Solving MISCOP using CVX


(Ahmed Al-baidhani) #1

Hi,

Given a MISCOP problem which was solved using Gurobi. Does this mean we can solve this MISOCP problem using CVX by setting cvx_solver Gurobi?
I have academic licenses for both CVX and Gurobi and both are running properly.

Regards.
Ahmed


Using CVX to solvie MISCOP
(Ahmed Al-baidhani) #2

Hi,
I’m trying to solve the following MISCOP problem. This problem has been solved using Gurobi optimizer according to the following paper ref section 4.1.
The probelm is formulated as follows
\text{min.}\ \ \sum_{i\in\mathcal{N}} P_i
\text{s.t.} \ \ \ \ \ \ \ x_{0,1} x_{0,2}\geq x_{1,1}^2
\ \ \ \ \ \ \ \ \ \ \ \ x_{0,3} x_{0,4}\geq x_{1,2}^2
\ \ \ \ \ \ \ \ \ \ \ \ x_{0,5} x_{0,6}\geq x_{1,3}^2
\ \ \ \ \ \ \ \ \ \ \ \ x_{0,7} x_{0,8}\geq x_{1,4}^2
\ \ \ \ \ \ \ \ \ \ \ \ x_{1,1} x_{1,2}\geq x_{2,1}^2
\ \ \ \ \ \ \ \ \ \ \ \ x_{1,3} x_{1,4}\geq x_{2,2}^2
\ \ \ \ \ \ \ \ \ \ \ \ x_{2,1} x_{2,2}\geq x_{3,1}^2
\ \ \ \ \ \ \ \ \ \ \ \ x_{3,1}\geq 2^{C/N}
\ \ \ \ \ \ \ \ \ \ \ \ x_{0,i} =1+y_is_{i}^2, \ \ \ i\in\mathcal{N}
\ \ \ \ \ \ \ \ \ \ \ \ x_{j,m}\geq 0 \ \ \ j=0,..,3 , \ \ \ \ m=1,...2^{3-j}
\ \ \ \ \ \ \ \ \ \ \ \ \sum_{i\in\mathcal{N}} P_is_i^2-y_is_i^2\geq E
\ \ \ \ \ \ \ \ \ \ \ \ 0 \leq y_i \leq P_i \leq P_{max}
\ \ \ \ \ \ \ \ \ \ \ \ y_i\leq \pi_i P_{max},\ \ y_i\geq P_i-(1-\pi_i)P_{max} ,\ \ \pi_i \in \{0,1\}

where \mathcal{N}=\{1,\ldots,8\} and s_i is the i-th singular value of a matrix \mathbf{H} \in \mathbb{R}^{8\times 8}

I’m trying to solve via CVX via the following code:
N=8; H=randn(N);
s=svd(H);l=s.^2;
C=10;E=5;
Pmax=2;
P_UB=Pmax*ones(N,1);
P_LB=zeros(N,1);k=0;
cvx_solver Gurobi
cvx_begin
variables P(N) y(N) x0(N) xn(N-1)
variable w(N)
x0=1+l.*y;
minimize sum _P
subject to
xn>=zeros(N-1,1);
for ii=1:2:N
x0(ii)*x0(ii+1)>=xn(k+1);
k=k+1;
end
xn(1)*xn(2)>=xn(5)^2;
xn(3)*xn(4)>=xn(6)^2;
xn(5)*xn(6)>=xn(7)^2;
xn(7)>=2^(C/N);
sum(P.*l-y.*l)>=E;
P_LB<=y<=P<=P_UB;
y<=w.*P_UB;
y>=P-(1-w).*P_UB;
cvx_end

but I have the following errors:
Error using .* (line 262)
Disciplined convex programming error:
Invalid quadratic form(s): not a square.

Error in * (line 36)
z = feval( oper, x, y );

Error in svd1_cvx (line 19)
x0(ii)*x0(ii+1)>=xn(k+1);

Thanks in advance


(Ahmed Al-baidhani) #3

Hi,
I have modified the code to be like this:
N=8; H=randn(N);
s=svd(H);l=s.^2;
C=10;E=5;
Pmax=1000;
P_UB=Pmax*ones(N,1);
P_LB=zeros(N,1);k=0;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
cvx_solver Gurobi
cvx_begin
variables P(N) y(N)
variable w(N) binary
expressions x0(N) xn(N-1)
%x0=1+l.*y;

minimize sum §
subject to
xn>=zeros(N-1,1);
x0>=ones(N,1)
for ii=1:2:N
x0(ii)*x0(ii+1)>=xn(k+1);
k=k+1;
end
xn(1)*xn(2)>=xn(5)^2;
xn(3)*xn(4)>=xn(6)^2;
xn(5)*xn(6)>=xn(7)^2;
xn(7)>=2^(C/N);
sum(P.*l-y.*l)>=E;
x0=1+y.*l;
for jj=1:N
P_LB(jj)<=y(jj)<=P(jj)<=P_UB(jj);
y(jj)<=w(jj).*P_UB(jj);
y(jj)>=P(jj)-(1-w(jj)).*P_UB(jj);
end
cvx_end

I got the following output:

Calling Gurobi 7.01: 51 variables, 27 equality constraints


Gurobi optimizer, licensed to CVX for CVX
Optimize a model with 27 rows, 51 columns and 99 nonzeros
Variable types: 43 continuous, 8 integer (8 binary)
Coefficient statistics:
Matrix range [9e-03, 1e+03]
Objective range [1e+00, 1e+00]
Bounds range [1e+00, 1e+00]
RHS range [1e+00, 5e+04]
Presolve removed 0 rows and 27 columns
Presolve time: 0.00s

Explored 0 nodes (0 simplex iterations) in 0.03 seconds
Thread count was 1 (of 4 available processors)

Solution count 0
Pool objective bound 0

Model is infeasible or unbounded

Best objective -, best bound -, gap -


Status: Failed
Optimal value (cvx_optval): NaN

I’m sure the problem is feasible, but I guess the reason of the failure because the problem is unbounded. How can I fix this?

Thanks

Ahmed


(Ahmed Al-baidhani) #4

Hi again,
I have managed to solve this problem via this code

N=8; H=randn(N);
s=svd(H);l=s.^2;
C=10;E=5;
Pmax=2;
P_UB=Pmax*ones(N,1);
P_LB=zeros(N,1);k=0;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

cvx_begin
cvx_solver Gurobi_2
variables P(N) y(N)
variable w(N) binary
expression x0(N)
x0=1+l.*y;
minimize sum§
subject to

geo_mean(x0)>=2^(C/N);
sum(P.*l-y.*l)>=E;

for jj=1:N
P_LB(jj)<=y(jj)<=P(jj)<=P_UB(jj);
y(jj)<=w(jj).*P_UB(jj);
y(jj)>=P(jj)-(1-w(jj)).*P_UB(jj);
end
cvx_end

as you can see the first 7 constraints associated with x_{i,j} converted to geo_mean(x0)>=2^(C/N) so no need to go through all these 7 constraints.

Thanks
Ahmed