The problem status is ill-posed

my parameters are as follows
beta=[1.3526,0.1405];
sigma1 = [9485,187];
eta = [80.2606,12.9895];
phi = [157,2456,25.4488];
a = [0.0.316;0,0];
my code is
cvx_begin
cvx_solver mosek
variables l(1,K) p(1,K)
expressions I(1,K) I_old(1,K) dc(K,K) c1(1,K) c2(1,K)
for k=1:K
I(k) = 0;
I_old(k) = 0;
dc(k,k) =1/(log(2)(p_old(k)+eta(k)+p_mphi(k)));
for j=k+1:K
I(k) = I(k)+ a(k,j).p(j);
I_old(k) = I_old(k)+ a(k,j).p_old(j);
c1(k) = log((p(k)+sigma1(k)+I(k)+p_mbeta(k))(eta(k)+p_mphi(k)))/log(2);
c2(k) = log((sigma1(k)+I_old(k)+p_mbeta(k))(eta(k)+p_mphi(k)+p_old(k)))/log(2);
dc(k,j) = a(k,j)/(log(2)(sigma1(k)+ I_old(k)+p_mbeta(k)));
end
end
obj = sum(zeta1.pow_abs(l,3)+p.T)+p_mT;%
minimize obj
subject to
for k=1:K
c1(k) - (c2(k)+ dc(k,(p-p_old)’) >= (L-l(k))/10^5;
l(k)>=0;
l(k)<=L;
p(k)>=0;
end
cvx_end

but, the cvx result is failed
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 1.0e+00 9.5e+03 4.0e+01 0.00e+00 3.937977890e+01 -1.000000000e-08 1.0e+00 0.03
1 1.6e-01 1.5e+03 1.6e+01 -1.00e+00 3.401089412e+01 4.626146472e-02 1.6e-01 0.09
2 2.5e-02 2.4e+02 6.4e+00 -1.00e+00 9.800189297e-01 -1.453999747e-02 2.5e-02 0.11
3 2.3e-03 2.1e+01 1.9e+00 -1.00e+00 -4.078197358e+02 -7.138912494e-01 2.3e-03 0.11
4 5.3e-04 5.0e+00 9.3e-01 -1.00e+00 -1.862107308e+03 -4.426419268e+00 5.3e-04 0.11
5 1.3e-04 1.2e+00 4.4e-01 -9.86e-01 -7.597812859e+03 -5.537082432e+01 1.3e-04 0.11
6 2.5e-05 2.4e-01 1.7e-01 -8.91e-01 -2.812936830e+04 -2.713137350e+02 2.5e-05 0.11
7 4.0e-06 3.8e-02 2.8e-02 -3.93e-01 -2.942013587e+04 -7.285984717e+02 4.0e-06 0.11
8 4.6e-07 4.4e-03 1.1e-03 7.40e-01 -4.542864525e+03 -9.191886108e+02 4.6e-07 0.13
9 4.7e-08 4.5e-04 2.8e-05 1.12e+00 -1.164514789e+03 -9.458344869e+02 4.7e-08 0.13
10 1.5e-09 1.4e-05 1.7e-07 1.07e+00 -9.558061482e+02 -9.484863179e+02 1.5e-09 0.13
11 4.4e-10 4.2e-06 3.9e-08 6.61e-01 -9.533771678e+02 -9.485377730e+02 4.4e-10 0.13
12 2.1e-10 2.0e-06 1.7e-08 1.21e-01 -9.524523543e+02 -9.485543006e+02 2.1e-10 0.13
13 1.1e-10 1.0e-06 7.2e-09 4.28e-01 -9.512977146e+02 -9.485672776e+02 1.1e-10 0.13
14 3.2e-11 3.1e-07 1.1e-09 8.36e-01 -9.492490198e+02 -9.485835280e+02 3.2e-11 0.14
15 3.1e-12 2.9e-08 3.4e-11 9.88e-01 -9.486619887e+02 -9.485876144e+02 3.1e-12 0.14
16 5.4e-13 5.1e-09 2.8e-12 9.33e-01 -9.486036717e+02 -9.485875142e+02 5.4e-13 0.14
17 2.5e-13 2.4e-09 2.4e-12 5.23e-02 -9.486632818e+02 -9.486061563e+02 2.5e-13 0.14
18 4.2e-14 3.9e-10 8.1e-13 -4.61e-01 -9.488941629e+02 -9.486677484e+02 4.2e-14 0.16
19 5.2e-15 5.0e-11 2.5e-13 -8.63e-01 -9.501816197e+02 -9.488940270e+02 5.2e-15 0.16
20 2.3e-15 1.9e-11 1.3e-13 -1.04e+00 -9.539981112e+02 -9.501582074e+02 2.0e-15 0.16
21 4.4e-16 4.2e-12 3.7e-14 -1.00e+00 -9.730059164e+02 -9.557071441e+02 4.4e-16 0.16
22 2.6e-16 2.4e-12 4.6e-14 -9.83e-01 -9.884220671e+02 -9.592154512e+02 2.6e-16 0.16
23 5.0e-17 1.1e-12 3.0e-14 -9.92e-01 -1.150534663e+03 -9.993406460e+02 4.9e-17 0.16
24 4.5e-17 1.0e-12 1.0e-13 -9.93e-01 -1.172323512e+03 -1.004251462e+03 4.4e-17 0.17
25 4.4e-17 6.0e-13 1.2e-13 -9.94e-01 -1.184077933e+03 -1.006936014e+03 4.2e-17 0.17
26 3.8e-17 7.4e-13 1.4e-13 -9.94e-01 -1.210243690e+03 -1.012937214e+03 3.8e-17 0.19
27 3.6e-17 1.3e-12 6.3e-14 -9.94e-01 -1.230621425e+03 -1.017734071e+03 3.5e-17 0.19
28 3.5e-17 1.1e-12 8.4e-14 -9.94e-01 -1.245547595e+03 -1.021168097e+03 3.3e-17 0.19
29 1.6e-17 3.3e-11 4.1e-14 -9.95e-01 -2.545004528e+03 -1.320751512e+03 6.1e-18 0.19
30 1.6e-17 3.3e-11 1.5e-13 -1.01e+00 -2.550004682e+03 -1.321951622e+03 6.1e-18 0.19
31 1.6e-17 3.3e-11 1.5e-13 -1.01e+00 -2.550004682e+03 -1.321951622e+03 6.1e-18 0.19
32 1.6e-17 3.3e-11 8.9e-14 -1.01e+00 -2.555514940e+03 -1.322903505e+03 6.1e-18 0.20
33 1.6e-17 3.3e-11 2.5e-13 -1.00e+00 -2.556673699e+03 -1.323215915e+03 6.1e-18 0.20
34 1.6e-17 3.3e-11 2.9e-13 -1.00e+00 -2.576210429e+03 -1.328192233e+03 6.0e-18 0.20
35 1.6e-17 3.3e-11 4.3e-14 -1.01e+00 -2.581575291e+03 -1.329362000e+03 6.0e-18 0.20
36 1.6e-17 3.3e-11 1.0e-13 -1.01e+00 -2.587099570e+03 -1.330522972e+03 6.0e-18 0.20
37 1.5e-17 3.0e-11 3.0e-13 -1.00e+00 -2.626800495e+03 -1.340854480e+03 5.8e-18 0.20
38 1.5e-17 3.0e-11 3.0e-13 -1.00e+00 -2.626800495e+03 -1.340854480e+03 5.8e-18 0.22
39 1.5e-17 3.0e-11 3.0e-13 -9.59e-01 -2.626800495e+03 -1.340854480e+03 5.8e-18 0.22
Optimizer terminated. Time: 0.25

Interior-point solution summary
Problem status : ILL_POSED
Solution status : DUAL_ILLPOSED_CER
Primal. obj: -1.6684035842e-07 nrm: 2e+01 Viol. con: 4e-12 var: 0e+00 cones: 4e-18
Optimizer summary
Optimizer - time: 0.25
Interior-point - iterations : 40 time: 0.22
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

How to solve this problem?

The dual problem is not robust feasible.

So assuming the problem has a feasible solution it is close to being unbounded.

However, the problem works with SDPT3; it gets an error when using Mosek. Following is the information about my optimization problem. What it the reason?

               cvx_begin 
              cvx_solver mosek
               
               variable phi_ut(N,1) complex;
               variable phi_ur(N,1) complex;
               variable Phi_ut(N,N) hermitian;
               variable Phi_ur(N,N) hermitian;
               variable Y nonnegative;
               variable Lambda1_t(N,N) hermitian;
               variable Lambda1_r(N,N) hermitian;
               variable Lambda2_t(N,N) hermitian;
               variable Lambda2_r(N,N) hermitian;
            
               maximize Y;
               subject to
            
               for i=1:N
                   AA(i) = Phi_ut(i,i);
                   BB(i) = Phi_ur(i,i);
               end

               AA + BB == ones(1,N);
               Phi_ut == hermitian_semidefinite(N);
               Phi_ur == hermitian_semidefinite(N);
               M_t = [Lambda1_t, Phi_ut, phi_ut; Phi_ut', Lambda2_t, phi_ut; phi_ut', phi_ut', 1];
               M_r = [Lambda1_r, Phi_ur, phi_ur; Phi_ur', Lambda2_r, phi_ur; phi_ur', phi_ur', 1];            
               M_t == hermitian_semidefinite(2*N+1); 
               M_r == hermitian_semidefinite(2*N+1);
               trace(Lambda1_t) <= 2*real(trace(phi_tin*phi_ut')) - trace(phi_tin*phi_tin');
               trace(Lambda1_r) <= 2*real(trace(phi_rin*phi_ur')) - trace(phi_rin*phi_rin');

               for q =1:K
                    Y + real(phi_ut'*Omega_t(:,:,q) * phi_ut) + real(phi_ur'*Omega_r(:,:,q) * phi_ur) + norm(lambda(q),2)^2*norm((U_in(:,q)'*sigma),2)^2 <=  2*real(omega_t(q, :) * phi_ut);
                    Y + real(phi_ut'*Omega_t(:,:,q + K) * phi_ut) + real(phi_ur'*Omega_r(:,:,q + K) * phi_ur) + norm(lambda(q + K),2)^2*norm((U_in(:,q +K)'*sigma),2)^2 <=  2*real(omega_r(q, :) * phi_ur);
               end 
            
            cvx_end

Calling Mosek 10.1.25: 767 variables, 350 equality constraints

MOSEK Version 10.1.25 (Build date: 2024-2-14 13:03:04)
Copyright (c) MOSEK ApS, Denmark WWW: mosek.com
Platform: MACOSX/aarch64

Problem
Name :
Objective sense : minimize
Type : CONIC (conic optimization problem)
Constraints : 350
Affine conic cons. : 0
Disjunctive cons. : 0
Cones : 8
Scalar variables : 61
Matrix variables : 4 (scalarized: 1462)
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 : 0.00
Lin. dep. - primal attempts : 1 successes : 1
Lin. dep. - dual attempts : 0 successes : 0
Lin. dep. - primal deps. : 0 dual deps. : 0
Presolve terminated. Time: 0.00
GP based matrix reordering started.
GP based matrix reordering terminated.
Optimizer - threads : 8
Optimizer - solved problem : the primal
Optimizer - Constraints : 350
Optimizer - Cones : 8
Optimizer - Scalar variables : 61 conic : 54
Optimizer - Semi-definite variables: 4 scalarized : 1462
Factor - setup time : 0.00
Factor - dense det. time : 0.00 GP order time : 0.00
Factor - nonzeros before factor : 3.10e+04 after factor : 3.18e+04
Factor - dense dim. : 0 flops : 5.56e+06
Factor - GP saved nzs : 768 GP saved flops : 1.78e+05
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 1.9e+01 1.0e+00 1.0e+00 0.00e+00 0.000000000e+00 0.000000000e+00 1.0e+00 0.01
1 5.6e+00 2.9e-01 1.6e-01 6.97e-01 -9.735666047e-03 -2.848319462e-02 2.9e-01 0.02
2 1.4e+00 7.3e-02 2.5e-02 9.00e-01 -9.812045177e-03 1.655822563e-02 7.3e-02 0.02
3 3.9e-01 2.0e-02 3.6e-03 9.08e-01 -1.058976275e-02 -4.438749441e-03 2.0e-02 0.03
4 7.7e-02 3.9e-03 4.4e-04 7.43e-01 -9.223025061e-03 -3.355987353e-03 3.9e-03 0.03
5 1.8e-02 9.3e-04 8.4e-05 4.22e-01 -9.068630337e-03 -3.258421548e-03 9.3e-04 0.04
6 4.8e-03 2.5e-04 2.1e-05 1.38e-01 -8.154930303e-03 -1.860620426e-03 2.5e-04 0.04
7 1.5e-03 7.9e-05 6.6e-06 7.70e-02 -7.550765275e-03 -1.245310561e-03 7.9e-05 0.05
8 3.8e-04 1.9e-05 1.6e-06 2.65e-02 -7.137179862e-03 -4.857049099e-04 1.9e-05 0.05
9 1.4e-04 7.0e-06 5.7e-07 1.95e-02 -7.082203765e-03 -6.035528185e-04 7.0e-06 0.05
10 3.3e-05 1.7e-06 1.4e-07 6.31e-03 -6.890862619e-03 -1.950633496e-04 1.7e-06 0.06
11 1.0e-05 5.2e-07 4.3e-08 6.86e-04 -6.862637214e-03 -2.433397002e-04 5.2e-07 0.06
12 2.8e-06 1.5e-07 1.2e-08 -2.34e-04 -6.783869060e-03 -1.139713588e-04 1.5e-07 0.06
13 8.4e-07 4.3e-08 3.5e-09 -3.98e-03 -6.717101921e-03 -3.565424631e-06 4.3e-08 0.07
14 2.5e-07 1.3e-08 1.1e-09 -2.89e-03 -6.683644639e-03 6.890483990e-06 1.3e-08 0.07
15 6.9e-08 3.5e-09 2.9e-10 -5.35e-03 -6.608625400e-03 1.636262144e-04 3.5e-09 0.07
16 2.1e-08 1.1e-09 8.8e-11 -1.71e-03 -6.610632220e-03 8.125847520e-05 1.1e-09 0.08
17 5.8e-09 3.0e-10 2.5e-11 -3.38e-03 -6.553608148e-03 2.035108058e-04 3.0e-10 0.08
18 1.7e-09 8.7e-11 7.1e-12 -4.74e-03 -6.523806337e-03 2.390025368e-04 8.6e-11 0.08
19 5.0e-10 2.6e-11 2.1e-12 -2.12e-03 -6.513960480e-03 2.189800641e-04 2.6e-11 0.08
20 1.4e-10 7.4e-12 5.9e-13 -4.76e-03 -6.462113055e-03 3.391022877e-04 7.1e-12 0.09
21 4.1e-11 2.7e-12 1.7e-13 -2.84e-03 -6.465222091e-03 2.854683177e-04 2.1e-12 0.09
22 1.1e-11 9.3e-13 4.9e-14 -3.21e-03 -6.432695812e-03 3.533778300e-04 6.0e-13 0.09
23 3.8e-12 9.0e-13 1.4e-14 -4.05e-03 -6.415665633e-03 3.739228611e-04 1.7e-13 0.10
24 1.5e-12 2.3e-12 4.2e-15 -2.77e-03 -6.407477832e-03 3.702339501e-04 5.1e-14 0.10
25 4.1e-13 1.0e-12 1.2e-15 -1.22e-03 -6.387665888e-03 4.073142815e-04 1.4e-14 0.10
26 1.3e-13 2.4e-12 3.8e-16 -2.02e-03 -6.369981340e-03 4.395992433e-04 4.2e-15 0.10
27 3.5e-14 2.5e-12 3.5e-18 1.09e-04 -6.357052556e-03 4.598049456e-04 1.2e-15 0.11
28 2.1e-14 3.8e-12 1.3e-16 -2.32e-04 -6.337928068e-03 5.018704626e-04 3.5e-16 0.11
29 1.7e-14 1.3e-12 1.4e-17 -1.92e-03 -6.323580263e-03 5.309988383e-04 1.1e-16 0.11
30 4.8e-15 2.7e-12 6.0e-17 7.62e-04 -6.309760934e-03 5.598625327e-04 3.0e-17 0.12
31 1.4e-15 9.3e-12 3.6e-18 -2.04e-03 -6.294831911e-03 5.944773133e-04 8.8e-18 0.12
32 4.1e-16 1.4e-11 6.4e-17 1.76e-04 -6.284070802e-03 6.166282576e-04 2.6e-18 0.13
33 1.2e-16 6.6e-12 4.5e-17 -9.99e-04 -6.270661401e-03 6.493123923e-04 7.4e-19 0.13
Optimizer terminated. Time: 0.13

Interior-point solution summary
Problem status : ILL_POSED
Solution status : PRIMAL_ILLPOSED_CER
Dual. obj: 7.7055280375e-12 nrm: 4e+00 Viol. con: 0e+00 var: 1e-08 barvar: 7e-12 cones: 0e+00
Optimizer summary
Optimizer - time: 0.13
Interior-point - iterations : 33 time: 0.13
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

My reply above applies to this case.

Btw if two different optimizers provide different conclusions for a problem then it implies

• 1. one or both optimizers are broken
• or 2) your problem is nasty.

After seeing the Mosek output I am pretty sure option 2) applies in this case.

Thanks for your prompt reply.

What can I do for that? I appreciate your advice.

Fixing this is often tricky and requires an in-depth knowledge of the problem. Moreover, one has to understand the theory of conic optimization. I am not saying that is easy but there is no way around it IMO.

For a starter for instance

Theorem 2.1

https://www2.isye.gatech.edu/~nemirovs/ICMNemirovski.pdf

tells you that the lack of strong feasibility of either the primal or dual problem most likely is the cause of the issue.

There is something called

facial reduction

pioneered by Borwein and Wolkowitz that may be worthwhile to study too.

Enjoy.

Thanks for your advice