I used Mosek solver. The result for K=2 is as follows:
Note that i run the CVX for 5 times, and average over all realizations(=5).
Calling Mosek 9.1.9: 32 variables, 10 equality constraints
For improved efficiency, Mosek is solving the dual problem.
MOSEK Version 9.1.9 (Build date: 2019-11-21 11:34:40)
Copyright © MOSEK ApS, Denmark. WWW: mosek.com
Platform: Windows/64-X86
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 10
Cones : 4
Scalar variables : 32
Matrix variables : 0
Integer variables : 0
Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator started.
Freed constraints in eliminator : 6
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: 0.02
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 10
Cones : 4
Scalar variables : 32
Matrix variables : 0
Integer variables : 0
Optimizer - threads : 2
Optimizer - solved problem : the primal
Optimizer - Constraints : 4
Optimizer - Cones : 4
Optimizer - Scalar variables : 20 conic : 12
Optimizer - Semi-definite variables: 0 scalarized : 0
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 : 9 after factor : 10
Factor - dense dim. : 0 flops : 2.46e+02
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 8.2e+03 1.0e+00 3.0e+00 0.00e+00 1.999999999e+00 0.000000000e+00 1.0e+00 0.05
1 6.3e+02 7.8e-02 8.3e-01 -9.99e-01 4.942182388e+01 5.928395281e+01 7.8e-02 0.13
2 1.2e+01 1.4e-03 1.0e-01 -9.93e-01 2.310350387e+03 2.884738353e+03 1.4e-03 0.13
3 4.6e+00 5.6e-04 4.4e-02 -5.58e-01 3.012553752e+03 3.696342093e+03 5.6e-04 0.14
4 1.7e+00 2.0e-04 3.2e-03 8.29e-01 3.815868327e+02 4.093445805e+02 2.0e-04 0.14
5 4.4e-02 5.4e-06 1.4e-05 8.03e-01 2.362313261e+01 2.432621877e+01 5.4e-06 0.14
6 2.2e-03 2.7e-07 1.6e-07 1.00e+00 8.651567221e-01 9.007351459e-01 2.7e-07 0.14
7 8.8e-04 1.1e-07 5.4e-08 9.61e-01 2.616489753e-01 2.893341733e-01 1.1e-07 0.16
8 5.4e-04 6.6e-08 3.8e-08 6.15e-01 -3.141143559e-03 3.272972751e-02 6.6e-08 0.17
9 1.7e-04 2.0e-08 9.4e-09 5.92e-01 -3.468586474e-01 -3.231969212e-01 2.0e-08 0.17
10 5.9e-05 7.2e-09 3.1e-09 9.66e-02 -7.333085016e-01 -7.123205550e-01 7.2e-09 0.17
11 1.0e-05 1.3e-09 2.5e-10 9.00e-01 -9.956292703e-01 -9.915235489e-01 1.3e-09 0.17
12 1.5e-06 1.8e-10 1.5e-11 9.80e-01 -1.053817253e+00 -1.053075772e+00 1.8e-10 0.17
13 5.3e-07 6.4e-11 3.4e-12 1.01e+00 -1.061157348e+00 -1.060850572e+00 6.4e-11 0.19
14 9.4e-08 1.1e-11 2.5e-13 1.02e+00 -1.063919087e+00 -1.063866544e+00 1.1e-11 0.19
15 1.9e-08 2.2e-12 2.2e-14 1.03e+00 -1.064390713e+00 -1.064379727e+00 2.2e-12 0.19
16 1.2e-08 1.1e-13 1.7e-16 1.01e+00 -1.064510737e+00 -1.064510313e+00 8.5e-14 0.19
17 2.1e-09 1.1e-13 8.1e-19 1.00e+00 -1.064515574e+00 -1.064515570e+00 8.0e-16 0.20
Optimizer terminated. Time: 0.27
Interior-point solution summary
Problem status : PRIMAL_AND_DUAL_FEASIBLE
Solution status : OPTIMAL
Primal. obj: -1.0645155743e+00 nrm: 3e+04 Viol. con: 2e-09 var: 0e+00 cones: 0e+00
Dual. obj: -1.0645155703e+00 nrm: 1e+00 Viol. con: 0e+00 var: 3e-10 cones: 0e+00
Optimizer summary
Optimizer - time: 0.27
Interior-point - iterations : 17 time: 0.20
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: Solved
Optimal value (cvx_optval): +1.06452
Calling Mosek 9.1.9: 32 variables, 10 equality constraints
For improved efficiency, Mosek is solving the dual problem.
MOSEK Version 9.1.9 (Build date: 2019-11-21 11:34:40)
Copyright © MOSEK ApS, Denmark. WWW: mosek.com
Platform: Windows/64-X86
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 10
Cones : 4
Scalar variables : 32
Matrix variables : 0
Integer variables : 0
Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator started.
Freed constraints in eliminator : 6
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 : 10
Cones : 4
Scalar variables : 32
Matrix variables : 0
Integer variables : 0
Optimizer - threads : 2
Optimizer - solved problem : the primal
Optimizer - Constraints : 4
Optimizer - Cones : 4
Optimizer - Scalar variables : 19 conic : 12
Optimizer - Semi-definite variables: 0 scalarized : 0
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 : 9 after factor : 10
Factor - dense dim. : 0 flops : 2.40e+02
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 4.1e+03 1.0e+00 3.0e+00 0.00e+00 1.999999999e+00 0.000000000e+00 1.0e+00 0.03
1 3.3e+01 8.1e-03 2.7e-01 -1.00e+00 4.903035667e+02 6.103948392e+02 8.1e-03 0.11
2 8.2e+00 2.0e-03 1.3e-01 -9.62e-01 1.760297032e+03 2.193933657e+03 2.0e-03 0.11
3 3.7e+00 8.9e-04 3.9e-02 -3.23e-01 1.116764873e+03 1.326921494e+03 8.9e-04 0.11
4 3.9e-01 9.5e-05 1.2e-03 4.74e-01 2.394489778e+02 2.569244382e+02 9.5e-05 0.13
5 2.3e-03 5.6e-07 4.8e-07 1.04e+00 7.463056147e-01 8.264297379e-01 5.6e-07 0.13
6 5.5e-04 1.3e-07 6.1e-08 9.87e-01 6.235316085e-02 8.551034976e-02 1.3e-07 0.13
7 2.4e-04 5.8e-08 2.8e-08 7.54e-01 -7.574319903e-02 -4.938319655e-02 5.8e-08 0.13
8 9.0e-05 2.2e-08 1.4e-08 2.98e-01 -1.891179260e-01 -1.436236838e-01 2.2e-08 0.13
9 1.3e-05 3.2e-09 1.0e-08 -1.12e+00 -7.118921736e-01 3.699800526e-01 3.2e-09 0.13
10 2.1e-08 4.8e-12 2.5e-10 -9.11e-01 -2.989154374e+02 2.577776937e-02 4.8e-12 0.14
11 4.6e-10 5.1e-20 1.3e-10 -1.00e+00 -3.338692711e+10 1.672495727e-02 4.8e-20 0.14
12 4.4e-16 8.8e-26 1.5e-10 -1.00e+00 -2.969966148e-05 2.767397087e-26 8.1e-30 0.14
Optimizer terminated. Time: 0.17
Interior-point solution summary
Problem status : DUAL_INFEASIBLE
Solution status : DUAL_INFEASIBLE_CER
Primal. obj: -2.9699661481e-05 nrm: 5e+00 Viol. con: 4e-16 var: 0e+00 cones: 0e+00
Optimizer summary
Optimizer - time: 0.17
Interior-point - iterations : 12 time: 0.14
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
Calling Mosek 9.1.9: 32 variables, 10 equality constraints
For improved efficiency, Mosek is solving the dual problem.
MOSEK Version 9.1.9 (Build date: 2019-11-21 11:34:40)
Copyright © MOSEK ApS, Denmark. WWW: mosek.com
Platform: Windows/64-X86
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 10
Cones : 4
Scalar variables : 32
Matrix variables : 0
Integer variables : 0
Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator started.
Freed constraints in eliminator : 6
Eliminator terminated.
Eliminator started.
Freed constraints in eliminator : 0
Eliminator terminated.
Eliminator - tries : 2 time : 0.00
Lin. dep. - tries : 1 time : 0.01
Lin. dep. - number : 0
Presolve terminated. Time: 0.03
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 10
Cones : 4
Scalar variables : 32
Matrix variables : 0
Integer variables : 0
Optimizer - threads : 2
Optimizer - solved problem : the primal
Optimizer - Constraints : 4
Optimizer - Cones : 4
Optimizer - Scalar variables : 20 conic : 12
Optimizer - Semi-definite variables: 0 scalarized : 0
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 : 9 after factor : 10
Factor - dense dim. : 0 flops : 2.46e+02
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 2.0e+03 1.0e+00 3.0e+00 0.00e+00 1.999999999e+00 0.000000000e+00 1.0e+00 0.03
1 3.4e+01 1.7e-02 3.8e-01 -9.98e-01 2.251777964e+02 2.790814263e+02 1.7e-02 0.11
2 6.8e+00 3.3e-03 1.4e-01 -8.79e-01 8.320392205e+02 1.032763735e+03 3.3e-03 0.11
3 2.2e+00 1.1e-03 1.0e-02 -3.03e-02 1.847148333e+02 1.936482058e+02 1.1e-03 0.13
4 1.0e-01 5.1e-05 9.4e-05 8.00e-01 1.504078917e+01 1.538758372e+01 5.1e-05 0.13
5 2.0e-03 9.8e-07 2.5e-07 1.01e+00 1.713581927e-01 1.778206823e-01 9.8e-07 0.13
6 7.7e-04 3.8e-07 8.7e-08 9.87e-01 2.566820890e-02 3.125321368e-02 3.8e-07 0.13
7 4.3e-04 2.1e-07 5.5e-08 6.23e-01 -2.358459714e-02 -1.603890853e-02 2.1e-07 0.13
8 9.2e-05 4.5e-08 1.1e-08 3.73e-01 -1.139105409e-01 -1.067212387e-01 4.5e-08 0.13
9 2.1e-05 1.0e-08 1.6e-09 2.59e-01 -2.422997654e-01 -2.395794652e-01 1.0e-08 0.14
10 3.8e-06 1.9e-09 1.3e-10 9.03e-01 -2.797273792e-01 -2.791873727e-01 1.9e-09 0.14
11 1.1e-06 5.5e-10 2.2e-11 9.90e-01 -2.861373766e-01 -2.859676605e-01 5.5e-10 0.14
12 2.0e-07 1.0e-10 1.9e-12 1.01e+00 -2.885866513e-01 -2.885455033e-01 1.0e-10 0.14
13 4.0e-08 2.2e-11 1.9e-13 1.04e+00 -2.889303957e-01 -2.889216995e-01 2.2e-11 0.14
14 7.5e-09 2.6e-12 7.6e-15 1.03e+00 -2.890214532e-01 -2.890204579e-01 2.5e-12 0.16
15 3.8e-09 8.4e-14 5.1e-18 1.00e+00 -2.890332790e-01 -2.890332758e-01 8.6e-15 0.16
Optimizer terminated. Time: 0.19
Interior-point solution summary
Problem status : PRIMAL_AND_DUAL_FEASIBLE
Solution status : OPTIMAL
Primal. obj: -2.8903327904e-01 nrm: 7e+03 Viol. con: 2e-09 var: 0e+00 cones: 0e+00
Dual. obj: -2.8903327577e-01 nrm: 1e+00 Viol. con: 0e+00 var: 6e-12 cones: 0e+00
Optimizer summary
Optimizer - time: 0.19
Interior-point - iterations : 15 time: 0.16
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: Solved
Optimal value (cvx_optval): +0.289033
Calling Mosek 9.1.9: 32 variables, 10 equality constraints
For improved efficiency, Mosek is solving the dual problem.
MOSEK Version 9.1.9 (Build date: 2019-11-21 11:34:40)
Copyright © MOSEK ApS, Denmark. WWW: mosek.com
Platform: Windows/64-X86
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 10
Cones : 4
Scalar variables : 32
Matrix variables : 0
Integer variables : 0
Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator started.
Freed constraints in eliminator : 6
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 : 10
Cones : 4
Scalar variables : 32
Matrix variables : 0
Integer variables : 0
Optimizer - threads : 2
Optimizer - solved problem : the primal
Optimizer - Constraints : 4
Optimizer - Cones : 4
Optimizer - Scalar variables : 20 conic : 12
Optimizer - Semi-definite variables: 0 scalarized : 0
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 : 9 after factor : 10
Factor - dense dim. : 0 flops : 2.46e+02
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 1.6e+04 1.0e+00 3.0e+00 0.00e+00 1.999999999e+00 0.000000000e+00 1.0e+00 0.03
1 3.5e+01 2.2e-03 1.4e-01 -1.00e+00 1.744664877e+03 2.178446970e+03 2.2e-03 0.11
2 1.0e+01 6.3e-04 6.3e-02 -8.64e-01 4.590998145e+03 5.705970792e+03 6.3e-04 0.11
3 4.7e+00 2.9e-04 9.1e-03 5.83e-02 1.152452861e+03 1.261706834e+03 2.9e-04 0.11
4 3.8e-01 2.3e-05 2.3e-04 5.99e-01 2.264695650e+02 2.373581419e+02 2.3e-05 0.13
5 3.7e-03 2.3e-07 2.1e-07 1.01e+00 1.351291529e+00 1.448390172e+00 2.3e-07 0.13
6 9.5e-04 5.8e-08 3.2e-08 1.00e+00 9.076172173e-02 1.245863735e-01 5.8e-08 0.13
7 5.0e-04 3.0e-08 1.8e-08 8.15e-01 -1.542997652e-01 -1.161458241e-01 3.0e-08 0.13
8 2.0e-04 1.2e-08 7.1e-09 6.02e-01 -3.905111144e-01 -3.517166426e-01 1.2e-08 0.13
9 7.9e-05 4.8e-09 3.3e-09 -2.44e-01 -9.773429117e-01 -9.235279259e-01 4.8e-09 0.14
10 7.0e-06 4.3e-10 8.5e-11 8.45e-01 -1.602306755e+00 -1.597896469e+00 4.3e-10 0.14
11 1.5e-06 9.1e-11 8.9e-12 9.77e-01 -1.659023834e+00 -1.657967195e+00 9.1e-11 0.14
12 4.7e-07 2.9e-11 1.7e-12 1.01e+00 -1.672049689e+00 -1.671646985e+00 2.9e-11 0.14
13 8.5e-08 5.4e-12 1.4e-13 1.03e+00 -1.675882757e+00 -1.675809755e+00 5.4e-12 0.14
14 3.1e-08 1.1e-12 1.2e-14 1.04e+00 -1.676616507e+00 -1.676602298e+00 1.1e-12 0.14
15 5.0e-09 1.3e-14 4.2e-18 1.01e+00 -1.676799268e+00 -1.676799197e+00 5.0e-15 0.16
16 4.9e-09 6.6e-14 7.5e-18 1.00e+00 -1.676800156e+00 -1.676800157e+00 3.6e-17 0.16
Optimizer terminated. Time: 0.19
Interior-point solution summary
Problem status : PRIMAL_AND_DUAL_FEASIBLE
Solution status : OPTIMAL
Primal. obj: -1.6768001564e+00 nrm: 4e+04 Viol. con: 1e-08 var: 0e+00 cones: 0e+00
Dual. obj: -1.6768001572e+00 nrm: 2e+00 Viol. con: 0e+00 var: 3e-10 cones: 0e+00
Optimizer summary
Optimizer - time: 0.19
Interior-point - iterations : 16 time: 0.16
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: Solved
Optimal value (cvx_optval): +1.6768
Calling Mosek 9.1.9: 32 variables, 10 equality constraints
For improved efficiency, Mosek is solving the dual problem.
MOSEK Version 9.1.9 (Build date: 2019-11-21 11:34:40)
Copyright © MOSEK ApS, Denmark. WWW: mosek.com
Platform: Windows/64-X86
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 10
Cones : 4
Scalar variables : 32
Matrix variables : 0
Integer variables : 0
Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator started.
Freed constraints in eliminator : 6
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: 0.01
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 10
Cones : 4
Scalar variables : 32
Matrix variables : 0
Integer variables : 0
Optimizer - threads : 2
Optimizer - solved problem : the primal
Optimizer - Constraints : 4
Optimizer - Cones : 4
Optimizer - Scalar variables : 20 conic : 12
Optimizer - Semi-definite variables: 0 scalarized : 0
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 : 9 after factor : 10
Factor - dense dim. : 0 flops : 2.46e+02
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 1.6e+04 1.0e+00 3.0e+00 0.00e+00 1.999999999e+00 0.000000000e+00 1.0e+00 0.03
1 1.2e+01 7.5e-04 8.0e-02 -1.00e+00 4.963394117e+03 6.201872367e+03 7.5e-04 0.11
2 5.5e+00 3.3e-04 4.2e-02 -7.60e-01 7.503087814e+03 9.277537617e+03 3.3e-04 0.11
3 2.7e+00 1.7e-04 4.0e-03 9.23e-01 9.968654413e+02 1.059694099e+03 1.7e-04 0.11
4 1.6e-01 9.5e-06 5.9e-05 6.71e-01 1.461240985e+02 1.503426382e+02 9.5e-06 0.11
5 4.2e-03 2.6e-07 2.6e-07 1.01e+00 2.886806537e+00 2.999734966e+00 2.6e-07 0.13
6 8.3e-04 5.1e-08 3.2e-08 9.80e-01 3.542952004e-01 3.976504073e-01 5.1e-08 0.13
7 2.4e-04 1.5e-08 1.0e-08 7.49e-01 -2.792978649e-01 -2.294361343e-01 1.5e-08 0.13
8 1.3e-04 7.8e-09 8.1e-09 -1.76e-02 -5.614453852e-01 -4.413734050e-01 7.8e-09 0.13
9 4.1e-05 2.5e-09 2.9e-09 -8.87e-02 -1.531958641e+00 -1.378406643e+00 2.5e-09 0.13
10 8.8e-06 5.4e-10 2.5e-09 -1.13e+00 -3.530626158e+00 -1.124349954e+00 5.4e-10 0.14
11 1.1e-07 6.9e-12 3.3e-10 -1.01e+00 -2.606864433e+02 -1.366252618e+00 6.9e-12 0.14
12 2.2e-12 1.2e-17 1.8e-11 -1.00e+00 -2.015224479e+08 -1.497137394e+00 1.2e-17 0.14
13 4.4e-16 5.5e-23 8.0e-13 -1.00e+00 -1.813665146e-05 -3.212423233e-22 9.7e-27 0.14
Optimizer terminated. Time: 0.17
Interior-point solution summary
Problem status : DUAL_INFEASIBLE
Solution status : DUAL_INFEASIBLE_CER
Primal. obj: -1.8136651460e-05 nrm: 9e+00 Viol. con: 4e-16 var: 0e+00 cones: 0e+00
Optimizer summary
Optimizer - time: 0.17
Interior-point - iterations : 13 time: 0.14
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
opt_finded =
0.0874