The following code outputs an infeasible solution, but the problem is indeed feasible! May someone help me to figure it out and thanks!

For example, the optimized a and b are

a =

1.0e-08 *

```
0.5729
0.5629
```

b =

1.0e-08 *

```
0.5602 0.5713
0.5782 0.5653
```

The objective is 0.1284.

The result is obviously not correct since a and b violate the constraints. However, the problem is indeed feasible, for example when a = [0.5;0.5] and b=[0 1; 1 0], the problem is feasible.

**MATLAB CODE (this code can run without errors)**

```
clear all
rng(0)
%%
T=2;
L = 2;
alph = 1;
bta = 1;
D = ones(T, 1);
P = 10; % Watt`
N0 = 1e-3*10^(-174/10); % Watt/Hz, -174 dbm/Hz
W = 100e6; % W MHz
N = 10;
ksi = 0.5; % A/Watt
h = exprnd(1e-6, N, T); % channel gain, simplified
R = 1e7; % 10Mbps
tau = 1; % tau seconds video
miu = 1e6*[50, 150]'; % bitrate
%%
Delta = (exp(1)/(2*pi)*P^2/N0*(ksi*sum(h, 1)).^2)';
V = repmat((miu*tau)', T, 1);
%% BnB
cvx_solver mosek
cvx_precision high
cvx_begin
variable b(T, L)
variable a(T, 1)
obj = 1/T*sum(rel_entr(1, alph./D.*sum(b.*V, 2)));
minimize(obj)
subject to
0 <= b <= 1;
sum(b,2) == 1;
0 <= a <= 1;
sum(a) == 1
-tau*rel_entr(a.*W, a.*W+Delta) >= sum(b.*V, 2);
cvx_end
b = full(b);
qoe = -1/T*sum(rel_entr(1, alph./D.*sum(b.*V, 2)))
```

**Output**

CVX Warning:

Models involving “rel_entr” or other functions in the log, exp, and entropy

family are solved using an experimental successive approximation method.

This method is slower and less reliable than the method CVX employs for

other models. Please see the section of the user’s guide entitled

The successive approximation method

for more details about the approach, and for instructions on how to

suppress this warning message in the future.

## Calling Mosek 9.1.9: 27 variables, 9 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 © MOSEK ApS, Denmark. WWW: mosek.com

Platform: Windows/64-X86

MOSEK warning 57: A large value of 5.2e+10 has been specified in c for variable ‘’ (23).

MOSEK warning 57: A large value of 3.7e+10 has been specified in c for variable ‘’ (26).

Problem

Name :

Objective sense : min

Type : CONIC (conic optimization problem)

Constraints : 9

Cones : 3

Scalar variables : 27

Matrix variables : 0

Integer variables : 0

Optimizer started.

Presolve started.

Linear dependency checker started.

Linear dependency checker terminated.

Eliminator started.

Freed constraints in eliminator : 2

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: 0.03

Problem

Name :

Objective sense : min

Type : CONIC (conic optimization problem)

Constraints : 9

Cones : 3

Scalar variables : 27

Matrix variables : 0

Integer variables : 0

Optimizer - threads : 10

Optimizer - solved problem : the primal

Optimizer - Constraints : 6

Optimizer - Cones : 4

Optimizer - Scalar variables : 26 conic : 14

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 : 21 after factor : 21

Factor - dense dim. : 0 flops : 2.79e+02

ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME

0 1.0e+00 5.2e+10 1.1e+11 0.00e+00 1.141272097e+11 -4.025510008e-01 1.0e+00 0.03

1 1.2e-01 6.2e+09 4.0e+10 -1.00e+00 1.141244680e+11 -6.189938641e+00 1.2e-01 0.08

2 1.9e-02 1.0e+09 1.6e+10 -1.00e+00 1.141242944e+11 -3.586542216e+01 1.9e-02 0.08

3 6.8e-03 3.5e+08 9.4e+09 -1.00e+00 1.141242815e+11 -9.518210206e+01 6.8e-03 0.08

4 1.9e-03 9.7e+07 4.9e+09 -1.00e+00 1.141242762e+11 -3.554443936e+02 1.9e-03 0.09

5 5.3e-05 2.8e+06 8.3e+08 -1.00e+00 1.141238554e+11 -1.235443378e+04 5.3e-05 0.09

6 6.4e-07 3.3e+04 9.1e+07 -1.00e+00 1.140890586e+11 -1.094891031e+06 6.4e-07 0.09

7 7.8e-09 4.0e+02 9.9e+06 -9.99e-01 1.112885239e+11 -8.997440240e+07 7.8e-09 0.09

8 6.1e-09 1.4e+01 1.4e+06 -9.54e-01 6.426330097e+10 -1.561326813e+09 2.7e-10 0.09

9 3.1e-09 6.3e+00 5.9e+05 -1.59e-01 4.126593215e+10 -1.512412990e+09 1.2e-10 0.11

10 3.0e-09 5.9e+00 5.4e+05 3.00e-01 3.932531344e+10 -1.444784481e+09 1.1e-10 0.11

11 2.0e-09 1.5e+00 6.7e+04 3.17e-01 1.211608038e+10 -5.291480873e+08 3.0e-11 0.11

12 5.8e-10 1.7e-01 2.9e+03 7.50e-01 1.501952555e+09 -8.676488459e+07 3.6e-12 0.11

13 5.8e-10 1.7e-01 2.9e+03 9.73e-01 1.501952555e+09 -8.676488459e+07 3.6e-12 0.13

14 5.8e-10 1.7e-01 2.9e+03 9.73e-01 1.501952555e+09 -8.676488459e+07 3.6e-12 0.13

Optimizer terminated. Time: 0.16

Interior-point solution summary

Problem status : UNKNOWN

Solution status : UNKNOWN

Primal. obj: 1.5019525550e+09 nrm: 6e+08 Viol. con: 7e+08 var: 0e+00 cones: 0e+00

Dual. obj: -8.6764884591e+07 nrm: 5e+10 Viol. con: 0e+00 var: 8e+08 cones: 0e+00

Optimizer summary

Optimizer - time: 0.16

Interior-point - iterations : 15 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: Inaccurate/Unbounded

Optimal value (cvx_optval): -Inf

qoe =

`0.1284`