Hi, I have a simple set of LMI’s for a feedback control problem that I have trouble solving due to numerics. I can’t seem to find a good way to scale the problem to make it easier for the solver. I am using Mosek. I have tried using slack variables which seemed help for some combinations of k1, k2, c1, c2, but I run into problems with the k1, k2, c1, and c2 below (mostly due to k2 and c2). Thank you for your help.

Code:

```
alpha = 0.01; %decay rate design variable
A = @(ks) [zeros(2,3);0,-ks,0];
B = @(cs) [eye(2);[0,-cs]];
k1 = 300;
k2 = 2.7e6;
c1 = 0.35;
c2 = 1000;
cvx_begin sdp
variable Q(3,3) symmetric
variable Y(2,3)
A(k1)*Q + Q*A(k1)' + B(c1)*Y + Y'*B(c1)' + 2*alpha*Q <= -eye(3);
A(k1)*Q + Q*A(k1)' + B(c2)*Y + Y'*B(c2)' + 2*alpha*Q <= -eye(3);
A(k2)*Q + Q*A(k2)' + B(c1)*Y + Y'*B(c1)' + 2*alpha*Q <= -eye(3);
A(k2)*Q + Q*A(k2)' + B(c2)*Y + Y'*B(c2)' + 2*alpha*Q <= -eye(3);
Q >= eye(3);
cvx_end
K = Y/Q %feedback controller
```

Mosek output:

```
Calling Mosek_2 9.2.40: 30 variables, 12 equality constraints
For improved efficiency, Mosek_2 is solving the dual problem.
------------------------------------------------------------
MOSEK Version 9.2.40 (Build date: 2021-3-10 14:33:40)
Copyright (c) MOSEK ApS, Denmark. WWW: mosek.com
Platform: Windows/64-X86
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 12
Cones : 0
Scalar variables : 0
Matrix variables : 5
Integer variables : 0
Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator started.
Freed constraints in eliminator : 0
Eliminator terminated.
Eliminator - tries : 1 time : 0.00
Lin. dep. - tries : 1 time : 0.00
Lin. dep. - number : 0
Presolve terminated. Time: 0.03
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 12
Cones : 0
Scalar variables : 0
Matrix variables : 5
Integer variables : 0
Optimizer - threads : 6
Optimizer - solved problem : the primal
Optimizer - Constraints : 12
Optimizer - Cones : 0
Optimizer - Scalar variables : 0 conic : 0
Optimizer - Semi-definite variables: 5 scalarized : 30
Factor - setup time : 0.00 dense det. time : 0.00
Factor - ML order time : 0.00 GP order time : 0.00
Factor - nonzeros before factor : 72 after factor : 72
Factor - dense dim. : 0 flops : 3.20e+03
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 4.0e+00 2.7e+06 7.2e+00 0.00e+00 -8.160000000e+00 0.000000000e+00 1.0e+00 0.03
1 1.2e+00 8.1e+05 1.4e+01 -2.46e+00 -8.695714696e+01 0.000000000e+00 3.0e-01 0.08
2 1.1e-01 7.2e+04 2.2e+00 -2.18e+00 -1.477237605e+02 0.000000000e+00 2.7e-02 0.08
3 9.8e-03 6.6e+03 1.5e+00 -1.17e+00 -6.947140589e+03 0.000000000e+00 2.4e-03 0.09
4 1.5e-03 9.9e+02 5.4e-01 -1.00e+00 -4.291752444e+04 0.000000000e+00 3.7e-04 0.09
5 3.8e-04 2.5e+02 2.8e-01 -1.10e+00 -1.749550417e+05 0.000000000e+00 9.4e-05 0.09
6 2.9e-05 1.9e+01 4.6e-02 -7.03e-01 -8.031846519e+05 0.000000000e+00 7.1e-06 0.09
7 4.5e-06 3.1e+00 1.7e-02 -9.29e-01 -4.638722612e+06 0.000000000e+00 1.1e-06 0.09
8 9.2e-07 6.2e-01 8.7e-03 -9.65e-01 -2.771301660e+07 0.000000000e+00 2.3e-07 0.09
9 2.1e-07 1.4e-01 4.6e-03 -1.01e+00 -1.514640916e+08 0.000000000e+00 5.2e-08 0.09
10 4.4e-08 3.0e-02 1.9e-03 -8.66e-01 -5.714168838e+08 0.000000000e+00 1.1e-08 0.11
11 1.4e-08 9.8e-03 1.3e-03 -1.16e+00 -2.591403085e+09 0.000000000e+00 3.6e-09 0.11
Optimizer terminated. Time: 0.13
Interior-point solution summary
Problem status : DUAL_INFEASIBLE
Solution status : DUAL_INFEASIBLE_CER
Primal. obj: -3.0762116129e+00 nrm: 1e+00 Viol. con: 4e-02 barvar: 0e+00
Optimizer summary
Optimizer - time: 0.13
Interior-point - iterations : 11 time: 0.11
Basis identification - time: 0.00
Primal - iterations : 0 time: 0.00
Dual - iterations : 0 time: 0.00
Clean primal - iterations : 0 time: 0.00
Clean dual - iterations : 0 time: 0.00
Simplex - time: 0.00
Primal simplex - iterations : 0 time: 0.00
Dual simplex - iterations : 0 time: 0.00
Mixed integer - relaxations: 0 time: 0.00
------------------------------------------------------------
Status: Infeasible
Optimal value (cvx_optval): +Inf
```