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;

`%======================================================================`

clear;

%

`M=[-1,-1;3,3];`

%

cvx_begin sdp

variables ro d1 d2

%

minimize ro

% 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;`

cvx_end

%

`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.

Best regards.