You apparently didn’t take my advice to heart from your previous question Invalid quadratic : not a square ti improve the scaling of the input data. SDPT3 is not robust enough to handle your problem, and encountered severe numerical difficulties.
Mosek is much more robust, and claimed to solve the problem I ran (see below), with optimal objective value = 6.1905, despite issuing warnings for large input numbers.
CVX made me change the names of beta and gamma due to name conflict, so I added 1 to the end of those variable names. I also had to use inv_pos on the LHS of the 2nd constraint. How was your code ever accepted by CVX?
SeDuMi provided a different answer.
Bottom line: Don’t use very large or small magnitude numbers in your input data. Until that is fixed, I don’t trust any of these results.
rng('default') % For reproducibility
I = 50;
eps1 = 10^-2;
eps2 = 10^-2;
fmax = 20 * 10^9;
s = normrnd(1000, 100, I, 1) * 8000 ;
c = normrnd(1000, 100, I, 1) * 10^6 ;
fl = c;
B = 10 * 10^6;
rfix = [0.5, 1, 1.5, 2, 2.5, 2.5, 2, 1.5, 1, 0.5];
cvx_begin
variable a(I) nonnegative
variable beta1(I) nonnegative
variable fs(I) nonnegative
variable b(I) nonnegative
variable mu(I) nonnegative
variable omega(I) nonnegative
variable gamma1(I) nonnegative
maximize sum(a)
subject to
0 <= a <= 1;
inv_pos(a * fmax + eps2) <= gamma1 <= 1/ eps1; % Reformulated so that it would be accepted by CVX
sum(inv_pos(gamma1)) <= (fmax + I * eps2);
inv_pos(a .* B + eps1) <= beta1 <= 1/ eps1;
sum(inv_pos(beta1)) <= B + I * eps1;
mu - a./ (B +eps1) >= 0;
beta1 - 1/(B + eps1) - mu + a/(B + eps1) >= 0;
a/eps1 - mu >= 0;
1/eps1 - beta1 - a/eps1 + mu >= 0;
omega - a/(fmax + eps2) >= 0;
gamma1 - 1/(fmax + eps2) - omega + a/(fmax +eps2) >=0;
a/eps2 - omega >= 0;
1/eps2 - gamma1 - a/eps2 + omega >=0;
(s .* mu / rfix(1) + c .* omega) + (1 - a) .* c ./ fl <= 4;
cvx_end
MOSEK Version 9.3.10 (Build date: 2021-11-5 08:42:07)
Copyright © MOSEK ApS, Denmark. WWW: mosek.com
Platform: Windows/64-X86
MOSEK warning 62: The A matrix contains a large value of 2.8e+10 in constraint ‘’ (98) at variable ‘’ (1002).
MOSEK warning 62: The A matrix contains a large value of 2.8e+10 in constraint ‘’ (99) at variable ‘’ (1005).
MOSEK warning 62: The A matrix contains a large value of 2.8e+10 in constraint ‘’ (100) at variable ‘’ (1008).
MOSEK warning 62: The A matrix contains a large value of 2.8e+10 in constraint ‘’ (101) at variable ‘’ (1011).
MOSEK warning 62: The A matrix contains a large value of 2.8e+10 in constraint ‘’ (102) at variable ‘’ (1014).
MOSEK warning 62: The A matrix contains a large value of 2.8e+10 in constraint ‘’ (103) at variable ‘’ (1017).
MOSEK warning 62: The A matrix contains a large value of 2.8e+10 in constraint ‘’ (104) at variable ‘’ (1020).
MOSEK warning 62: The A matrix contains a large value of 2.8e+10 in constraint ‘’ (105) at variable ‘’ (1023).
MOSEK warning 62: The A matrix contains a large value of 2.8e+10 in constraint ‘’ (106) at variable ‘’ (1026).
MOSEK warning 62: The A matrix contains a large value of 2.8e+10 in constraint ‘’ (107) at variable ‘’ (1029).
Warning number 62 is disabled.
MOSEK warning 57: A large value of 2.8e+10 has been specified in c for variable ‘’ (1154).
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 450
Cones : 200
Scalar variables : 1602
Matrix variables : 0
Integer variables : 0
Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator started.
Freed constraints in eliminator : 102
Eliminator terminated.
Eliminator started.
Freed constraints in eliminator : 50
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 : 450
Cones : 200
Scalar variables : 1602
Matrix variables : 0
Integer variables : 0
Optimizer - threads : 8
Optimizer - solved problem : the primal
Optimizer - Constraints : 298
Optimizer - Cones : 200
Optimizer - Scalar variables : 1250 conic : 600
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 : 3096 after factor : 4770
Factor - dense dim. : 0 flops : 2.30e+05
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 1.0e+00 3.4e+10 2.0e+10 0.00e+00 2.001002000e+10 0.000000000e+00 1.0e+00 0.09
1 1.1e-01 3.9e+09 6.3e+09 -1.00e+00 1.873998922e+10 1.208292594e-08 1.1e-01 0.23
2 2.6e-02 8.9e+08 3.0e+09 -1.00e+00 1.877367541e+10 8.644864668e-08 2.6e-02 0.23
3 3.0e-03 1.0e+08 1.0e+09 -1.00e+00 1.875236684e+10 8.177545440e-07 3.0e-03 0.23
4 2.2e-04 7.6e+06 2.8e+08 -1.00e+00 1.874846805e+10 1.207606514e-05 2.2e-04 0.25
5 8.8e-06 3.0e+05 5.5e+07 -1.00e+00 1.874741792e+10 3.397746678e-04 8.7e-06 0.25
6 2.4e-07 4.8e+03 7.0e+06 -1.00e+00 1.872665095e+10 2.243146617e-02 1.4e-07 0.25
7 8.7e-08 1.2e+03 3.5e+06 -9.98e-01 1.864933013e+10 8.516546811e-02 3.5e-08 0.27
8 4.9e-08 3.9e+02 1.9e+06 -9.85e-01 1.807153993e+10 2.357801337e-01 1.1e-08 0.27
9 3.6e-09 2.4e+01 3.7e+05 -9.28e-01 1.077487966e+10 2.454925923e+00 7.0e-10 0.28
10 3.3e-10 1.8e+00 2.5e+04 -1.52e-01 1.209778163e+09 5.398326420e+00 5.3e-11 0.28
11 1.5e-12 5.5e-03 6.3e+00 8.64e-01 1.454646902e+06 5.851989772e+00 1.6e-13 0.28
12 5.7e-15 2.1e-05 1.5e-03 1.00e+00 5.689460865e+03 5.853456439e+00 6.2e-16 0.30
13 2.2e-17 8.4e-08 3.8e-07 1.00e+00 2.801616259e+01 5.853493334e+00 2.4e-18 0.30
14 4.3e-20 4.5e-08 3.6e-11 9.99e-01 5.870572687e+00 5.859033543e+00 5.8e-21 0.30
15 6.0e-20 4.1e-08 6.9e-12 8.15e-01 6.182111863e+00 6.167744039e+00 1.9e-21 0.31
16 7.5e-20 4.6e-08 3.0e-16 9.91e-01 6.189899110e+00 6.189599844e+00 7.6e-24 0.31
17 7.5e-20 2.8e-08 2.1e-22 1.00e+00 6.190498339e+00 6.190498339e+00 9.9e-28 0.33
Optimizer terminated. Time: 0.39
Interior-point solution summary
Problem status : PRIMAL_AND_DUAL_FEASIBLE
Solution status : OPTIMAL
Primal. obj: 6.1904983387e+00 nrm: 4e+00 Viol. con: 1e+00 var: 4e-10 cones: 7e-23
Dual. obj: 6.1904983390e+00 nrm: 4e+09 Viol. con: 0e+00 var: 6e+01 cones: 0e+00
Optimizer summary
Optimizer - time: 0.39
Interior-point - iterations : 17 time: 0.33
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): +6.1905
+++++++++++++++++++++++++++++++
Running it in SeDuMi provided optimal objective value of 50.0054.
Calling SeDuMi 1.3.4: 1602 variables, 450 equality constraints
For improved efficiency, SeDuMi is solving the dual problem.
SeDuMi 1.3.4 by AdvOL, 2005-2008 and Jos F. Sturm, 1998-2003.
Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
eqs m = 450, order n = 1403, dim = 1603, blocks = 201
nnz(A) = 2694 + 204, nnz(ADA) = 1746, nnz(L) = 1098
Handling 4 + 2 dense columns.
it : by gap delta rate t/tP t/tD* feas cg cg prec
0 : 4.18E+08 0.000
1 : 2.06E+01 1.78E+08 0.000 0.4259 0.9000 0.9000 1.30 92 1 2.4E+10
2 : 3.22E+01 1.49E+07 0.000 0.0834 0.9281 0.9000 0.84 55 39 2.0E+09
3 : 2.35E+01 4.30E+06 0.000 0.2895 0.0000 0.9000 2.83 16 23 2.0E+09
4 : 2.35E+01 6.26E+05 0.305 0.1456 0.9000 0.0000 4.83 8 15 4.8E+08
5 : 1.66E+01 4.16E+05 0.000 0.6634 0.9000 0.9000 2.00 16 16 1.6E+08
6 : 1.66E+01 1.99E+04 0.375 0.0479 0.9000 0.0000 2.47 15 15 2.3E+07
7 : 1.66E+01 1.01E+04 0.081 0.5082 0.9000 0.9000 1.82 15 15 6.2E+06
8 : 1.98E+01 6.93E+03 0.000 0.6848 0.4500 0.4500 1.71 15 15 3.5E+06
9 : 2.81E+01 3.28E+03 0.046 0.4734 0.9000 0.9000 1.69 8 8 1.0E+06
10 : 1.61E+01 2.68E+03 0.098 0.8157 0.9000 0.9000 4.93 10 10 3.1E+05
11 : 1.58E+01 1.11E+02 0.366 0.0414 0.9000 0.0000 9.99 11 11 1.6E+04
12 : 7.73E+00 7.77E+01 0.000 0.7008 0.9000 0.9000 5.01 8 8 3.5E+03
13 : 7.73E+00 2.47E+00 0.045 0.0318 0.9000 0.0000 6.47 10 10 2.2E+02
14 : 7.36E+00 5.56E-01 0.000 0.2253 0.9000 0.9000 1.38 7 8 3.8E+01
15 : 6.17E+00 5.15E-02 0.000 0.0927 0.9900 0.9900 1.67 3 3 2.4E+00
16 : 1.59E+01 3.20E-03 0.023 0.0622 0.9000 0.4384 -0.11 2 2 2.4E+00
17 : 7.57E+01 6.75E-04 0.000 0.2108 0.9000 0.9000 -0.99 1 2 2.4E+00
18 : 2.46E+03 1.73E-05 0.000 0.0256 0.9900 0.9900 -0.99 1 1 2.0E+00
19 : 7.68E+02 4.19E-06 0.000 0.2427 0.9005 0.9000 0.61 1 1 4.6E-01
20 : 1.87E+02 4.82E-08 0.000 0.0115 0.9285 0.9000 3.44 1 1 7.1E-03
21 : 7.77E+01 2.69E-09 0.000 0.0559 0.9210 0.9000 2.96 1 1 3.9E-04
22 : 7.77E+01 1.50E-10 0.000 0.0559 0.9000 0.0000 1.69 1 1 6.5E-05
23 : 5.80E+01 4.76E-11 0.000 0.3162 0.9000 0.9000 1.20 1 1 1.9E-05
24 : 5.80E+01 5.46E-12 0.148 0.1148 0.9000 0.0000 1.12 1 1 6.0E-06
25 : 5.19E+01 1.70E-13 0.000 0.0311 0.9146 0.9000 1.07 1 1 4.8E-07
26 : 5.01E+01 1.10E-14 0.000 0.0645 0.9900 0.9900 1.14 1 9 2.7E-08
27 : 5.00E+01 9.06E-16 0.000 0.0826 0.9900 0.9900 1.17 7 7 1.6E-09
iter seconds digits cx by
27 1.6 Inf 5.0000835079e+01 5.0005424578e+01
|Ax-b| = 1.0e-04, [Ay-c]_+ = 6.6E-03, |x|= 6.9e-02, |y|= 1.7e+09
Detailed timing (sec)
Pre IPM Post
2.230E-01 1.172E+00 2.500E-02
Max-norms: ||b||=1, ||c|| = 2.000000e+10,
Cholesky |add|=49, |skip| = 50, ||L.L|| = 98.0484.
Status: Solved
Optimal value (cvx_optval): +50.0054