# Solvers fail on nonlinear problem (which has solutions)
**URL:** https://discourse.julialang.org/t/solvers-fail-on-nonlinear-problem-which-has-solutions/20051
**Category:** Optimization (Mathematical)
**Tags:** jump
**Created:** [January 24, 2019, 5:52pm UTC](https://discourse.julialang.org/t/solvers-fail-on-nonlinear-problem-which-has-solutions/20051 "2019-01-24T17:52:43Z")
**Posts on this page:** 11
**Page:** 2
### Author:  [@Torkel](https://discourse.julialang.org/u/Torkel)
#### Post date: [January 25, 2019, 4:39pm UTC](https://discourse.julialang.org/t/solvers-fail-on-nonlinear-problem-which-has-solutions/20051/21 "2019-01-25T16:39:35Z")
This sounds promising, I run into a problem with KNITRO installation, but got help which should fix it. I hope to test this tomorrow.
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### Author:  [@Torkel](https://discourse.julialang.org/u/Torkel)
#### Post date: [January 25, 2019, 4:41pm UTC](https://discourse.julialang.org/t/solvers-fail-on-nonlinear-problem-which-has-solutions/20051/22 "2019-01-25T16:41:08Z")
Just a second thanks to everyone, and a note why I love the community around Julia. I go to bed and the next morning I have loads of really good ideas on how to fix my problem, looking forward to looking deeper into this the following few days.
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### Author:  [@ChrisRackauckas](https://discourse.julialang.org/u/ChrisRackauckas)
#### Post date: [January 25, 2019, 4:44pm UTC](https://discourse.julialang.org/t/solvers-fail-on-nonlinear-problem-which-has-solutions/20051/23 "2019-01-25T16:44:42Z")
> [@Torkel](#):
>
> You mean one could use a transformation with arctan to ensure that non-negative values are not considered (I think remember you suggested using arctan for a very different problem).
`arctan` was for removing a discontinuity. `x^2` or `exp` transformations are for positivity constraints. In fact, we may want to bake that into DiffEqBiological somehow.
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### Author:  [@dpsanders](https://discourse.julialang.org/u/dpsanders)
#### Post date: [January 25, 2019, 5:22pm UTC](https://discourse.julialang.org/t/solvers-fail-on-nonlinear-problem-which-has-solutions/20051/24 "2019-01-25T17:22:06Z")
The problem with this is that it might create a lot of “extra” roots that have some `x_i < 0`.
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### Author:  [@tkoolen](https://discourse.julialang.org/u/tkoolen)
#### Post date: [January 25, 2019, 5:30pm UTC](https://discourse.julialang.org/t/solvers-fail-on-nonlinear-problem-which-has-solutions/20051/25 "2019-01-25T17:30:04Z")
You could try [https://github.com/rdeits/CouenneNL.jl](https://github.com/rdeits/CouenneNL.jl), which just downloads Couenne binaries instead of building them from source and hides the AmplNLWriter layer. Note that Couenne is not the fastest global solver, but it’s free.
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### Author:  [@dpsanders](https://discourse.julialang.org/u/dpsanders)
#### Post date: [January 25, 2019, 5:30pm UTC](https://discourse.julialang.org/t/solvers-fail-on-nonlinear-problem-which-has-solutions/20051/26 "2019-01-25T17:30:33Z")
Indeed, doing a simple Newton method, I always end up finding roots that have some component negative:
```julia
p = [3600, 18, 18, 3600, 3600, 3600, 1800, 3600, 18, 18, 18, 1800, 18, 36, 11, 180, 0.7, 0.4, 30, 0.2, 4, 4.5, 0.4]
using ForwardDiff
function newton(f, x, n=1000)
for i in 1:n
J = ForwardDiff.jacobian(f, x)
δ = J \ (-f(x))
x += δ
end
return x
end
x0 = 100 * rand(10)
root = newton(x -> f(x, p), x0)
@show maximum(abs.(f(root, p)))
root
```
gives
```julia
maximum(abs.(f(root, p))) = 4.547473508864641e-13
10-element Array{Float64,1}:
0.11482204904345317
0.00032484595992817946
0.0406348879740191
6.852413274079044
33.75115705986466
-0.06847078576137997
16.87400939428231
0.1015090297557086
2.062941412150575
-1.6629414121505748
```
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### Author:  [@dpsanders](https://discourse.julialang.org/u/dpsanders)
#### Post date: [January 25, 2019, 5:32pm UTC](https://discourse.julialang.org/t/solvers-fail-on-nonlinear-problem-which-has-solutions/20051/27 "2019-01-25T17:32:00Z")
OK, just had to run it a few times from random initial conditions until I happened upon a non-negative solution.
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### Author:  [@dpsanders](https://discourse.julialang.org/u/dpsanders)
#### Post date: [January 25, 2019, 5:32pm UTC](https://discourse.julialang.org/t/solvers-fail-on-nonlinear-problem-which-has-solutions/20051/28 "2019-01-25T17:32:27Z")
How do you prove that there is a unique non-negative root?
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### Author:  [@Torkel](https://discourse.julialang.org/u/Torkel)
#### Post date: [January 26, 2019, 3:49pm UTC](https://discourse.julialang.org/t/solvers-fail-on-nonlinear-problem-which-has-solutions/20051/29 "2019-01-26T15:49:23Z")
This worked well, for those problems that `Ipopt` also can solve it is slower (`Ipopt`~0.025s and `CouenneNL`~0.25s). However it do work well, thanks.
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### Author:  [@Torkel](https://discourse.julialang.org/u/Torkel)
#### Post date: [January 26, 2019, 4:45pm UTC](https://discourse.julialang.org/t/solvers-fail-on-nonlinear-problem-which-has-solutions/20051/30 "2019-01-26T16:45:22Z")
Yes, using your approach is much better. This should probably be fast enough (or rather it is, I will see what happens when I increase complexity of the problem, but I think it should work well, especially if you are about to release an improved version).
I’d be more than happy to help you with the blog post.
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### Author:  [@Torkel](https://discourse.julialang.org/u/Torkel)
#### Post date: [January 26, 2019, 4:51pm UTC](https://discourse.julialang.org/t/solvers-fail-on-nonlinear-problem-which-has-solutions/20051/31 "2019-01-26T16:51:36Z")
I don’t have any proof really, but I know that the system neither can cross into negative values, nor approach infinity (due to the nature of the model and both of these alternatives are biologically weird). Also, the system should be quite well behaved and I do not expect any strange cases. There will be at least one root.
In principle there could be more than one root, however, after playing around with this model for a while I am quite certain that this is not the case (and for this specific application I am satisfied with finding a single solution).
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