help!