Scaling problem in LMI

22

Calling Mosek_3 9.1.9: 145 variables, 49 equality constraints
For improved efficiency, Mosek_3 is solving the dual problem.

MOSEK Version 9.3.18 (Build date: 2022-3-17 11:29:16)
Copyright © MOSEK ApS, Denmark. WWW: mosek.com
Platform: Windows/64-X86

MOSEK warning 710: #1 (nearly) zero elements are specified in sparse col ‘’ (12) of matrix ‘A’.
MOSEK warning 710: #2 (nearly) zero elements are specified in sparse col ‘’ (19) of matrix ‘A’.
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 49
Cones : 1
Scalar variables : 45
Matrix variables : 5
Integer variables : 0

Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator started.
Freed constraints in eliminator : 1
Eliminator terminated.
Eliminator started.
Freed constraints in eliminator : 0
Eliminator terminated.
Eliminator - tries : 2 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 : 49
Cones : 1
Scalar variables : 45
Matrix variables : 5
Integer variables : 0

Optimizer - threads : 16
Optimizer - solved problem : the primal
Optimizer - Constraints : 47
Optimizer - Cones : 2
Optimizer - Scalar variables : 28 conic : 24
Optimizer - Semi-definite variables: 5 scalarized : 222
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 : 1128 after factor : 1128
Factor - dense dim. : 0 flops : 2.61e+05
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 3.9e+00 2.6e+04 1.2e+03 0.00e+00 1.199903036e+03 0.000000000e+00 1.0e+00 0.06
1 7.4e-01 4.9e+03 5.2e+02 -1.00e+00 1.198996839e+03 3.918354910e+00 1.9e-01 0.16
2 2.8e-01 1.9e+03 3.2e+02 -9.99e-01 1.190051798e+03 4.628700884e+00 7.1e-02 0.17
3 3.5e-02 2.4e+02 1.1e+02 -9.97e-01 1.099413301e+03 2.037696737e+01 9.0e-03 0.17
4 6.1e-03 4.1e+01 4.5e+01 -9.79e-01 5.582323279e+02 1.969795699e+01 1.6e-03 0.19
5 1.1e-03 7.4e+00 1.6e+01 -8.76e-01 -1.373858940e+03 1.432104284e+01 2.8e-04 0.19
6 2.7e-04 1.8e+00 3.7e+00 -3.07e-01 -1.641948497e+03 3.336921296e+00 6.9e-05 0.19
7 1.2e-04 8.3e-01 4.8e-01 1.31e+00 -3.545303998e+01 1.368713621e+00 3.2e-05 0.19
8 5.5e-05 3.7e-01 7.6e-02 2.76e+00 3.170106970e+00 5.967568655e-01 1.4e-05 0.20
9 3.1e-05 2.0e-01 2.5e-02 2.18e+00 1.677772080e+00 4.732629616e-01 7.8e-06 0.20
10 5.7e-06 3.8e-02 1.9e-03 1.36e+00 7.040987173e-01 5.038592509e-01 1.5e-06 0.20
11 7.6e-07 5.0e-03 7.2e-05 1.10e+00 1.745262335e+00 1.679276280e+00 1.9e-07 0.22
12 2.5e-07 1.6e-03 1.2e-05 1.15e+00 2.949251500e+00 2.926724291e+00 6.2e-08 0.22
13 1.1e-07 6.6e-04 2.4e-06 1.17e+00 3.268379028e+00 3.258080593e+00 1.9e-08 0.22
14 4.1e-08 1.6e-04 3.7e-07 1.25e+00 3.351485165e+00 3.349876775e+00 7.3e-09 0.22
15 3.1e-08 5.0e-05 6.6e-08 9.79e-01 3.386744752e+00 3.386261537e+00 2.2e-09 0.23
16 3.1e-08 5.0e-05 6.6e-08 9.92e-01 3.386744752e+00 3.386261537e+00 2.2e-09 0.23
17 3.1e-08 5.0e-05 6.6e-08 9.95e-01 3.386744752e+00 3.386261537e+00 2.2e-09 0.23
Optimizer terminated. Time: 0.30

Interior-point solution summary
Problem status : UNKNOWN
Solution status : UNKNOWN
Primal. obj: 3.3867447516e+00 nrm: 1e+01 Viol. con: 3e+01 var: 5e-01 barvar: 0e+00 cones: 0e+00
Dual. obj: 3.3862615370e+00 nrm: 6e+02 Viol. con: 0e+00 var: 1e-07 barvar: 3e-04 cones: 0e+00
Optimizer summary
Optimizer - time: 0.30
Interior-point - iterations : 18 time: 0.25
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: Failed
Optimal value (cvx_optval): NaN

微信截图_20220527221614
微信截图_202205272216141697×323 18.9 KB

23

CVX Warning:
Models involving “log” or other functions in the log, exp, and entropy
family are solved using an experimental successive approximation method.
This method is slower and less reliable than the method CVX employs for
other models. Please see the section of the user’s guide entitled
The successive approximation method
for more details about the approach, and for instructions on how to
suppress this warning message in the future.

Calling Mosek 9.3.18: 145 variables, 49 equality constraints
For improved efficiency, Mosek is solving the dual problem.

MOSEK Version 9.3.18 (Build date: 2022-3-17 11:29:16)
Copyright © MOSEK ApS, Denmark. WWW: mosek.com
Platform: Windows/64-X86

MOSEK warning 710: #1 (nearly) zero elements are specified in sparse col ‘’ (12) of matrix ‘A’.
MOSEK warning 710: #2 (nearly) zero elements are specified in sparse col ‘’ (19) of matrix ‘A’.
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 49
Cones : 1
Scalar variables : 45
Matrix variables : 5
Integer variables : 0

Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator started.
Freed constraints in eliminator : 1
Eliminator terminated.
Eliminator started.
Freed constraints in eliminator : 0
Eliminator terminated.
Eliminator - tries : 2 time : 0.00
Lin. dep. - tries : 1 time : 0.00
Lin. dep. - number : 0
Presolve terminated. Time: 0.05
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 49
Cones : 1
Scalar variables : 45
Matrix variables : 5
Integer variables : 0

Optimizer - threads : 8
Optimizer - solved problem : the primal
Optimizer - Constraints : 47
Optimizer - Cones : 2
Optimizer - Scalar variables : 28 conic : 24
Optimizer - Semi-definite variables: 5 scalarized : 222
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 : 1128 after factor : 1128
Factor - dense dim. : 0 flops : 2.61e+05
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 3.9e+00 2.6e+04 1.2e+03 0.00e+00 1.199903036e+03 0.000000000e+00 1.0e+00 0.06
1 7.4e-01 4.9e+03 5.2e+02 -1.00e+00 1.198996839e+03 3.918354910e+00 1.9e-01 0.16
2 2.8e-01 1.9e+03 3.2e+02 -9.99e-01 1.190051798e+03 4.628700884e+00 7.1e-02 0.16
3 3.5e-02 2.4e+02 1.1e+02 -9.97e-01 1.099413301e+03 2.037696737e+01 9.0e-03 0.17
4 6.1e-03 4.1e+01 4.5e+01 -9.79e-01 5.582323279e+02 1.969795699e+01 1.6e-03 0.17
5 1.1e-03 7.4e+00 1.6e+01 -8.76e-01 -1.373858940e+03 1.432104284e+01 2.8e-04 0.17
6 2.7e-04 1.8e+00 3.7e+00 -3.07e-01 -1.641948497e+03 3.336921296e+00 6.9e-05 0.19
7 1.2e-04 8.3e-01 4.8e-01 1.31e+00 -3.545303998e+01 1.368713621e+00 3.2e-05 0.19
8 5.5e-05 3.7e-01 7.6e-02 2.76e+00 3.170106970e+00 5.967568655e-01 1.4e-05 0.19
9 3.1e-05 2.0e-01 2.5e-02 2.18e+00 1.677772080e+00 4.732629616e-01 7.8e-06 0.20
10 5.7e-06 3.8e-02 1.9e-03 1.36e+00 7.040987173e-01 5.038592509e-01 1.5e-06 0.20
11 7.6e-07 5.0e-03 7.2e-05 1.10e+00 1.745262335e+00 1.679276280e+00 1.9e-07 0.20
12 2.5e-07 1.6e-03 1.2e-05 1.15e+00 2.949251500e+00 2.926724291e+00 6.2e-08 0.22
13 1.1e-07 6.6e-04 2.4e-06 1.17e+00 3.268379028e+00 3.258080593e+00 1.9e-08 0.22
14 4.1e-08 1.6e-04 3.7e-07 1.25e+00 3.351485165e+00 3.349876775e+00 7.3e-09 0.22
15 3.1e-08 5.0e-05 6.6e-08 9.79e-01 3.386744752e+00 3.386261537e+00 2.2e-09 0.22
16 3.1e-08 5.0e-05 6.6e-08 9.92e-01 3.386744752e+00 3.386261537e+00 2.2e-09 0.23
17 3.1e-08 5.0e-05 6.6e-08 9.95e-01 3.386744752e+00 3.386261537e+00 2.2e-09 0.23
Optimizer terminated. Time: 0.33

Interior-point solution summary
Problem status : UNKNOWN
Solution status : UNKNOWN
Primal. obj: 3.3867447516e+00 nrm: 1e+01 Viol. con: 3e+01 var: 5e-01 barvar: 0e+00 cones: 0e+00
Dual. obj: 3.3862615370e+00 nrm: 6e+02 Viol. con: 0e+00 var: 1e-07 barvar: 3e-04 cones: 0e+00
Optimizer summary
Optimizer - time: 0.33
Interior-point - iterations : 18 time: 0.25
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: Failed
Optimal value (cvx_optval): NaN

24

When ratio = 10, Mosek was able to handle your problem, despite the near zero input data. When ratio = 100, Mosek was not able to confidently find an optimal solution, and so reported UNKNOWN status. CVX’s design seems to be (at least sometimes) reporting the problem as failed and populating cvx_optval and variables with NaN when Mosek reports UNKNOWN. CVX could have been designed to report the result produced by Mosek, but it was not designed to. It can be seen that the Mosek final primal and dual; objective values are a little different between the two problems, and due to transformations by CVX, do not reflect what the cvx_optval would be.

25

ok, i got it, thank you very much

26

Mosek struggles with solving your problem.

Since CVX dualize then this

Primal. obj: 3.3867447516e+00 nrm: 1e+01 Viol. con: 3e+01 var: 5e-01 barvar: 0e+00 cones: 0e+00
Dual. obj: 3.3862615370e+00 nrm: 6e+02 Viol. con: 0e+00 var: 1e-07 barvar: 3e-04 cones: 0e+00

shows the feasibility of the dual solution is not that good. I cannot say from the log output why your problem is numerically hard.