# JuMP: solver errors in reformulating Complex NLP to Real NLP

**URL:** <https://discourse.julialang.org/t/jump-solver-errors-in-reformulating-complex-nlp-to-real-nlp/14977>\
**Category:** Optimization (Mathematical)\
**Tags:** question\
**Created:** [September 14, 2018, 8:25pm UTC](https://discourse.julialang.org/t/jump-solver-errors-in-reformulating-complex-nlp-to-real-nlp/14977 "2018-09-14T20:25:46Z")\
**Posts on this page:** 1\
**Page:** 1

<div class="post-metadata">

**Author:** ![jacob-roth](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/jacob-roth/32/1862_2.png) [@jacob-roth](https://discourse.julialang.org/u/jacob-roth)\
**Post date:** [September 14, 2018, 8:25pm UTC](https://discourse.julialang.org/t/jump-solver-errors-in-reformulating-complex-nlp-to-real-nlp/14977/1 "2018-09-14T20:25:46Z")

</div>

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:

```julia-auto
## 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`:

```julia-auto
eig(full(Q))[1]
145-element Array{Float64,1}:
  1.17301e-15
  0.0574616 
  ⋮          
 16.0423     
 22.1229     

```

and using [`DegeneracyHunter`](https://github.com/adowling2/DegeneracyHunter.jl)’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:

```julia-auto
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.

1. I removed the c\_2^T \log(m) term and was able to solve. This makes sense to me.

2. I introduced an auxiliary variable `logm` and a constraint `m[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 does `Ipopt` catch that the problem is unbounded in the this case but not the original formulation?