The cvx status is infeasible, but i’m sure there is one feasible solution for the considered problem

The main code is as follows, and the commented code is the feasible solution that I have tested.

M               = 50;
K               = 7;
H_bs            = 30;
H_uav           = 100;
P_bs_TX         = 0.2;
P_bs_sta        = 3000;
P_bs_sleep      = 600;
P_uav_max_sum   = 2;
noise           = 10^-14;
N               = 120;
S_max           = 100;
R_th            = 1.5;
Rho             = 10^-6;
error           = 100;
P_on=P_bs_sta+P_bs_TX*M/K;
P_off=P_bs_sleep;
a=15;
f_a=138.5433;
beta=[0.0087,0.4759,5.4100];
UE_position=[-2074,2004,391,492,-3614,-2618,-2992,554,3003,1132,4206,2994,1473,-2460,429,-2008,1216,4536,-2630,2632,1814,4652,-2218,3681,-4335,-4428,-2148,-1219,3559,4486,877,1179,1958,-4595,3969,-157,-1699,521,-1252,-3157,-1335,4078,2578,1103,-1207,98,4574,3210,2480,-110;-3546,-3926,990,2091,2302,-1249,2807,11,-3040,992,-936,3788,1882,-3566,4700,-4454,4379,-179,839,-1222,-3434,-806,4076,1539,-256,3235,-162,2395,1466,-3236,-4095,-3434,1273,4435,3136,-4072,1702,-2161,1796,2437,2838,-1482,1570,-2726,-1833,-1291,4010,-523,-1844,1282];
BS_position=[0,4000,-4000,2000,-2000,-2000,2000;0,0,0,3464,3464,-3464,-3464];
constraint2=[1,0.5,0.5,0.5,0.5,0.5,0.5];
ua_array=[6,7,1,4,5,3,5,1,7,1,2,4,4,6,4,6,4,2,3,2,7,2,5,2,3,5,3,5,2,7,7,7,4,5,4,6,5,7,5,5,5,2,4,7,6,1,4,2,7,1];
L=[114,463,1065,2039,1989,1863,1191,555,1089,1505,959,1046,1667,472,1999,990,1205,566,1607,1834,191,1037,650,1572,423,2439,1860,1324,1531,2497,1289,822,2192,2771,1997,1940,1788,1971,1829,1548,914,1484,1980,1162,1814,1295,2632,948,1690,1287];
theta = [0,2*pi/(N-1):2*pi/(N-1):2*pi/(N-1)*(N-2),0];

cons2_Circle1 =  S_max*(N-1)/3/pi*cos(theta);
cons2_Circle2 =  S_max*(N-1)/3/pi*sin(theta);
q_nr=[cons2_Circle1;cons2_Circle2];
%% Feasible solution
% P_uav=P_uav_max_sum/M*ones(M,N);
% q_n=q_nr;
% S_str=ones(1,K);
%% CVX  Mosek 9.1.9
cvx_begin
% 变量
variable S_str(1,K) nonnegative
variable q_n(2,N)
variable P_uav(M,N) nonnegative
expression P_tot(1,1)
expression P_tot_1(1,1)
expression P_tot_2(1,1)
expression R_bs(1,M)
expression r_uu_n(2,N)
expression f0_n(1,N)
expression f1(1,N)
expression Taylor(M,N)
expression R_uav(1,M)
%             variable V(1,N-1)
% 优化目标
P_tot_1=sum(S_str*P_on+(1-S_str)*P_off)+sum(sum(P_uav))/N;
P_tot_2 = 0;
for i=1:N-1
    V = norm((q_n(:,i+1)-q_n(:,i)),2);
    P_tot_2 = P_tot_2...
        +f_a...
        +beta(1)*pow_pos(V-a,3)+beta(2)*square_pos(V-a)+beta(3)*max([V-a,0]);
end
P_tot=P_tot_1+1/(N-1)*P_tot_2;
R_bs=S_str(ua_array).*log(1+P_bs_TX*Rho./L./L/noise);
for i=1:M
    r_uu=q_nr-UE_position(:,i);
    for j=1:N
        r_uu_n(:,j)=q_n(:,j)-UE_position(:,i);
    end
    f0=r_uu(1,:).^2+r_uu(2,:).^2;
    f0_n=r_uu_n(1,:).^2+r_uu_n(2,:).^2;
    f1=2*r_uu.*(q_n-q_nr);
    Taylor(i,:)=f0+f1(1,:)+f1(2,:);
    R_uav(1,i) = 1/N*sum(...
        -log(f0+H_uav^2)...
        -(f0_n-f0)./(f0+H_uav^2)...
        +log(Taylor(i,:)+H_uav^2+P_uav(i,:)*Rho/noise)...
        );
end
minimize(P_tot)
subject to
for i = 1:K
    if (constraint2(i) == 0)
        S_str(i) == 0;
    elseif (constraint2(i) == 1)
        S_str(i) == 1;
    else
        0 <= S_str(i) <= 1;
    end
end
for i=1:N
    sum(P_uav(:,i)) <= P_uav_max_sum;
end

for i=1:2
    q_n(i,1) == q_n(i,N);
end
for i = 1:N-1
    norm((q_n(:,i+1)-q_n(:,i)),2) <= S_max;
end
R_th*log(2) <= R_bs+R_uav;
cvx_end

Your feasible solution

P_uav=P_uav_max_sum/M*ones(M,N);
q_n=q_nr; 
S_str=ones(1,K);

does appear to be feasible, with a maximum violation of 0.88e-16, which is well within solver tolerance.

CVX’s Successive Approximation Method (used due to exponential cones) failed using SeDuMi and is still running using SDPT.3 The probl3em was reported infeasible using Mosek 9.3.21. I haven’t tried the new stable release of Mosek 10. Some of the input numbers look suspiciously low, but Mosek didn’t issue any warnings. I will defer to Mosek people to assess the output or do whatever they deem appropriate.

Calling Mosek 9.3.15: 62809 variables, 25436 equality constraints
For improved efficiency, Mosek is solving the dual problem.

MOSEK Version 9.3.21 (Build date: 2022-8-8 14:59:43)
Copyright © MOSEK ApS, Denmark. WWW: mosek.com
Platform: Windows/64-X86

Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 25436
Cones : 18595
Scalar variables : 62809
Matrix variables : 0
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 : 0.03
Lin. dep. - number : 0
Presolve terminated. Time: 0.14
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 25436
Cones : 18595
Scalar variables : 62809
Matrix variables : 0
Integer variables : 0

Optimizer - threads : 8
Optimizer - solved problem : the dual
Optimizer - Constraints : 19003
Optimizer - Cones : 18595
Optimizer - Scalar variables : 62387 conic : 55785
Optimizer - Semi-definite variables: 0 scalarized : 0
Factor - setup time : 0.70 dense det. time : 0.00
Factor - ML order time : 0.50 GP order time : 0.00
Factor - nonzeros before factor : 1.19e+06 after factor : 1.97e+06
Factor - dense dim. : 0 flops : 2.22e+08
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 2.4e+03 1.6e+04 2.9e+04 0.00e+00 2.896861201e+04 -2.423000129e-01 1.0e+00 1.00
1 8.0e+02 5.5e+03 1.7e+04 -1.00e+00 2.022654653e+04 -8.740431645e+03 3.3e-01 1.25
2 6.9e+02 4.7e+03 1.6e+04 -1.00e+00 1.791539674e+04 -1.105114458e+04 2.9e-01 1.36
3 5.5e+02 3.8e+03 1.4e+04 -1.00e+00 1.859516798e+04 -1.037051509e+04 2.3e-01 1.50
4 4.5e+02 3.1e+03 1.3e+04 -1.00e+00 2.254400054e+04 -6.420782011e+03 1.9e-01 1.62
5 2.7e+02 1.8e+03 9.6e+03 -1.00e+00 2.541603143e+04 -3.545168804e+03 1.1e-01 1.75
6 3.8e+01 2.6e+02 3.6e+03 -1.00e+00 2.466742803e+04 -4.240599614e+03 1.6e-02 1.86
7 7.4e+00 5.1e+01 1.6e+03 -1.00e+00 2.241998953e+04 -6.229502045e+03 3.1e-03 1.98
8 1.5e+00 1.0e+01 7.1e+02 -1.00e+00 1.020266212e+04 -1.712385512e+04 6.0e-04 2.09
9 3.5e-01 2.4e+00 3.5e+02 -1.00e+00 -2.821968531e+04 -5.025489548e+04 1.4e-04 2.20
10 7.3e-02 5.0e-01 1.6e+02 -1.00e+00 -2.084365893e+05 -2.046375959e+05 3.1e-05 2.32
11 1.9e-02 1.3e-01 8.1e+01 -9.97e-01 -8.662784321e+05 -7.678837760e+05 7.8e-06 2.43
12 3.9e-03 2.7e-02 3.6e+01 -9.85e-01 -4.110157793e+06 -3.547107340e+06 1.6e-06 2.56
13 9.5e-04 6.6e-03 1.6e+01 -9.26e-01 -1.505924744e+07 -1.308076181e+07 4.0e-07 2.68
14 6.2e-04 4.3e-03 1.2e+01 -6.78e-01 -2.090924691e+07 -1.827357004e+07 2.6e-07 2.84
15 6.1e-04 4.2e-03 1.2e+01 -4.89e-01 -2.113482330e+07 -1.848015161e+07 2.5e-07 3.00
16 5.8e-04 4.0e-03 1.2e+01 -4.80e-01 -2.194578777e+07 -1.922368659e+07 2.4e-07 3.15
17 4.4e-04 3.0e-03 9.2e+00 -4.48e-01 -2.624387215e+07 -2.318104460e+07 1.8e-07 3.31
18 3.8e-04 2.6e-03 8.1e+00 -2.71e-01 -2.837807052e+07 -2.521265790e+07 1.6e-07 3.46
19 3.7e-04 2.6e-03 8.0e+00 -1.73e-01 -2.856576977e+07 -2.539506768e+07 1.6e-07 3.62
20 3.2e-04 2.2e-03 6.8e+00 -1.64e-01 -3.017235228e+07 -2.713295432e+07 1.3e-07 3.74
21 3.1e-04 2.2e-03 6.6e+00 -7.31e-02 -3.054510015e+07 -2.750188363e+07 1.3e-07 3.90
22 2.5e-04 1.7e-03 5.4e+00 -5.59e-02 -3.352986057e+07 -3.046326910e+07 1.1e-07 4.07
23 2.5e-04 1.7e-03 5.3e+00 7.88e-02 -3.366751490e+07 -3.060451887e+07 1.0e-07 4.24
24 2.5e-04 1.7e-03 5.2e+00 8.49e-02 -3.394358275e+07 -3.088818625e+07 1.0e-07 4.40
25 1.8e-04 1.3e-03 3.6e+00 9.72e-02 -3.688216238e+07 -3.422010838e+07 7.6e-08 4.52
26 1.8e-04 1.2e-03 3.5e+00 2.14e-01 -3.704738044e+07 -3.439332774e+07 7.5e-08 4.70
27 1.8e-04 1.2e-03 3.5e+00 2.20e-01 -3.721125940e+07 -3.456535738e+07 7.4e-08 4.87
28 7.8e-05 5.3e-04 1.3e+00 2.25e-01 -4.527993697e+07 -4.346630592e+07 3.2e-08 4.99
29 7.5e-05 5.1e-04 1.2e+00 4.87e-01 -4.559029757e+07 -4.381020925e+07 3.1e-08 5.16
30 7.0e-05 4.8e-04 1.1e+00 5.12e-01 -4.605835350e+07 -4.433525435e+07 2.9e-08 5.33
31 3.9e-05 2.7e-04 4.7e-01 5.54e-01 -4.508266959e+07 -4.409532866e+07 1.6e-08 5.46
32 3.5e-05 2.4e-04 4.0e-01 9.78e-01 -4.446415806e+07 -4.357282737e+07 1.5e-08 5.74
33 3.0e-05 2.2e-04 3.1e-01 1.02e+00 -4.332466533e+07 -4.256488139e+07 1.2e-08 6.04
34 1.8e-05 1.8e-04 1.5e-01 1.08e+00 -3.957802505e+07 -3.912309455e+07 7.6e-09 6.33
35 9.9e-06 1.3e-04 5.6e-02 1.21e+00 -3.274794566e+07 -3.252778574e+07 4.1e-09 6.64
36 5.3e-06 3.6e-05 2.5e-03 1.26e+00 -1.033245495e+07 -1.030691521e+07 5.5e-10 6.91
37 1.6e-06 1.6e-05 4.0e-04 1.05e+00 -3.419610835e+06 -3.412243266e+06 1.6e-10 7.16
38 1.4e-06 1.4e-05 3.6e-04 1.02e+00 -3.208435424e+06 -3.201583328e+06 1.5e-10 7.50
39 1.4e-06 1.4e-05 3.5e-04 1.02e+00 -3.133866300e+06 -3.127189283e+06 1.5e-10 7.82
40 1.2e-06 1.1e-05 2.6e-04 1.02e+00 -2.648979417e+06 -2.643486349e+06 1.2e-10 8.11
41 9.6e-07 9.4e-06 2.0e-04 1.01e+00 -2.237540906e+06 -2.233014104e+06 1.0e-10 8.50
42 7.9e-07 7.8e-06 1.5e-04 1.01e+00 -1.888556396e+06 -1.884821508e+06 8.3e-11 8.80
43 2.5e-07 2.8e-06 2.5e-05 1.01e+00 -7.084512516e+05 -7.073142669e+05 2.6e-11 9.08
44 2.4e-07 2.7e-06 2.4e-05 1.01e+00 -6.819801503e+05 -6.808864852e+05 2.5e-11 9.41
45 8.8e-08 1.5e-06 5.4e-06 1.01e+00 -2.401273864e+05 -2.397225777e+05 9.2e-12 9.66
46 8.7e-08 1.5e-06 5.3e-06 1.00e+00 -2.386702797e+05 -2.382684507e+05 9.1e-12 9.97
47 3.2e-08 1.3e-06 1.2e-06 1.00e+00 -1.138921977e+05 -1.137442978e+05 3.4e-12 10.22
48 3.2e-08 1.3e-06 1.2e-06 1.00e+00 -1.133808599e+05 -1.132336800e+05 3.3e-12 10.51
49 8.6e-09 3.5e-07 1.6e-07 1.00e+00 -3.812254160e+04 -3.808314562e+04 9.0e-13 10.83
50 8.6e-09 3.5e-07 1.6e-07 1.00e+00 -3.812045133e+04 -3.808105893e+04 9.0e-13 11.22
51 8.6e-09 3.5e-07 1.6e-07 1.00e+00 -3.812045133e+04 -3.808105893e+04 9.0e-13 11.61
52 8.6e-09 3.5e-07 1.6e-07 1.00e+00 -3.812045133e+04 -3.808105893e+04 9.0e-13 12.01
Optimizer terminated. Time: 12.54

Interior-point solution summary
Problem status : DUAL_INFEASIBLE
Solution status : DUAL_INFEASIBLE_CER
Primal. obj: -5.7488560802e-03 nrm: 2e-02 Viol. con: 3e-18 var: 4e-04 cones: 4e-10
Optimizer summary
Optimizer - time: 12.54
Interior-point - iterations : 53 time: 12.46
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

Mosek delivers a certificate of infeasibility that has low quality. So most likely a small perturbation in the problem data makes the problem infeasible.

When I see

noise = 10^-14;

plays role in a model, then I conclude there is something wrong with the scaling of the model.
I refer to the 10^-14 number.

1 Like

Thanks for your reply and for forwarding the question to Mosek people for further analysis.
I refer to your topic post about CVXQUAD, replace log(cvx_expression) with -rel_entr(1,cvx_expresion), and use SDPT3 to solve. The result is still infeasible.

Thank you very much for your reply. Rho/noise=10^8 is always used together in this model. I tried combining them into one parameter, the cvx result is still infeasible. Would even this order of magnitude make the problem infeasible?
When I make M=40, and reduce the length of the corresponding array, cvx gets the optimal solution. However I didn’t change the base model, so I don’t think noise=10^-14 is the main reason.

Of course, the noise=1.0e-14 may not be a problem depending on how you use it.

For instance

noise1e12x+y=1

is innocent.

However, seeing a number like 1.0e-14 is a warning sign of potential problems.

My run using CVX’s Successive Approximation Method using SDPT3 (not with CVXQUAD) finished overnight. It claims to have found a solution.

Successive approximation method to be employed.
For improved efficiency, SDPT3 is solving the dual problem.
SDPT3 will be called several times to refine the solution.
Original size: 62809 variables, 25436 equality constraints
6000 exponentials add 48000 variables, 30000 equality constraints

Cones | Errors |
Mov/Act | Centering Exp cone Poly cone | Status
--------±--------------------------------±--------
6000/6000 | 8.000e+00 1.412e+01 0.000e+00 | Unbounded
5703/5703 | 8.000e+00 5.212e+00 0.000e+00 | Unbounded
4938/4938 | 7.754e+00 3.601e+00 0.000e+00 | Solved
4670/4675 | 4.675e+00 1.250e+00 0.000e+00 | Solved
4164/4241 | 3.384e+00 7.123e-01 0.000e+00 | Solved
2668/3271 | 7.004e-01 4.274e-02 0.000e+00 | Solved
148/592 | 6.340e-02 1.456e-04 0.000e+00 | Solved
0/ 12 | 3.164e-04 2.066e-09 0.000e+00 | Solved

Status: Solved
Optimal value (cvx_optval): +11932.5

Yes, this further proves that there is indeed an optimal solution to this problem. However, considering that I will repeat a large number of similar simulations based on this model, such solution efficiency is unacceptable. So I prefer to use the more efficient Mosek.
I found that Mosek solved the optimal solution when I tried to downsize some data. The specific way is to add the following code before cvx_begin of the initial code.

M=40;
UE_position=UE_position(1:M);
ua_array=ua_array(1:M);
L=L(1:M);

Obviously this doesn’t change the structure of the model, so I think the problem Erling mentioned might not be the main reason. What do you think about this?

    Calling Mosek 9.1.9: 50799 variables, 20636 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 (c) MOSEK ApS, Denmark. WWW: mosek.com
Platform: Windows/64-X86

Problem
  Name                   :                 
  Objective sense        : min             
  Type                   : CONIC (conic optimization problem)
  Constraints            : 20636           
  Cones                  : 14995           
  Scalar variables       : 50799           
  Matrix variables       : 0               
  Integer variables      : 0               

Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator started.
Freed constraints in eliminator : 119
Eliminator terminated.
Eliminator started.
Freed constraints in eliminator : 0
Eliminator terminated.
Eliminator - tries                  : 2                 time                   : 0.00            
Lin. dep.  - tries                  : 1                 time                   : 0.02            
Lin. dep.  - number                 : 0               
Presolve terminated. Time: 0.06    
Problem
  Name                   :                 
  Objective sense        : min             
  Type                   : CONIC (conic optimization problem)
  Constraints            : 20636           
  Cones                  : 14995           
  Scalar variables       : 50799           
  Matrix variables       : 0               
  Integer variables      : 0               

Optimizer  - threads                : 16              
Optimizer  - solved problem         : the primal      
Optimizer  - Constraints            : 19761
Optimizer  - Cones                  : 14995
Optimizer  - Scalar variables       : 50155             conic                  : 44985           
Optimizer  - Semi-definite variables: 0                 scalarized             : 0               
Factor     - setup time             : 0.22              dense det. time        : 0.11            
Factor     - ML order time          : 0.02              GP order time          : 0.00            
Factor     - nonzeros before factor : 1.63e+05          after factor           : 2.80e+05        
Factor     - dense dim.             : 120               flops                  : 1.17e+07        
ITE PFEAS    DFEAS    GFEAS    PRSTATUS   POBJ              DOBJ              MU       TIME  
0   1.0e+01  6.6e+07  3.3e+10  0.00e+00   3.277427897e+10   -2.423000000e-01  1.0e+00  0.34  
1   3.1e+00  2.0e+07  1.8e+10  -1.00e+00  3.277427419e+10   8.535825032e+00   3.0e-01  0.50  
2   5.6e-01  3.6e+06  7.6e+09  -1.00e+00  3.277418600e+10   -1.744871856e+01  5.4e-02  0.55  
3   1.4e-01  9.2e+05  3.9e+09  -1.00e+00  3.277387965e+10   -1.186024007e+02  1.4e-02  0.58  
4   4.1e-02  2.6e+05  2.1e+09  -1.00e+00  3.277283403e+10   -4.590938754e+02  3.9e-03  0.61  
5   1.3e-02  8.1e+04  1.1e+09  -1.00e+00  3.276961409e+10   -1.506098731e+03  1.2e-03  0.67  
6   3.9e-03  2.5e+04  6.4e+08  -1.00e+00  3.275901179e+10   -4.939223650e+03  3.8e-04  0.72  
7   1.0e-03  6.6e+03  3.3e+08  -9.99e-01  3.271641181e+10   -1.864036591e+04  1.0e-04  0.77  
8   2.1e-04  1.3e+03  1.4e+08  -9.96e-01  3.248333886e+10   -9.336111515e+04  2.0e-05  0.80  
9   5.3e-05  3.4e+02  7.2e+07  -9.83e-01  3.171717446e+10   -3.513703852e+05  5.1e-06  0.84  
10  1.5e-05  9.6e+01  3.6e+07  -9.40e-01  2.960891102e+10   -1.161792149e+06  1.4e-06  0.88  
11  3.9e-06  2.5e+01  1.6e+07  -8.28e-01  2.411708978e+10   -3.617884731e+06  3.8e-07  0.92  
12  1.2e-06  7.4e+00  5.3e+06  -4.90e-01  1.465751327e+10   -7.433186756e+06  1.1e-07  0.95  
13  4.6e-07  2.9e+00  1.5e+06  2.23e-01   7.193203733e+09   -9.107699118e+06  4.4e-08  1.00  
14  2.0e-07  1.2e+00  4.0e+05  7.72e-01   3.140749071e+09   -9.084655343e+06  1.9e-08  1.05  
15  2.8e-08  1.8e-01  1.9e+04  1.02e+00   4.181802162e+08   -7.834157307e+06  2.7e-09  1.08  
16  2.6e-09  1.6e-02  5.5e+02  1.06e+00   3.372060006e+07   -4.067069279e+06  2.5e-10  1.11  
17  4.4e-10  2.8e-03  3.9e+01  1.02e+00   5.691113470e+06   -6.832161112e+05  4.2e-11  1.16  
18  1.3e-10  8.2e-04  6.3e+00  1.01e+00   1.659687633e+06   -2.109891516e+05  1.2e-11  1.19  
19  5.1e-11  3.2e-04  1.6e+00  1.01e+00   6.415471181e+05   -9.637279609e+04  4.9e-12  1.23  
20  1.7e-11  1.1e-04  3.2e-01  1.00e+00   2.148255688e+05   -3.802694661e+04  1.7e-12  1.30  
21  5.6e-12  3.6e-05  5.9e-02  1.00e+00   5.866591616e+04   -2.250822364e+04  5.4e-13  1.34  
22  1.7e-12  1.1e-05  9.9e-03  1.00e+00   7.803879281e+03   -1.699852152e+04  1.6e-13  1.38  
23  1.5e-12  9.3e-06  7.8e-03  1.01e+00   4.594784014e+03   -1.653613154e+04  1.4e-13  1.42  
24  1.2e-12  7.4e-06  5.6e-03  1.01e+00   2.384165027e+03   -1.441100362e+04  1.1e-13  1.45  
25  7.1e-13  4.5e-06  2.6e-03  1.01e+00   -2.757530575e+03  -1.296492639e+04  6.8e-14  1.50  
26  6.2e-13  3.9e-06  2.1e-03  1.01e+00   -2.602665408e+03  -1.142180099e+04  5.9e-14  1.53  
27  4.7e-13  3.0e-06  1.4e-03  1.01e+00   -4.038620811e+03  -1.070402437e+04  4.5e-14  1.58  
28  4.0e-13  2.6e-06  1.1e-03  1.02e+00   -3.395130810e+03  -9.105522798e+03  3.9e-14  1.61  
29  3.0e-13  1.9e-06  7.3e-04  1.02e+00   -4.055789840e+03  -8.332047605e+03  2.9e-14  1.64  
30  2.6e-13  1.6e-06  5.6e-04  1.03e+00   -3.823072660e+03  -7.408385182e+03  2.4e-14  1.69  
31  1.2e-13  7.4e-07  1.7e-04  1.03e+00   -4.207849034e+03  -5.832909867e+03  1.1e-14  1.72  
32  8.4e-14  5.3e-07  1.0e-04  1.03e+00   -4.118604103e+03  -5.280307876e+03  8.1e-15  1.76  
33  7.7e-14  4.8e-07  9.0e-05  1.02e+00   -4.194003288e+03  -5.246421835e+03  7.3e-15  1.80  
34  2.4e-14  1.5e-07  1.6e-05  1.01e+00   -4.026227140e+03  -4.361337354e+03  2.3e-15  1.84  
35  1.5e-14  9.4e-08  7.7e-06  1.00e+00   -3.983945002e+03  -4.186385434e+03  1.4e-15  1.88  
36  6.1e-15  2.8e-08  1.3e-06  1.00e+00   -3.919524983e+03  -3.980271569e+03  4.2e-16  1.92  
37  4.3e-15  6.7e-09  1.5e-07  1.01e+00   -3.893474243e+03  -3.907959309e+03  1.0e-16  1.95  
38  8.8e-15  8.0e-09  2.6e-08  1.01e+00   -3.886702210e+03  -3.891204688e+03  3.2e-17  2.00  
39  4.9e-15  8.6e-09  8.9e-09  1.05e+00   -3.884724377e+03  -3.886921623e+03  1.7e-17  2.13  
40  1.9e-15  1.1e-08  2.2e-09  1.02e+00   -3.883542139e+03  -3.884412301e+03  6.9e-18  2.27  
41  4.6e-15  1.6e-08  4.1e-10  1.01e+00   -3.882968643e+03  -3.883246598e+03  2.3e-18  2.39  
42  2.2e-14  1.7e-08  4.1e-10  1.00e+00   -3.882968159e+03  -3.883245655e+03  2.3e-18  2.50  
43  2.1e-14  1.7e-08  4.1e-10  1.00e+00   -3.882968040e+03  -3.883245422e+03  2.2e-18  2.64  
44  2.3e-14  1.8e-08  4.1e-10  1.00e+00   -3.882967980e+03  -3.883245304e+03  2.2e-18  2.78  
45  2.3e-14  1.8e-08  4.1e-10  1.00e+00   -3.882967739e+03  -3.883244835e+03  2.2e-18  2.91  
46  2.2e-14  1.8e-08  4.1e-10  1.00e+00   -3.882967259e+03  -3.883243897e+03  2.2e-18  3.03  
47  2.3e-14  1.8e-08  4.1e-10  1.00e+00   -3.882967139e+03  -3.883243663e+03  2.2e-18  3.17  
48  2.3e-14  1.9e-08  4.1e-10  1.00e+00   -3.882967124e+03  -3.883243634e+03  2.2e-18  3.31  
49  2.3e-14  1.9e-08  4.1e-10  1.00e+00   -3.882967064e+03  -3.883243517e+03  2.2e-18  3.47  
50  2.3e-14  1.9e-08  4.1e-10  1.00e+00   -3.882967064e+03  -3.883243517e+03  2.2e-18  3.58  
51  2.0e-14  3.7e-08  1.9e-11  1.00e+00   -3.882725998e+03  -3.882762164e+03  3.0e-19  3.64  
52  2.2e-14  3.5e-08  1.9e-11  1.00e+00   -3.882725864e+03  -3.882761906e+03  2.9e-19  3.73  
53  2.2e-14  3.7e-08  1.9e-11  1.00e+00   -3.882725730e+03  -3.882761648e+03  2.9e-19  3.86  
54  2.2e-14  3.8e-08  1.9e-11  1.00e+00   -3.882725592e+03  -3.882761382e+03  2.9e-19  4.00  
55  2.3e-14  3.9e-08  1.9e-11  1.00e+00   -3.882725526e+03  -3.882761254e+03  2.9e-19  4.16  
56  2.3e-14  4.0e-08  1.9e-11  1.00e+00   -3.882725509e+03  -3.882761222e+03  2.9e-19  4.33  
57  2.2e-14  4.1e-08  1.9e-11  1.00e+00   -3.882725377e+03  -3.882760967e+03  2.9e-19  4.41  
58  2.3e-14  4.0e-08  1.9e-11  1.00e+00   -3.882725245e+03  -3.882760714e+03  2.9e-19  4.50  
59  1.8e-14  3.9e-08  1.9e-11  1.00e+00   -3.882725108e+03  -3.882760448e+03  2.9e-19  4.67  
60  2.2e-14  3.8e-08  1.8e-11  1.00e+00   -3.882724977e+03  -3.882760197e+03  2.9e-19  4.80  
61  2.3e-14  3.7e-08  1.8e-11  1.00e+00   -3.882724841e+03  -3.882759934e+03  2.9e-19  4.94  
62  2.3e-14  3.8e-08  1.8e-11  1.00e+00   -3.882724812e+03  -3.882759879e+03  2.9e-19  5.09  
63  2.4e-14  3.8e-08  1.8e-11  1.00e+00   -3.882724808e+03  -3.882759871e+03  2.9e-19  5.31  
64  2.3e-14  3.8e-08  1.8e-11  1.00e+00   -3.882724776e+03  -3.882759808e+03  2.9e-19  5.41  
65  2.3e-14  3.9e-08  1.8e-11  1.00e+00   -3.882724711e+03  -3.882759684e+03  2.9e-19  5.50  
66  2.3e-14  4.1e-08  1.8e-11  1.00e+00   -3.882724646e+03  -3.882759560e+03  2.8e-19  5.64  
67  2.3e-14  4.1e-08  1.8e-11  1.00e+00   -3.882724518e+03  -3.882759312e+03  2.8e-19  5.76  
68  2.3e-14  4.2e-08  1.8e-11  1.00e+00   -3.882724261e+03  -3.882758817e+03  2.8e-19  5.89  
69  2.3e-14  4.4e-08  1.8e-11  1.00e+00   -3.882724134e+03  -3.882758572e+03  2.8e-19  6.03  
70  2.3e-14  4.5e-08  1.8e-11  1.00e+00   -3.882723880e+03  -3.882758084e+03  2.8e-19  6.16  
71  2.3e-14  4.4e-08  1.8e-11  1.00e+00   -3.882723747e+03  -3.882757827e+03  2.8e-19  6.30  
72  2.3e-14  4.2e-08  1.8e-11  1.00e+00   -3.882723739e+03  -3.882757812e+03  2.8e-19  6.39  
73  2.3e-14  4.1e-08  1.8e-11  1.00e+00   -3.882723677e+03  -3.882757692e+03  2.8e-19  6.53  
74  2.3e-14  4.1e-08  1.7e-11  1.00e+00   -3.882723614e+03  -3.882757571e+03  2.8e-19  6.66  
75  2.3e-14  4.0e-08  1.7e-11  1.00e+00   -3.882723489e+03  -3.882757331e+03  2.8e-19  6.80  
76  2.4e-14  4.2e-08  1.7e-11  1.00e+00   -3.882723427e+03  -3.882757211e+03  2.8e-19  6.92  
77  2.4e-14  4.0e-08  1.7e-11  1.00e+00   -3.882723425e+03  -3.882757207e+03  2.8e-19  7.01  
78  2.3e-14  4.0e-08  1.7e-11  1.00e+00   -3.882723363e+03  -3.882757088e+03  2.8e-19  7.11  
79  2.3e-14  3.7e-08  1.7e-11  1.00e+00   -3.882723347e+03  -3.882757058e+03  2.8e-19  7.26  
80  2.3e-14  3.7e-08  1.7e-11  1.00e+00   -3.882723332e+03  -3.882757028e+03  2.7e-19  7.41  
81  2.3e-14  3.7e-08  1.7e-11  1.00e+00   -3.882723301e+03  -3.882756968e+03  2.7e-19  7.55  
82  2.3e-14  3.9e-08  1.7e-11  1.00e+00   -3.882723293e+03  -3.882756954e+03  2.7e-19  7.72  
83  2.3e-14  3.9e-08  1.7e-11  1.00e+00   -3.882723262e+03  -3.882756894e+03  2.7e-19  7.81  
84  2.4e-14  3.8e-08  1.7e-11  1.00e+00   -3.882723247e+03  -3.882756864e+03  2.7e-19  7.91  
85  2.4e-14  3.9e-08  1.7e-11  1.00e+00   -3.882723243e+03  -3.882756857e+03  2.7e-19  8.06  
86  2.4e-14  3.9e-08  1.7e-11  1.00e+00   -3.882723243e+03  -3.882756857e+03  2.7e-19  8.25  
87  1.9e-14  1.3e-07  1.4e-12  1.00e+00   -3.882699783e+03  -3.882706221e+03  5.3e-20  8.38  
88  2.3e-14  1.2e-07  1.4e-12  1.00e+00   -3.882699759e+03  -3.882706176e+03  5.3e-20  8.45  
89  2.4e-14  1.2e-07  1.4e-12  1.00e+00   -3.882699758e+03  -3.882706174e+03  5.3e-20  8.53  
90  2.2e-14  1.5e-07  1.1e-12  1.00e+00   -3.882698448e+03  -3.882703941e+03  4.5e-20  8.67  
91  2.3e-14  1.5e-07  1.1e-12  1.00e+00   -3.882698445e+03  -3.882703936e+03  4.5e-20  8.80  
92  2.3e-14  1.6e-07  1.1e-12  1.00e+00   -3.882698445e+03  -3.882703935e+03  4.5e-20  8.94  
93  2.3e-14  1.6e-07  1.1e-12  1.00e+00   -3.882698444e+03  -3.882703934e+03  4.5e-20  9.09  
94  2.2e-14  1.6e-07  1.1e-12  1.00e+00   -3.882698442e+03  -3.882703929e+03  4.5e-20  9.25  
95  2.2e-14  1.6e-07  1.1e-12  1.00e+00   -3.882698439e+03  -3.882703924e+03  4.5e-20  9.38  
96  2.3e-14  1.6e-07  1.1e-12  1.00e+00   -3.882698439e+03  -3.882703923e+03  4.5e-20  9.47  
97  2.4e-14  1.7e-07  1.1e-12  1.00e+00   -3.882698433e+03  -3.882703913e+03  4.5e-20  9.63  
98  2.4e-14  1.7e-07  1.1e-12  1.00e+00   -3.882698433e+03  -3.882703913e+03  4.5e-20  9.72  
99  2.4e-14  1.7e-07  1.1e-12  1.00e+00   -3.882698434e+03  -3.882703912e+03  4.5e-20  9.81  
100 2.3e-14  1.7e-07  1.1e-12  1.00e+00   -3.882698440e+03  -3.882703894e+03  4.5e-20  9.94  
101 2.4e-14  1.7e-07  1.1e-12  1.00e+00   -3.882698441e+03  -3.882703892e+03  4.5e-20  10.08 
102 2.3e-14  1.7e-07  1.1e-12  1.00e+00   -3.882698454e+03  -3.882703857e+03  4.5e-20  10.20 
103 2.3e-14  1.7e-07  1.1e-12  1.00e+00   -3.882698456e+03  -3.882703853e+03  4.5e-20  10.34 
104 2.3e-14  1.7e-07  1.1e-12  1.00e+00   -3.882698458e+03  -3.882703848e+03  4.4e-20  10.47 
105 2.4e-14  1.7e-07  1.1e-12  1.00e+00   -3.882698458e+03  -3.882703846e+03  4.4e-20  10.61 
106 2.4e-14  1.9e-07  9.9e-13  1.00e+00   -3.882698565e+03  -3.882703568e+03  4.1e-20  10.72 
107 2.4e-14  1.9e-07  9.9e-13  1.00e+00   -3.882698565e+03  -3.882703568e+03  4.1e-20  10.88 
108 2.4e-14  1.9e-07  9.9e-13  1.00e+00   -3.882698565e+03  -3.882703568e+03  4.1e-20  11.05 
109 2.4e-14  1.9e-07  9.9e-13  1.00e+00   -3.882698565e+03  -3.882703568e+03  4.1e-20  11.20 
Optimizer terminated. Time: 11.42   


Interior-point solution summary
  Problem status  : PRIMAL_AND_DUAL_FEASIBLE
  Solution status : OPTIMAL
  Primal.  obj: -3.8826985647e+03   nrm: 2e+03    Viol.  con: 2e+00    var: 8e-06    cones: 0e+00  
  Dual.    obj: -3.8827035680e+03   nrm: 4e+07    Viol.  con: 0e+00    var: 8e-01    cones: 0e+00  
Optimizer summary
  Optimizer                 -                        time: 11.42   
    Interior-point          - iterations : 110       time: 11.41   
      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): +10622.8

Here is the result of solving the original problem with Mosek, but with the objective removed.

Calling Mosek 9.3.15: 62809 variables, 25436 equality constraints
For improved efficiency, Mosek is solving the dual problem.

MOSEK Version 9.3.21 (Build date: 2022-8-8 14:59:43)
Copyright © MOSEK ApS, Denmark. WWW: mosek.com
Platform: Windows/64-X86

Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 25436
Cones : 18595
Scalar variables : 62809
Matrix variables : 0
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 : 0.03
Lin. dep. - number : 0
Presolve terminated. Time: 0.22
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 25436
Cones : 18595
Scalar variables : 62809
Matrix variables : 0
Integer variables : 0

Optimizer - threads : 8
Optimizer - solved problem : the dual
Optimizer - Constraints : 18765
Optimizer - Cones : 18476
Optimizer - Scalar variables : 61673 conic : 55428
Optimizer - Semi-definite variables: 0 scalarized : 0
Factor - setup time : 0.70 dense det. time : 0.00
Factor - ML order time : 0.48 GP order time : 0.00
Factor - nonzeros before factor : 1.19e+06 after factor : 1.97e+06
Factor - dense dim. : 0 flops : 2.22e+08
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 1.3e+00 1.6e+04 2.9e+04 0.00e+00 2.884961201e+04 0.000000000e+00 1.0e+00 1.23
1 4.3e-01 5.5e+03 1.7e+04 -1.00e+00 2.884773389e+04 0.000000000e+00 3.3e-01 1.45
2 3.7e-01 4.8e+03 1.5e+04 -1.00e+00 2.884729669e+04 0.000000000e+00 2.9e-01 1.57
3 3.0e-01 3.8e+03 1.4e+04 -1.00e+00 2.884643689e+04 0.000000000e+00 2.3e-01 1.72
4 2.4e-01 3.1e+03 1.3e+04 -1.00e+00 2.884554718e+04 0.000000000e+00 1.9e-01 1.84
5 1.4e-01 1.8e+03 9.6e+03 -1.00e+00 2.884196113e+04 0.000000000e+00 1.1e-01 1.98
6 2.1e-02 2.7e+02 3.7e+03 -1.00e+00 2.879139949e+04 0.000000000e+00 1.6e-02 2.11
7 4.2e-03 5.4e+01 1.6e+03 -1.00e+00 2.854850433e+04 0.000000000e+00 3.3e-03 2.23
8 6.8e-04 8.7e+00 6.6e+02 -1.00e+00 2.696699405e+04 0.000000000e+00 5.3e-04 2.35
9 1.8e-04 2.3e+00 3.4e+02 -1.00e+00 2.178411370e+04 0.000000000e+00 1.4e-04 2.50
10 4.8e-05 6.1e-01 1.8e+02 -1.00e+00 1.732684687e+03 0.000000000e+00 3.7e-05 2.62
11 1.1e-05 1.4e-01 8.3e+01 -9.98e-01 -9.119991786e+04 0.000000000e+00 8.3e-06 2.75
12 2.3e-06 2.9e-02 3.8e+01 -9.87e-01 -5.065173344e+05 0.000000000e+00 1.8e-06 2.87
13 2.0e-05 6.8e-03 1.7e+01 -9.34e-01 -1.948777193e+06 0.000000000e+00 4.1e-07 2.98
14 3.8e-04 6.6e-03 1.6e+01 -6.99e-01 -1.976463751e+06 0.000000000e+00 4.0e-07 3.09
15 3.4e-04 5.6e-03 1.4e+01 -6.88e-01 -2.133444606e+06 0.000000000e+00 3.4e-07 3.23
16 3.6e-04 4.5e-03 1.2e+01 -6.22e-01 -2.451711626e+06 0.000000000e+00 2.7e-07 3.38
17 3.3e-04 4.0e-03 1.1e+01 -5.21e-01 -2.602783642e+06 0.000000000e+00 2.4e-07 3.54
18 4.0e-04 3.9e-03 1.1e+01 -4.59e-01 -2.637793620e+06 0.000000000e+00 2.3e-07 3.71
19 3.6e-04 3.4e-03 1.0e+01 -4.42e-01 -2.777873857e+06 0.000000000e+00 2.1e-07 3.85
20 1.0e-04 9.8e-04 2.9e+00 -3.72e-01 -2.844758795e+06 0.000000000e+00 5.9e-08 3.98
21 1.2e-04 9.8e-04 2.9e+00 3.20e-01 -2.844404685e+06 0.000000000e+00 5.9e-08 4.13
22 1.2e-04 9.5e-04 2.8e+00 3.20e-01 -2.817506672e+06 0.000000000e+00 5.8e-08 4.29
23 1.2e-04 9.3e-04 2.7e+00 3.27e-01 -2.796479628e+06 0.000000000e+00 5.6e-08 4.45
24 9.9e-05 7.6e-04 2.2e+00 3.31e-01 -2.605548614e+06 0.000000000e+00 4.6e-08 4.60
25 1.0e-04 7.6e-04 2.2e+00 3.70e-01 -2.603549178e+06 0.000000000e+00 4.6e-08 4.76
26 6.4e-05 4.7e-04 1.2e+00 3.71e-01 -2.012560510e+06 0.000000000e+00 2.8e-08 4.88
27 6.6e-05 4.7e-04 1.2e+00 5.91e-01 -2.012395102e+06 0.000000000e+00 2.8e-08 5.04
28 5.7e-05 4.1e-04 9.7e-01 5.91e-01 -1.846881225e+06 0.000000000e+00 2.5e-08 5.33
29 5.7e-05 4.1e-04 9.7e-01 6.86e-01 -1.842900105e+06 0.000000000e+00 2.5e-08 5.51
30 5.3e-05 3.8e-04 8.8e-01 6.88e-01 -1.757630525e+06 0.000000000e+00 2.3e-08 5.66
31 5.2e-05 3.7e-04 8.6e-01 7.38e-01 -1.727299135e+06 0.000000000e+00 2.3e-08 5.99
32 3.5e-05 2.6e-04 5.0e-01 7.56e-01 -1.277462442e+06 0.000000000e+00 1.5e-08 6.29
33 3.4e-05 2.5e-04 4.8e-01 1.06e+00 -1.239851298e+06 0.000000000e+00 1.5e-08 6.46
34 3.4e-05 2.5e-04 4.7e-01 1.07e+00 -1.219791868e+06 0.000000000e+00 1.5e-08 6.79
35 2.4e-05 1.7e-04 2.7e-01 1.08e+00 -8.210423074e+05 0.000000000e+00 1.0e-08 7.05
36 2.3e-05 1.7e-04 2.6e-01 1.19e+00 -7.948894343e+05 0.000000000e+00 9.9e-09 7.44
37 6.8e-06 4.9e-05 3.3e-02 1.20e+00 -1.523718089e+05 0.000000000e+00 2.9e-09 7.57
38 2.1e-06 4.1e-05 5.1e-03 1.19e+00 -3.642542255e+04 0.000000000e+00 9.3e-10 7.86
39 1.8e-06 3.5e-05 3.9e-03 1.11e+00 -3.014260894e+04 0.000000000e+00 7.8e-10 8.17
40 1.1e-06 2.6e-05 1.8e-03 1.10e+00 -1.735632800e+04 0.000000000e+00 4.7e-10 8.47
41 2.3e-07 2.1e-05 6.6e-05 1.06e+00 -1.578709691e+03 0.000000000e+00 5.7e-11 8.75
42 2.3e-07 2.1e-05 6.5e-05 1.01e+00 -1.568349824e+03 0.000000000e+00 5.7e-11 9.09
43 1.3e-07 1.2e-05 2.8e-05 1.01e+00 -8.982006181e+02 0.000000000e+00 3.3e-11 9.39
44 1.1e-07 1.0e-05 2.1e-05 1.01e+00 -7.493805442e+02 0.000000000e+00 2.7e-11 9.64
45 2.9e-10 8.3e-07 1.5e-10 1.00e+00 -2.758889497e-01 0.000000000e+00 9.5e-15 9.89
46 1.4e-10 4.2e-07 5.1e-11 1.00e+00 -1.379621807e-01 0.000000000e+00 4.8e-15 10.14
47 1.4e-10 4.2e-07 5.1e-11 1.00e+00 -1.379621807e-01 0.000000000e+00 4.8e-15 10.58
48 1.4e-10 4.2e-07 5.1e-11 1.00e+00 -1.379621807e-01 0.000000000e+00 4.8e-15 10.98
Optimizer terminated. Time: 11.53

Interior-point solution summary
Problem status : UNKNOWN
Solution status : UNKNOWN
Primal. obj: -1.3796151038e-01 nrm: 1e+03 Viol. con: 2e-13 var: 0e+00 cones: 9e-06
Dual. obj: 0.0000000000e+00 nrm: 5e+07 Viol. con: 0e+00 var: 3e+00 cones: 0e+00
Optimizer summary
Optimizer - time: 11.53
Interior-point - iterations : 49 time: 11.43
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: Inaccurate/Solved
Optimal value (cvx_optval): +0.137962

The objective value of the feasibility problem (P_tot, computed starting from CVX variable values) turns out to be 2.204089626565311e+04, which is greater than the Successive Approximation method using SDPT3 optimal objective value of 11932.5; so at least they are not inconsistent.

Although I don’t know what inaccurate solved (or Mosek UNKNOWN) really means for a feasibility problem. Perhaps that there are some constraint violations greater than solver tolerance?

Thanks for your continued attention. This further proves the existence of feasible solutions. I also don’t know what inaccurate solved (or Mosek UNKNOWN) means. But at least the feasible solution I provided fits the constraints. The problem that the original problem cannot get the optimal solution is still not solved.