Hello, everyone. When I use cvx, I encounter some problems, as follows
Calling Mosek 9.1.9: 6250 variables, 884 equality constraints
For improved efficiency, Mosek is solving the dual problem.
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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
MOSEK warning 710: #12 (nearly) zero elements are specified in sparse col '' (32) of matrix 'A'.
MOSEK warning 710: #3 (nearly) zero elements are specified in sparse col '' (46) of matrix 'A'.
MOSEK warning 710: #3 (nearly) zero elements are specified in sparse col '' (100) of matrix 'A'.
MOSEK warning 710: #4 (nearly) zero elements are specified in sparse col '' (133) of matrix 'A'.
MOSEK warning 710: #1 (nearly) zero elements are specified in sparse col '' (147) of matrix 'A'.
MOSEK warning 710: #1 (nearly) zero elements are specified in sparse col '' (201) of matrix 'A'.
MOSEK warning 710: #8 (nearly) zero elements are specified in sparse col '' (430) of matrix 'A'.
MOSEK warning 710: #2 (nearly) zero elements are specified in sparse col '' (444) of matrix 'A'.
MOSEK warning 710: #2 (nearly) zero elements are specified in sparse col '' (498) of matrix 'A'.
MOSEK warning 710: #4 (nearly) zero elements are specified in sparse col '' (531) of matrix 'A'.
Warning number 710 is disabled.
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 884
Cones : 16
Scalar variables : 1482
Matrix variables : 20
Integer variables : 0
Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator started.
Freed constraints in eliminator : 24
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 : 884
Cones : 16
Scalar variables : 1482
Matrix variables : 20
Integer variables : 0
Optimizer - threads : 16
Optimizer - solved problem : the primal
Optimizer - Constraints : 855
Optimizer - Cones : 17
Optimizer - Scalar variables : 743 conic : 735
Optimizer - Semi-definite variables: 20 scalarized : 9832
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 : 6.38e+04 after factor : 6.46e+04
Factor - dense dim. : 2 flops : 6.95e+07
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 1.3e+02 1.3e+00 1.3e+01 0.00e+00 1.531135369e+01 3.509105336e+00 1.0e+00 0.05
1 3.5e+01 3.5e-01 5.6e+00 -6.88e-01 4.678028163e+01 3.862443819e+01 2.7e-01 0.13
2 5.1e+00 5.1e-02 2.1e+00 -8.83e-01 3.321001950e+02 3.385156753e+02 4.0e-02 0.14
3 6.6e-01 6.6e-03 7.1e-01 -9.45e-01 2.187548819e+03 2.293596718e+03 5.1e-03 0.16
4 1.1e-01 1.1e-03 2.8e-01 -8.77e-01 1.198185847e+04 1.261379843e+04 8.6e-04 0.19
5 2.3e-02 2.3e-04 1.2e-01 -9.35e-01 5.195964428e+04 5.470518257e+04 1.8e-04 0.20
6 2.3e-03 2.3e-05 3.8e-02 -9.87e-01 5.213111896e+05 5.486210202e+05 1.8e-05 0.23
7 2.8e-04 2.8e-06 1.3e-02 -9.98e-01 4.418116376e+06 4.645422545e+06 2.2e-06 0.25
8 9.6e-05 9.7e-07 7.7e-03 -9.93e-01 1.444187640e+07 1.509070424e+07 7.5e-07 0.27
9 2.3e-05 2.3e-07 3.8e-03 -9.91e-01 6.798052006e+07 7.070005298e+07 1.8e-07 0.30
Optimizer terminated. Time: 0.31
Interior-point solution summary
Problem status : PRIMAL_INFEASIBLE
Solution status : PRIMAL_INFEASIBLE_CER
Dual. obj: 1.8023539098e+01 nrm: 4e+04 Viol. con: 0e+00 var: 4e+04 barvar: 2e-07 cones: 0e+00
Optimizer summary
Optimizer - time: 0.31
Interior-point - iterations : 9 time: 0.30
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
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Status: Unbounded
Optimal value (cvx_optval): +Inf
My code
cvx_clear
cvx_begin
% cvx_solver mosek
cvx_save_prefs
variable W(M,M,K) hermitian semidefinite
variables relax_S1_lamda(L) relax_S2_lamda(L) tau_max(K) gammaa(K,L) R(K,K)
expressions LMI_S1(M+N_all+1,M+N_all+1,K) LMI_S2(M+N_all+1,M+N_all+1,K)
P=0;
S=0;
for k=1:K
for l=1:L
E(:,l)=[h_rE(:,l)',h_BE(:,l)'];
A(:,:,k)=[Theta*G*sum(W(:,:,k+1:K),3)*G'*Theta',Theta*G*sum(W(:,:,k+1:K),3);...
sum(W(:,:,k+1:K),3)*G'*Theta',sum(W(:,:,k+1:K),3)];
a(:,:,k,l)=A(:,:,k)+relax_S1_lamda(l)*eye(M+N_all);
b(:,k,l)=E(:,l)'*A(:,:,k);
c(k,l)=E(:,l)'*A(:,:,k)*E(:,l)+noise-gammaa(k,l)-relax_S1_lamda(l)*epsilong(l);
LMI_S1(:,:,k,l)=[a(:,:,k,l) b(:,k,l);...
b(:,k,l)' c(k,l)];
right_up(k,l)=(exp(tau_max_t(1,k))-1)*gammaa(k,l)+gamma_t(k,l)*exp(tau_max_t(1,k))*...
(tau_max(k)-real(tau_max_t(1,k)));
B(:,:,k)=[Theta*G*W(:,:,k)*G'*Theta',Theta*G*W(:,:,k);...
W(:,:,k)*G'*Theta',W(:,:,k)];
a0(:,:,k,l)=relax_S2_lamda(l)*eye(M+N_all)-B(:,:,k);
b0(:,k,l)=-E(:,l)'*B(:,:,k);
c0(k,l)=-E(:,l)'*B(:,:,k)*E(:,l)+right_up(k,l)-relax_S2_lamda(l)*epsilong(l);
LMI_S2(:,:,k,l)=[a0(:,:,k,l) b0(:,k,l);...
b0(:,k,l)' c0(k,l)];
end
S=S-rel_entr(1,R(k,k))/log(2)-tau_max(k)*log2(exp(1));
end
maximize real(trace(R))
subject to
for k=1:K
for l=1:L
LMI_S1(:,:,k,l) ==hermitian_semidefinite(M+N_all+1);
LMI_S2(:,:,k,l) ==hermitian_semidefinite(M+N_all+1);
relax_S1_lamda(l)>0;
relax_S2_lamda(l)>0;
end
P=P+trace(W(:,:,k));
tau_max(k)*log2(exp(1))<=M_E;
end
for j=2:K
for k=1:j-1
W_temp3=0;
for i=k+1:K
W_temp3=W_temp3+h_k(:,j)'*W(:,:,i)*h_k(:,j);
end
pow_pos((R(j,k)-1)*real(A_t(j,k)),2)+pow_pos((real(W_temp3)+noise)/real(A_t(j,k)),2)<=...
2*real(h_k(:,j)'*W(:,:,k)*h_k(:,j));%Convert R (j, k) to convex constraint by AMG
R(j,k)>=R(k,k);
end
end
cvx_end
Some initial values
What is the reason for the unbounded？