lmi constraint interpreted as elementwise inequality constraint

[Chang Liu]

Hi Johan,

I want to add LMI constraints to my problem. However, such constraints are sometimes interpreted as the element-wise inequality constraints. Here is a MWE:

clear

num = 1;

N = 1;

% define variables

x = sdpvar(2*num,N+1,'full');

t = sdpvar(num*num,N+1); % dummy variable for LMI

P = cell(num,N+1);

for ii = 1:N+1

    for jj = 1:num

        P{jj,ii} = sdpvar(2,2,'full');

    end

end

% objective

obj = sum(sum(t));

% constraints

constr = [t>=0];

% psd constraint

for ii = 1:N+1

    for jj = 1:num

        constr = [constr,[P{jj,ii} >= 0]:'psd of P']; 

    end

end

% LMI constraint

for ii = 1:N

    for jj = 1:num

        tmp = 0;

        for ll = 1:num

            constr = [constr,[[P{ll,ii+1} x(2*jj-1:2*jj,ii+1)-x(2*ll-1:2*ll,ii+1);

                (x(2*jj-1:2*jj,ii+1)-x(2*ll-1:2*ll,ii+1))' t(num*(jj-1)+ll,ii+1)]>=0]:'LMI'];

        end

    end

end

% intentionally add this constraint to show, via the optimization result,

% that LMI is interpreted as something else

constr = [constr, P{1,1}(2,1) == -1, P{1,1}(1,2) == -1];

opt = sdpsettings('solver','mosek','verbose',3,'debug',1,'showprogress',1);

sol = optimize(constr,obj,opt);

The solver gives an infeasible error. And 

check(constr)

gives the following result:

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

|   ID|               Constraint|   Primal residual|   Dual residual|        Tag|

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

|   #1|   Elementwise inequality|                 0|               0|           |

|   #2|   Elementwise inequality|                 0|               0|   psd of P|

|   #3|   Elementwise inequality|                 0|               0|   psd of P|

|   #4|   Elementwise inequality|                 0|               0|        LMI|

|   #5|      Equality constraint|                -1|              -1|           |

|   #6|      Equality constraint|                -1|               0|           |

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

As it shows, LMI is interpreted as elementwise inequality.

However, if I define P in the following way:

P{jj,ii} = sdpvar(2,2)

then LMI is correctly interpreted. But this causes problem in my other part of code (which I don't include here).

I'm wondering what is the best way to define LMI in Yalmip, as adding "full" seems to be a safe to define matrices in general but here ruins the LMI interpretation.

Thanks!

[Johan Löfberg]

You are not adding any LMIs, as X>=0 for a full matrix is interpreted as an elementwise constraint. For psd to make sense in a convex optimization framework and yalmip, the matrix has to be structurally symmetric (not just constrained to be symmetric)

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

If your code fails, you must have some other misconception in your view of semidefinite programming. Psd only makes sense for structurally symmetric parameterizations

[Chang Liu]

Thanks! It turns that the problems of my code, besides the structurally non-symmetric, are the numerical inaccuracy and badly scaled variables. I post them here in case other people encounter the same problem:

It seems that Matlab treats -0 and 0 as different values and thus even matrix A=[1 -0; 0 1] is not symmetric, causing a problem when imposing the equality constraint (P == A). So I just use triu in the equality constraint.

The badly scaled variables causes infeasibility issue when using Mosek, even the primal problem is feasible.