Thanks, Mark, I have tried your first approach, starting with a realizable value of ‘gam’ and decreasing the value successively until the problem is unrealizable, finding the minimum value.
One last question and do not bother anymore, I explain;
I have tried to do the same problem, without iterative loop, trying to transform the inequalities constraints, so that CVX can minimize ‘gam’ without loop interaction, I have tried this from two different approaches;
- The first approach is to rewrite the inequalities in a scalar way, in this way;
variables ro d1 d2
% scalar afine constrains
d1*M(1,1)^2 + d2*M(2,1)^2 <= d1*ro; d1*M(1,1)*M(1,2) + d2*M(2,1)*M(2,2) <= 0; d1*M(1,1)*M(1,2) + d2*M(2,1)*M(2,2) <= 0; d1*M(1,2)^2 + d2*M(2,2)^2 <= d2*ro;
E=[d1 0; 0 d2];
gam = sqrt(ro); % %=================================================================
Unfortunately, this does not work, since I can not avoid using the square and you get 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 example_scaling_EE363_Review_Session_4 (line 140) d1*M(1,1)^2 + d2*M(2,1)^2 <= d1*ro;
Is there any way to implement the problem from this approach, either scalar or vectorial, without disobeying DCP ruleset?
- And the second approach is to rewrite the inequalities in the form of LMI, but here I also have problems, I explain,
First, through Schur’s complement, I rewrite the inequalities in LMI form preserving symmetry in this way,
[-E^-1, M; M', -gam^2*E] <= 0; E > 0;
But in fact, this is a BMI due to the term (
-gam ^ 2 * E) and also the inverse of a matrix contradicts DCP ruleset.
Next to eliminate the inverse operation of a matrix, pre and post-multiplying by
diag(eye(n),E^-1) we obtain,
[-E^-1, M*E^-1; E^-1*M', -E^-1*gam^2] <= 0;
and with a change of variable
W = E^-1, we have,
[-W, M*W; W*M', -W*gam^2] <= 0;
Now the inverse operation is not there, but it is still a BMI, the question is, is there any way to transform the original inequality into LMI and solve it with CVX?
Thank you very much for everything beforehand.