I’m trying to convert the following complex NLP to a real NLP and I think I have a formulation issue.
Complex problem:
\underset{v, m, f, a} {min} \quad v^H Q v + f^T f + c_1^T \log(m) + c_2^T a
s.t. \qquad v = m \times \exp(j\, a)
with variables v \in \mathbb{C}^n, m \in \mathbb{R}^n, a \in \mathbb{R}^n, f \in \mathbb{R}^n, and data Q \succeq 0, c_1 \in \mathbb{R}^n, c_2 \in \mathbb{R}^n.
Real problem:
\underset{m, f, a} {min} \quad (m \cos(a))^T\, Q\, (m \cos(a)) + (m\sin(a))^T\, Q\, (m\sin(a)) + f^T f + c_1^T \log(m) + c_2^T a
Below is my code for the real problem splitting complex v into rectangular coordinates:
## inputs
# c1
# c2
# Q
## model
nlp = Model(solver=IpoptSolver()) ## or MosekSolver()
## variables
@variable(nlp, m[1:N])
@variable(nlp, 2.0*pi <= a[1:N] <= 2.0*pi)
@variable(nlp, f[1:N])
## initialize
setvalue(f, f_init)
setvalue(a, a_init)
setvalue(m, m_init)
## helpers
@NLexpression(nlp, vQv, sum(
sum(
(
m[i] * cos(a[i]) * m[j] * cos(a[j])
+ m[i] * sin(a[i]) * m[j] * sin(a[j])
) * Q[i,j]
for i=1:N)
for j=1:N))
@NLexpression(nlp, fIf, sum(f[i]^2 for i=1:N))
@NLexpression(nlp, c1logm, sum(c1[i] * log(m[i]) for i=1:N))
@NLexpression(nlp, c2a, sum(c2[i] * a[i] for i=1:N)) ## linear, but `NL` bc in NLobjective
## objective
@NLobjective(nlp, Min, vQv + fIf + c1logm + c2a)
a bit of info on Q
:
eig(full(Q))[1]
145element Array{Float64,1}:
1.17301e15
0.0574616
⋮
16.0423
22.1229
and using DegeneracyHunter
's printInfeasibleEquations(nlp, 0.001)
and printVariableDiagnostics(nlp)
functions gives me no issues from the beginning.
When I tried solving in Ipopt
, I get an error:
Warning: Cutting back alpha due to evaluation error
(scaled) (unscaled)
Objective...............: 1.3065000882646967e+02 1.3065000882646967e+02
Dual infeasibility......: 2.4882922922388329e+10 2.4882922922388329e+10
Constraint violation....: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 0.0000000000000000e+00 0.0000000000000000e+00
Overall NLP error.......: 2.4882922922388329e+10 2.4882922922388329e+10
EXIT: Error in step computation (regularization becomes too large?)!
I thought I had a formulation error, so I tried a couple of things.

I removed the c_2^T \log(m) term and was able to solve. This makes sense to me.

I introduced an auxiliary variable
logm
and a constraintm[i] == exp(logm[i])
and reintroduced the c_2^T \log(m) term as c_2^T logm. When solving, I get the following message:EXIT: Iterates diverging; problem might be unbounded.
Why doesIpopt
catch that the problem is unbounded in the this case but not the original formulation?