How to identify the ill-posed LMIs

[Zihao Cheng]

close all;

clear all;

%%system parameters

A=[-4 -3 0 -1; -3 -7 0 -3; 0 0 -13 -1; -1 -3 -1 -10];

B=[0;-4;2;5];

%% feasible LMI

% Pn=sdpvar(4,4);

% Kn=sdpvar(1,4);

% F1=([-Pn,(A*Pn+B*Kn)';A*Pn+B*Kn,-Pn]<0)+(Pn>0);

% optimize(F1)%mosek

% check(F1)

% Pn=value(Pn);

% Kn=value(Kn);

% Lm1=[-Pn,(A*Pn+B*Kn)';A*Pn+B*Kn,-Pn]

%%infeasible LMI

det=1;

P=sdpvar(4,4);

K=sdpvar(1,4);

F2=(P>eye(4))+([-P,(A+B*K)';A+B*K,det^2*P-2*det*eye(4)]<0);

optimize(F2)%mosek

check(F2)

P=value(P);

K=value(K);

Lm2=[-P,(A+B*K)';A+B*K,det^2*P-2*det*eye(4)]

F1 and F2 are from same problem but obtained by different conversion method. Even through both are LMIs, F1 is OK, F2 is not OK with solution of e1-5 level. I don't know why. If you have more experience on this problem, please tell me. I am very interested about it.  

[Johan Löfberg]

F1 is easily feasible and F2 is clearly infeasible, so they are not in any sense equivalent (which also is obvious from the fact that the models are identical except for the (2,2) block where they differ substantially 

[Zihao Cheng]

Johan,thank you for your reply, very much.

F2(2,2) is the key factor for infeasible, but why?  Is there some knowledge on the operation principle of solver? Or just we can judge the feasible from the Lmis expressions before simulation on computer. You are professional in this areas with much more experiences. What kinds of ill-posed constraints you have countered, like ill-LMIs and bad-parameters.

[Johan Löfberg]

what makes you say it is ill-posed. it simply looks infeasible to me

They are fundamentally different.

2*det*I - det^2*P >= (AP+BK)'P^-1(AP+BK)

vs

P >= (AP+BK)'P^-1(AP+BK)

in the first one, making P larger makes things worse as left-hand decreases, while in the second one at least the left-hand sides increases and thus improves odds of being feasible