Hi everyone!
The following problem can be solved normally when K=1 (ie., when running the commented out part). However, when K=2, the status of the output is failed and the optimal value is NAN.
I would really appreciate if anybody has a solution for this.
The algorithm is
M=9;
N=2;
K=2;
%K=1;
p_tran=0.25ones(N,1);
order_b=[1;2]; %N1
order_e=[1,1;2,2]; %NK
% order_e=[1;2]; %NK
h_b=[1.09e-09 - 2.03e-16i,1.00e-09 - 1.96e-16i;
1.09e-09 - 2.03e-16i,1.05e-09 + 6.25e-10i;
1.09e-09 - 2.03e-16i,6.60e-10 + 1.11e-09i;
8.47e-10 - 3.98e-11i,7.49e-10 + 2.28e-11i;
8.47e-10 - 3.98e-11i,3.07e-10 + 3.43e-10i;
8.47e-10 - 3.98e-11i,-2.33e-10 + 2.7e-10i;
1.13e-09 - 9.63e-11i,1.12e-09 + 5.93e-11i;
1.13e-09 - 9.63e-11i,1.10e-09 + 7.14e-10i;
1.13e-09 - 9.63e-11i,6.40e-10 + 1.17e-09i;
4.89e-05 + 1.11e-05i,4.73e-05 + 1.08e-05i];% (M+1)N
h_e=[1.01e-09 - 1.55e-16i,9.41e-10 - 1.51e-16i;
1.01e-09 - 1.55e-16i,9.86e-10 + 5.84e-10i;
1.01e-09 - 1.55e-16i,6.16e-10 + 1.03e-09i;
9.52e-11 - 3.83e-10i,1.10e-10 - 3.31e-10i;
9.52e-11 - 3.83e-10i,1.94e-10 - 9.13e-11i;
9.52e-11 - 3.83e-10i,8.93e-11 + 1.40e-10i;
6.41e-11 - 1.21e-09i,2.25e-10 - 1.17e-09i;
6.41e-11 - 1.21e-09i,9.11e-10 - 1.05e-09i;
6.41e-11 - 1.21e-09i,1.33e-09 - 5.03e-10i;
4.73e-05 + 1.08e-05i,4.59e-05 + 1.04e-05i;
7.87e-10 - 1.00e-16i,7.28e-10 - 9.81e-17i;
7.87e-10 - 1.00e-16i,7.63e-10 + 4.51e-10i;
7.87e-10 - 1.00e-16i,4.77e-10 + 8.03e-10i;
-1.61e-10 - 2.28e-10i,-1.26e-10 - 2.12e-10i;
-1.61e-10 - 2.28e-10i,4.08e-11 - 1.46e-10i;
-1.61e-10 - 2.28e-10i,1.16e-10 + 1.70e-11i;
-3.80e-10 - 6.13e-10i,-2.88e-10 - 6.49e-10i;
-3.80e-10 - 6.13e-10i,8.56e-11 - 8.27e-10i;
-3.80e-10 - 6.13e-10i,4.79e-10 - 6.98e-10i;
4.55e-05 + 1.04e-05i,4.42e-05 + 1.01e-05i];%(K(M+1))N
% h_e=[1.01e-09 - 1.55e-16i,9.41e-10 - 1.51e-16i;
% 1.01e-09 - 1.55e-16i,9.86e-10 + 5.84e-10i;
% 1.01e-09 - 1.55e-16i,6.16e-10 + 1.03e-09i;
% 9.52e-11 - 3.83e-10i,1.10e-10 - 3.31e-10i;
% 9.52e-11 - 3.83e-10i,1.94e-10 - 9.13e-11i;
% 9.52e-11 - 3.83e-10i,8.93e-11 + 1.40e-10i;
% 6.41e-11 - 1.21e-09i,2.25e-10 - 1.17e-09i;
% 6.41e-11 - 1.21e-09i,9.11e-10 - 1.05e-09i;
% 6.41e-11 - 1.21e-09i,1.33e-09 - 5.03e-10i;
% 4.73e-05 + 1.08e-05i,4.59e-05 + 1.04e-05i];
S_b_0=[1.58e+05;1.69e+05];%N1
S_e_0=[1.69e+05,1.83e+05;1.80e+05,1.94e+05];%NK
% S_e_0=[1.69e+05;1.80e+05];%NK
I_b_0=[1.59e-05;9.99e-06];%N1
cvx_begin sdp
cvx_solver mosek
variable U((M+1),(M+1)) hermitian semidefinite
variables S_b(N) S_e(N,K) I_b(N) I_e(N,K) wide_R_b(N) wide_R_en(N,K) wide_R_e(N)
variable s(N,K)
variable z_1(N,K) nonnegative
variable z_2(N,K) nonnegative
expressions cvx_inter_b(N,1) cvx_inter_e(N,K) R_b_lb(N,1) d_S_e_lb(N,K)
for k=1:K
for n=1:N
cvx_inter_b(n,1)=10^(-5);
cvx_inter_e(n,k)=10^(-5);
for i=1:N
if order_b(n,1)<order_b(i,1)
cvx_inter_b(n,1)=cvx_inter_b(n,1)+10^4p_tran(i)real(trace(Uh_b(:,i)h_b(:,i)’));
end
if order_e(n,k)<order_e(i,k)
cvx_inter_e(n,k)=cvx_inter_e(n,k)+10^4p_tran(i)real(trace(Uh_e((k-1)(M+1)+1:k(M+1),i)h_e((k-1)(M+1)+1:k*(M+1),i)’));
end
end
end
end
for k=1:K
for n=1:N
R_b_lb(n,1)=-rel_entr(1,1+1/(S_b_0(n)I_b_0(n)))/log(2)…
-(log(exp(1))/log(2)(S_b(n,1)-S_b_0(n,1)))/(S_b_0(n,1)+S_b_0(n,1)^2I_b_0(n,1))…
-(log(exp(1))/log(2)(I_b(n,1)-I_b_0(n,1)))/(I_b_0(n,1)+I_b_0(n,1)^2S_b_0(n,1));
d_S_e_lb(n,k)=1/(S_e_0(n,k))-(S_e(n,k)-S_e_0(n,k))/(S_e_0(n,k)^2);
end
end
maximize sum(wide_R_b-wide_R_e);
subject to
for k=1:K
for n=1:N
wide_R_e(n,1)>=wide_R_en(n,k);
z_1(n,k)+z_2(n,k)<=1;
s(n,k)-log(2)wide_R_en(n,k)+rel_entr(1,z_1(n,k))<=0;
-log(2)wide_R_en(n,k)+rel_entr(1,z_2(n,k))<=0;
s(n,k)-rel_entr(1,S_e(n,k))-rel_entr(1,I_e(n,k))>=0;
d_S_e_lb(n,k)>=10^4p_tran(n)real(trace(Uh_e((k-1)(M+1)+1:k(M+1),n)h_e((k-1)(M+1)+1:k*(M+1),n)’));%44d
I_e(n,k)<=cvx_inter_e(n,k);
end
end
for n=1:N
wide_R_b(n,1)<=R_b_lb(n,1);
wide_R_b(n,1)>=wide_R_e(n,1);
inv_pos(S_b(n))<=10^4p_tran(n)real(trace(Uh_b(:,n)h_b(:,n)’));
I_b(n)>=cvx_inter_b(n,1);
end
for n=1:N
for i=1:N
if order_b(n,1)<order_b(i,1)
real(trace(Uh_b(:,n)h_b(:,n)’))>=real(trace(Uh_b(:,i)h_b(:,i)’));
end
end
end
for k=1:K
for n=1:N
for i=1:N
if order_e(n,1)<order_e(i,1)
real(trace(Uh_e((k-1)(M+1)+1:k*(M+1),n)h_e((k-1)(M+1)+1:k*(M+1),n)’))>=real(trace(Uh_e((k-1)(M+1)+1:k*(M+1),i)h_e((k-1)(M+1)+1:k*(M+1),i)’));%44f
end
end
end
end
for m=1:(M+1)
U(m,m)==1;%
end
cvx_end
The output is
警告: This linear matrix inequality appears to be unsymmetric. This is
very likely an error that will produce unexpected results. Please check
the LMI; and, if necessary, re-enter the model.
位置：cvxprob/newcnstr (第 241 行)
位置: variable (第 191 行)
位置: v1_0_question (第 58 行)
警告: This linear matrix inequality appears to be unsymmetric. This is
very likely an error that will produce unexpected results. Please check
the LMI; and, if necessary, re-enter the model.
位置：cvxprob/newcnstr (第 241 行)
位置: variable (第 191 行)
位置: v1_0_question (第 59 行)
=====================================
Using Pade approximation for exponential
cone with parameters m=3, k=3
=====================================
=====================================
Using Pade approximation for exponential
cone with parameters m=3, k=3
=====================================
=====================================
Using Pade approximation for exponential
cone with parameters m=3, k=3
=====================================
=====================================
Using Pade approximation for exponential
cone with parameters m=3, k=3
=====================================
=====================================
Using Pade approximation for exponential
cone with parameters m=3, k=3
=====================================
=====================================
Using Pade approximation for exponential
cone with parameters m=3, k=3
=====================================
=====================================
Using Pade approximation for exponential
cone with parameters m=3, k=3
=====================================
=====================================
Using Pade approximation for exponential
cone with parameters m=3, k=3
=====================================
=====================================
Using Pade approximation for exponential
cone with parameters m=3, k=3
=====================================
=====================================
Using Pade approximation for exponential
cone with parameters m=3, k=3
=====================================
=====================================
Using Pade approximation for exponential
cone with parameters m=3, k=3
=====================================
=====================================
Using Pade approximation for exponential
cone with parameters m=3, k=3
=====================================
=====================================
Using Pade approximation for exponential
cone with parameters m=3, k=3
=====================================
=====================================
Using Pade approximation for exponential
cone with parameters m=3, k=3
=====================================
=====================================
Using Pade approximation for exponential
cone with parameters m=3, k=3
=====================================
=====================================
Using Pade approximation for exponential
cone with parameters m=3, k=3
=====================================
Calling Mosek 9.1.9: 465 variables, 231 equality constraints
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 : 231
Cones : 100
Scalar variables : 365
Matrix variables : 1
Integer variables : 0
Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator started.
Freed constraints in eliminator : 12
Eliminator terminated.
Eliminator started.
Freed constraints in eliminator : 2
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 : 231
Cones : 100
Scalar variables : 365
Matrix variables : 1
Integer variables : 0
Optimizer - threads : 4
Optimizer - solved problem : the primal
Optimizer - Constraints : 114
Optimizer - Cones : 94
Optimizer - Scalar variables : 308 conic : 282
Optimizer - Semi-definite variables: 1 scalarized : 210
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 : 569 after factor : 702
Factor - dense dim. : 0 flops : 5.99e+04
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 1.5e+00 3.2e+01 2.2e+01 0.00e+00 0.000000000e+00 -2.100000000e+01 1.0e+00 0.23
1 6.9e-01 1.5e+01 1.3e+01 -8.13e-01 -6.049866145e+01 -7.796533521e+01 4.6e-01 0.31
2 3.2e-01 6.9e+00 6.0e+00 -4.27e-01 -7.759831382e+01 -8.917538075e+01 2.2e-01 0.31
3 1.5e-01 3.3e+00 2.4e+00 1.57e-01 -4.712137725e+01 -5.397972144e+01 1.0e-01 0.33
4 3.2e-02 6.9e-01 2.9e-01 5.00e-01 -1.322976745e+01 -1.494201117e+01 2.1e-02 0.33
5 1.1e-02 2.4e-01 7.7e-02 6.04e-01 -2.451566929e+00 -3.157555846e+00 7.4e-03 0.33
6 6.3e-03 1.4e-01 4.4e-02 4.23e-01 3.182314751e-01 -1.832996046e-01 4.3e-03 0.33
7 2.5e-03 5.4e-02 2.8e-02 -3.44e-01 -7.845865666e-01 -1.030064774e+00 1.7e-03 0.34
8 1.1e-03 2.4e-02 1.4e-02 -4.44e-01 -7.053375449e-01 -5.806517905e-01 7.5e-04 0.34
9 4.1e-04 8.9e-03 7.2e-03 -5.28e-01 -1.995062264e-01 6.761921119e-01 2.8e-04 0.34
10 2.2e-04 4.7e-03 4.2e-03 -5.41e-01 -8.021969961e-02 1.188324745e+00 1.5e-04 0.34
11 7.8e-05 1.7e-03 1.8e-03 -4.46e-01 6.145714697e-02 2.087616577e+00 5.3e-05 0.36
12 4.4e-05 9.6e-04 1.1e-03 -3.63e-01 1.232214056e-01 2.785795300e+00 3.0e-05 0.38
13 1.3e-05 2.8e-04 3.6e-04 -2.85e-01 1.090530498e-01 3.500678482e+00 8.7e-06 0.38
14 5.1e-06 1.1e-04 1.2e-04 9.37e-02 5.347310896e-02 2.403323878e+00 3.4e-06 0.38
15 1.9e-06 4.1e-05 3.1e-05 4.21e-01 1.366720906e-02 1.218482885e+00 1.3e-06 0.39
16 6.9e-07 1.5e-05 6.6e-06 8.86e-01 -1.622696297e-02 3.911105362e-01 4.7e-07 0.39
17 3.6e-07 7.8e-06 2.7e-06 9.66e-01 -2.396398873e-02 2.175431915e-01 2.4e-07 0.39
18 1.5e-07 3.2e-06 7.6e-07 9.05e-01 -2.996418383e-02 8.856004796e-02 9.9e-08 0.41
19 1.0e-07 2.3e-06 5.0e-07 7.23e-01 -3.226816609e-02 7.078032260e-02 7.1e-08 0.41
20 3.7e-08 7.9e-07 1.3e-07 7.06e-01 -3.734040569e-02 1.768640809e-02 2.5e-08 0.41
21 1.5e-08 3.3e-07 4.4e-08 5.55e-01 -4.374561880e-02 -5.653977367e-03 1.0e-08 0.41
22 6.6e-09 1.3e-06 1.5e-08 5.49e-01 -5.222919913e-02 -2.838822048e-02 4.5e-09 0.42
23 3.5e-09 4.3e-06 8.1e-09 3.52e-01 -6.254578314e-02 -3.839414554e-02 2.4e-09 0.42
24 1.2e-09 5.4e-06 1.7e-09 5.91e-01 -6.755192908e-02 -5.839973339e-02 8.2e-10 0.42
25 5.9e-10 8.3e-05 8.7e-10 3.47e-01 -3.270114401e-02 -2.314099474e-02 4.0e-10 0.42
26 2.1e-10 1.0e-05 2.4e-10 5.24e-01 -3.633813015e-03 2.285855995e-03 1.4e-10 0.44
27 5.5e-11 3.7e-04 9.9e-11 4.47e-02 -1.497986516e-02 -2.778156632e-04 3.7e-11 0.44
28 5.5e-13 1.8e-03 1.2e-11 -7.66e-01 -8.015772047e-02 2.067403664e+00 3.7e-13 0.44
29 1.2e-15 6.0e-04 5.5e-13 -9.96e-01 -8.136070812e-02 1.028624851e+03 7.9e-16 0.44
30 2.3e-18 2.0e-04 4.9e-14 -1.00e+00 -8.094180516e-02 5.163695079e+05 1.6e-18 0.45
31 6.9e-17 7.7e-09 5.3e-14 -1.00e+00 -8.267923481e-19 5.547779235e-07 2.2e-23 0.45
Optimizer terminated. Time: 0.48
Interior-point solution summary
Problem status : ILL_POSED
Solution status : PRIMAL_ILLPOSED_CER
Dual. obj: 5.5477778726e-07 nrm: 5e+03 Viol. con: 0e+00 var: 8e-09 barvar: 7e-20 cones: 0e+00
Optimizer summary
Optimizer - time: 0.48
Interior-point - iterations : 31 time: 0.45
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
Many thanks!