I use MOSEK to maximize my function. Here is the result.
Calling Mosek 9.1.9: 83006 variables, 4902 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
MOSEK warning 710: #1 (nearly) zero elements are specified in sparse col ‘’ (1225) of matrix ‘A’.
MOSEK warning 710: #1 (nearly) zero elements are specified in sparse col ‘’ (1244) of matrix ‘A’.
MOSEK warning 710: #1 (nearly) zero elements are specified in sparse col ‘’ (3026) of matrix ‘A’.
MOSEK warning 710: #1 (nearly) zero elements are specified in sparse col ‘’ (3057) of matrix ‘A’.
MOSEK warning 710: #4 (nearly) zero elements are specified in sparse col ‘’ (7082) of matrix ‘A’.
MOSEK warning 710: #5 (nearly) zero elements are specified in sparse col ‘’ (7084) of matrix ‘A’.
MOSEK warning 710: #6 (nearly) zero elements are specified in sparse col ‘’ (7096) of matrix ‘A’.
MOSEK warning 710: #7 (nearly) zero elements are specified in sparse col ‘’ (7098) of matrix ‘A’.
MOSEK warning 710: #29 (nearly) zero elements are specified in sparse col ‘’ (7114) of matrix ‘A’.
MOSEK warning 710: #7 (nearly) zero elements are specified in sparse col ‘’ (7117) of matrix ‘A’.
Warning number 710 is disabled.
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 4902
Cones : 0
Scalar variables : 24602
Matrix variables : 92
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 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: 0.11
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 4902
Cones : 0
Scalar variables : 24602
Matrix variables : 92
Integer variables : 0
Optimizer - threads : 32
Optimizer - solved problem : the primal
Optimizer - Constraints : 4902
Optimizer - Cones : 1
Optimizer - Scalar variables : 14565 conic : 7112
Optimizer - Semi-definite variables: 92 scalarized : 58404
Factor - setup time : 0.38 dense det. time : 0.00
Factor - ML order time : 0.06 GP order time : 0.00
Factor - nonzeros before factor : 4.95e+06 after factor : 5.88e+06
Factor - dense dim. : 2 flops : 8.73e+09
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 1.0e+00 1.7e+07 1.3e+03 0.00e+00 1.292000000e+03 0.000000000e+00 1.0e+00 0.61
1 4.2e-01 7.0e+06 8.4e+02 -1.00e+00 1.301869346e+03 -4.001241165e-03 4.2e-01 1.59
2 8.6e-02 1.4e+06 3.8e+02 -1.00e+00 1.291989178e+03 2.891663366e-02 8.6e-02 2.50
3 2.1e-02 3.6e+05 1.9e+02 -1.00e+00 1.256507267e+03 2.765749889e-01 2.1e-02 3.33
4 1.2e-02 2.0e+05 1.4e+02 -1.00e+00 1.219835468e+03 1.603252457e+00 1.2e-02 4.17
5 7.2e-03 1.2e+05 1.1e+02 -9.99e-01 1.171887681e+03 6.266038777e+00 7.2e-03 4.98
6 2.5e-03 4.2e+04 6.5e+01 -9.98e-01 9.483113883e+02 3.602691871e+01 2.5e-03 5.81
7 1.7e-03 2.8e+04 5.3e+01 -9.92e-01 7.852319715e+02 6.882045695e+01 1.7e-03 6.64
8 1.0e-03 1.7e+04 4.1e+01 -9.85e-01 4.737917326e+02 1.428949516e+02 1.0e-03 7.47
9 5.5e-04 9.2e+03 3.0e+01 -9.72e-01 -1.195692240e+02 3.134830689e+02 5.5e-04 8.27
10 4.1e-04 6.8e+03 2.5e+01 -9.42e-01 -5.372159941e+02 4.558551670e+02 4.1e-04 9.09
11 1.0e-04 1.7e+03 1.1e+01 -9.17e-01 -4.302355457e+03 1.929490256e+03 1.0e-04 10.05
12 6.8e-05 1.1e+03 8.4e+00 -6.70e-01 -5.172616643e+03 2.797391921e+03 6.8e-05 10.86
13 6.4e-05 1.1e+03 7.9e+00 -5.41e-01 -5.245498866e+03 2.944691078e+03 6.4e-05 11.67
14 3.8e-05 6.4e+02 5.2e+00 -5.20e-01 -5.628230174e+03 4.287843846e+03 3.8e-05 12.50
15 3.0e-05 5.0e+02 4.0e+00 -3.01e-01 -5.010073540e+03 5.009785132e+03 3.0e-05 13.33
16 1.3e-05 2.1e+02 1.6e+00 -1.75e-01 -1.736855066e+03 7.590836813e+03 1.3e-05 14.17
17 1.0e-05 1.7e+02 1.2e+00 2.81e-01 6.523874615e+01 8.129916233e+03 1.0e-05 14.98
18 4.6e-06 7.8e+01 4.3e-01 4.07e-01 4.850248794e+03 9.567360096e+03 4.6e-06 15.83
19 1.2e-06 2.0e+01 6.0e-02 7.16e-01 8.788237842e+03 1.015614868e+04 1.2e-06 16.78
20 4.4e-07 7.4e+00 1.4e-02 9.89e-01 9.866223379e+03 1.041511142e+04 4.4e-07 17.66
21 1.9e-07 3.2e+00 4.0e-03 1.05e+00 1.023074901e+04 1.047701834e+04 1.9e-07 18.45
22 7.2e-08 1.2e+00 9.4e-04 1.03e+00 1.034978690e+04 1.044599351e+04 7.2e-08 19.28
23 5.8e-10 9.8e-03 6.0e-07 1.02e+00 1.043351568e+04 1.043410377e+04 5.8e-10 20.23
24 4.3e-10 7.2e-03 3.8e-07 1.04e+00 1.130025121e+04 1.130069559e+04 4.3e-10 21.19
25 3.8e-10 6.3e-03 3.1e-07 1.02e+00 1.145643725e+04 1.145682961e+04 3.8e-10 22.19
26 1.7e-10 2.8e-03 9.5e-08 1.03e+00 1.250930691e+04 1.250949286e+04 1.7e-10 23.17
27 1.1e-10 1.9e-03 5.2e-08 1.01e+00 1.271644046e+04 1.271656523e+04 1.1e-10 24.14
28 1.0e-10 1.4e-03 3.6e-08 1.01e+00 1.280265856e+04 1.280275583e+04 8.6e-11 25.11
29 9.5e-11 1.4e-03 3.6e-08 9.12e-01 1.280278804e+04 1.280288527e+04 8.6e-11 26.11
30 9.5e-11 1.4e-03 3.6e-08 1.03e+00 1.280278804e+04 1.280288527e+04 8.6e-11 27.17
31 9.5e-11 1.4e-03 3.6e-08 1.03e+00 1.280278804e+04 1.280288527e+04 8.6e-11 28.20
32 9.5e-11 1.4e-03 3.6e-08 9.70e-01 1.280384525e+04 1.280394214e+04 8.6e-11 29.17
33 9.5e-11 1.4e-03 3.6e-08 9.70e-01 1.280384525e+04 1.280394214e+04 8.6e-11 30.19
34 9.5e-11 1.4e-03 3.6e-08 9.59e-01 1.280384525e+04 1.280394214e+04 8.6e-11 31.27
Optimizer terminated. Time: 32.36
Interior-point solution summary
Problem status : PRIMAL_AND_DUAL_FEASIBLE
Solution status : OPTIMAL
Primal. obj: 1.2803845249e+04 nrm: 4e+04 Viol. con: 1e-02 var: 9e+01 barvar: 0e+00
Dual. obj: 1.2803942138e+04 nrm: 2e+06 Viol. con: 0e+00 var: 2e+01 barvar: 3e-06
Optimizer summary
Optimizer - time: 32.36
Interior-point - iterations : 35 time: 32.27
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): +12804.9
However, after I add some other linear constraints, I obtain a larger answer. Here is the result.
Calling Mosek 9.1.9: 83086 variables, 4902 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
MOSEK warning 710: #1 (nearly) zero elements are specified in sparse col ‘’ (1225) of matrix ‘A’.
MOSEK warning 710: #1 (nearly) zero elements are specified in sparse col ‘’ (1244) of matrix ‘A’.
MOSEK warning 710: #1 (nearly) zero elements are specified in sparse col ‘’ (3026) of matrix ‘A’.
MOSEK warning 710: #1 (nearly) zero elements are specified in sparse col ‘’ (3057) of matrix ‘A’.
MOSEK warning 710: #4 (nearly) zero elements are specified in sparse col ‘’ (7082) of matrix ‘A’.
MOSEK warning 710: #5 (nearly) zero elements are specified in sparse col ‘’ (7084) of matrix ‘A’.
MOSEK warning 710: #6 (nearly) zero elements are specified in sparse col ‘’ (7096) of matrix ‘A’.
MOSEK warning 710: #7 (nearly) zero elements are specified in sparse col ‘’ (7098) of matrix ‘A’.
MOSEK warning 710: #29 (nearly) zero elements are specified in sparse col ‘’ (7114) of matrix ‘A’.
MOSEK warning 710: #7 (nearly) zero elements are specified in sparse col ‘’ (7117) of matrix ‘A’.
Warning number 710 is disabled.
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 4902
Cones : 0
Scalar variables : 24682
Matrix variables : 92
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 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: 0.08
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 4902
Cones : 0
Scalar variables : 24682
Matrix variables : 92
Integer variables : 0
Optimizer - threads : 32
Optimizer - solved problem : the primal
Optimizer - Constraints : 4902
Optimizer - Cones : 1
Optimizer - Scalar variables : 14603 conic : 7112
Optimizer - Semi-definite variables: 92 scalarized : 58404
Factor - setup time : 0.36 dense det. time : 0.00
Factor - ML order time : 0.06 GP order time : 0.00
Factor - nonzeros before factor : 4.95e+06 after factor : 5.88e+06
Factor - dense dim. : 2 flops : 8.73e+09
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 1.0e+00 1.7e+07 1.3e+03 0.00e+00 1.292000000e+03 0.000000000e+00 1.0e+00 0.56
1 4.2e-01 7.0e+06 8.3e+02 -1.00e+00 1.285927775e+03 -1.678563528e-03 4.2e-01 1.39
2 9.3e-02 1.6e+06 3.9e+02 -1.00e+00 1.272935249e+03 2.773555803e-02 9.3e-02 2.30
3 1.9e-02 3.2e+05 1.8e+02 -1.00e+00 1.231319505e+03 3.199987618e-01 1.9e-02 3.19
4 1.3e-02 2.3e+05 1.5e+02 -1.00e+00 1.210666644e+03 1.133788952e+00 1.3e-02 3.97
5 4.6e-03 7.8e+04 8.7e+01 -1.00e+00 1.081941884e+03 1.102545544e+01 4.6e-03 4.78
6 2.2e-03 3.7e+04 6.0e+01 -9.97e-01 8.848075504e+02 4.229377550e+01 2.2e-03 5.61
7 9.1e-04 1.5e+04 3.8e+01 -9.90e-01 3.664758219e+02 1.499671990e+02 9.1e-04 6.42
8 4.3e-04 7.3e+03 2.6e+01 -9.69e-01 -4.917036757e+02 3.920847142e+02 4.3e-04 7.20
9 2.0e-04 3.3e+03 1.7e+01 -9.25e-01 -2.092814166e+03 9.858129929e+02 2.0e-04 7.97
10 6.5e-05 1.1e+03 8.2e+00 -8.23e-01 -5.615291445e+03 2.845198976e+03 6.5e-05 8.91
11 5.1e-05 8.5e+02 6.6e+00 -5.23e-01 -5.875074684e+03 3.458664632e+03 5.1e-05 9.67
12 3.1e-05 5.1e+02 4.2e+00 -4.27e-01 -5.598383269e+03 4.842235142e+03 3.1e-05 10.47
13 1.1e-05 1.8e+02 1.4e+00 -1.88e-01 -1.621275509e+03 7.966606761e+03 1.1e-05 11.30
14 7.5e-06 1.3e+02 8.5e-01 3.80e-01 1.228978792e+03 8.680340296e+03 7.5e-06 12.09
15 3.0e-06 5.1e+01 2.4e-01 5.44e-01 6.331508202e+03 9.870239376e+03 3.0e-06 12.97
16 1.5e-06 2.6e+01 8.6e-02 8.60e-01 8.194159576e+03 1.005202845e+04 1.5e-06 13.75
17 4.5e-07 7.6e+00 1.5e-02 9.48e-01 9.819247469e+03 1.042217248e+04 4.5e-07 14.59
18 7.4e-08 1.2e+00 1.0e-03 1.04e+00 1.043301394e+04 1.054699964e+04 7.4e-08 15.48
19 3.9e-09 6.5e-02 1.2e-05 1.03e+00 1.046100888e+04 1.046641983e+04 3.9e-09 16.39
20 2.4e-09 4.0e-02 5.8e-06 1.01e+00 1.082146640e+04 1.082483386e+04 2.4e-09 17.34
21 2.1e-09 3.5e-02 4.6e-06 1.01e+00 1.096234546e+04 1.096523792e+04 2.1e-09 18.27
22 1.1e-09 1.9e-02 1.8e-06 1.01e+00 1.154264911e+04 1.154421150e+04 1.1e-09 19.17
23 8.9e-10 1.5e-02 1.3e-06 1.01e+00 1.177180375e+04 1.177305235e+04 8.9e-10 20.08
24 3.2e-10 5.3e-03 2.8e-07 1.01e+00 1.249002638e+04 1.249047115e+04 3.2e-10 20.98
25 1.2e-10 2.1e-03 6.7e-08 1.01e+00 1.286860212e+04 1.286877473e+04 1.2e-10 21.91
26 2.6e-11 4.4e-04 6.6e-09 1.00e+00 1.305340401e+04 1.305344085e+04 2.6e-11 22.81
27 2.6e-11 4.4e-04 6.5e-09 9.90e-01 1.305413762e+04 1.305417394e+04 2.6e-11 23.73
28 2.6e-11 4.3e-04 6.4e-09 9.96e-01 1.305431839e+04 1.305435459e+04 2.6e-11 24.70
29 2.6e-11 4.3e-04 6.4e-09 9.91e-01 1.305467830e+04 1.305471425e+04 2.6e-11 25.64
30 2.6e-11 4.3e-04 6.4e-09 9.90e-01 1.305472298e+04 1.305475890e+04 2.6e-11 26.58
31 2.6e-11 4.3e-04 6.3e-09 9.95e-01 1.305508007e+04 1.305511573e+04 2.6e-11 27.48
32 2.5e-11 4.3e-04 6.2e-09 9.91e-01 1.305543439e+04 1.305546981e+04 2.5e-11 28.42
33 2.5e-11 4.2e-04 6.2e-09 9.96e-01 1.305552237e+04 1.305555772e+04 2.5e-11 29.34
34 2.5e-11 4.2e-04 6.2e-09 1.00e+00 1.305552237e+04 1.305555772e+04 2.5e-11 30.34
35 2.5e-11 4.2e-04 6.2e-09 1.01e+00 1.305552237e+04 1.305555772e+04 2.5e-11 31.33
36 2.5e-11 4.1e-04 6.0e-09 9.93e-01 1.305692668e+04 1.305696106e+04 2.5e-11 32.25
37 2.5e-11 3.9e-04 5.5e-09 9.93e-01 1.305965289e+04 1.305968536e+04 2.3e-11 33.16
38 2.3e-11 3.9e-04 5.4e-09 9.94e-01 1.305997353e+04 1.306000578e+04 2.3e-11 34.09
39 2.3e-11 3.9e-04 5.4e-09 9.94e-01 1.305997353e+04 1.306000578e+04 2.3e-11 35.05
40 1.4e-11 2.2e-04 2.3e-09 9.93e-01 1.308037137e+04 1.308038938e+04 1.3e-11 35.98
41 9.6e-12 3.6e-05 1.5e-10 9.96e-01 1.310185654e+04 1.310185957e+04 2.1e-12 36.91
42 7.5e-12 2.6e-05 9.7e-11 9.99e-01 1.310293273e+04 1.310293495e+04 1.6e-12 37.81
43 6.8e-12 2.4e-05 8.5e-11 1.00e+00 1.310027750e+04 1.310027953e+04 1.4e-12 38.73
44 5.9e-12 1.3e-05 3.2e-11 1.00e+00 1.310160623e+04 1.310160730e+04 7.5e-13 39.67
45 5.9e-12 1.3e-05 3.2e-11 1.00e+00 1.310160623e+04 1.310160730e+04 7.5e-13 40.66
46 5.9e-12 1.3e-05 3.2e-11 1.00e+00 1.310160623e+04 1.310160730e+04 7.5e-13 41.61
Optimizer terminated. Time: 42.61
Interior-point solution summary
Problem status : PRIMAL_AND_DUAL_FEASIBLE
Solution status : OPTIMAL
Primal. obj: 1.3101606234e+04 nrm: 4e+04 Viol. con: 6e-02 var: 8e-01 barvar: 0e+00
Dual. obj: 1.3101607297e+04 nrm: 2e+06 Viol. con: 0e+00 var: 2e-01 barvar: 3e-08
Optimizer summary
Optimizer - time: 42.61
Interior-point - iterations : 47 time: 42.59
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): +13102.6
I think this situation do not make sense.
How do I fix it?