# Convex.jl and Icnn
**URL:** https://discourse.julialang.org/t/convex-jl-and-icnn/80942
**Category:** Optimization (Mathematical)
**Created:** [May 12, 2022, 9:28am UTC](https://discourse.julialang.org/t/convex-jl-and-icnn/80942 "2022-05-12T09:28:00Z")
**Posts on this page:** 5
**Page:** 1
### Author:  [@bienpierre](https://discourse.julialang.org/u/bienpierre)
#### Post date: [May 12, 2022, 9:28am UTC](https://discourse.julialang.org/t/convex-jl-and-icnn/80942/1 "2022-05-12T09:28:00Z")
Dear all,
I would like to implement an Input Convex Neural Networks [ICNN](http://proceedings.mlr.press/v70/amos17b/amos17b.pdf) with the awesome convex.jl tool. The purpose to minimize a cost function. However, in order to get convexity with Icnn, weighting matrices are nonnegative and activation function is convex and nonnegative. So within convex, there is the `pos(x)` and sound like relu activation function.
I did the following evaluation:
```julia
using Flux
using Convex
W = NNlib.relu.(rand(3,3))
b = rand(3,1)
x = Variable(3,1)
y = Variable(3,1)
expr = Convex.pos(W * x[:,1] + b)
```
Then I tried :
```julia
y[:,1] == Convex.pos(W * x[:,1] + b)
```
And I’ve got the warning
```julia
┌ Warning: Expression not DCP compliant. Trying to solve non-DCP compliant problems can lead to unexpected behavior.
```
I’m not a professional with DCP subject. However I do not understand why added the convex.jl constraint lead to “warming” as ‘y’ is a variable. Maybe a clever person can help me.
Warmest regards
---
### Author:  [@gdalle](https://discourse.julialang.org/u/gdalle)
#### Post date: [May 12, 2022, 10:26am UTC](https://discourse.julialang.org/t/convex-jl-and-icnn/80942/2 "2022-05-12T10:26:32Z")
Hi!
I don’t know anything about ICNNs but I think I see where your problem is.
When doing convex optimization, you are allowed two types of constraints: affine equality constraints and convex inequality constraints. This ensures that the feasible region you define is a convex set.
The function `f(x) -> pos(Wx+b)` is convex but not affine, so I guess that is why the equality constraint `y == f(x)` is not accepted by Convex.jl? Could you try with an inequality just to check?
---
### Author:  [@bienpierre](https://discourse.julialang.org/u/bienpierre)
#### Post date: [May 12, 2022, 11:35am UTC](https://discourse.julialang.org/t/convex-jl-and-icnn/80942/3 "2022-05-12T11:35:27Z")
Thanks for answering.
Please find the inequality:
```julia
constraint = Constraint[y[:,1] >= expr ]
1-element Vector{Constraint}:
>= constraint (convex)
├─ index (affine; real)
│ └─ 3-element real variable (id: 175…826)
└─ max (convex; positive)
├─ + (affine; real)
│ ├─ * (affine; real)
│ │ ├─ …
│ │ └─ …
│ └─ [0.34773; 0.713025; 0.600791;;]
└─ 0
```
and
```julia
julia> constraint = Constraint[y[:,1] <= expr ]
1-element Vector{Constraint}:
┌ Warning: Expression not DCP compliant. Trying to solve non-DCP compliant problems can lead to unexpected behavior.
```
So \<= provides a DCP compliant issue.
I also evaluate:
```julia
julia> constraint = Constraint[expr >= ones(3,1)]
1-element Vector{Constraint}:
┌ Warning: Expression not DCP compliant. Trying to solve non-DCP compliant problems can lead to unexpected behavior.
```
and
```julia
constraint = Constraint[expr <= ones(3,1)]
1-element Vector{Constraint}:
<= constraint (convex)
├─ max (convex; positive)
│ ├─ + (affine; real)
│ │ ├─ * (affine; real)
│ │ │ ├─ …
│ │ │ └─ …
│ │ └─ [0.34773; 0.713025; 0.600791;;]
│ └─ 0
└─ [1.0; 1.0; 1.0;;]
```
---
### Author:  [@ericphanson](https://discourse.julialang.org/u/ericphanson)
#### Post date: [May 12, 2022, 1:57pm UTC](https://discourse.julialang.org/t/convex-jl-and-icnn/80942/4 "2022-05-12T13:57:42Z")
Yeah, just as @gdalle said, expressions of the form `convex_function_of_variables >= 0` are not allowed (and if zero isn’t on one side, we subtract terms over to the other so it is), because that region is not a convex set. Think about a disk: the equation for `(x,y)` to be in a circular disk of radius `r` is `x^2 + y^2 <= r^2`:
For points to be outside the disk, `x^2 + y^2 > r^2`:
Clearly this shape is not convex.
See [The DCP ruleset — CVX Users' Guide](http://cvxr.com/cvx/doc/dcp.html) for all the DCP rules.
---
### Author:  [@gdalle](https://discourse.julialang.org/u/gdalle)
#### Post date: [May 12, 2022, 1:59pm UTC](https://discourse.julialang.org/t/convex-jl-and-icnn/80942/5 "2022-05-12T13:59:14Z")
Yeah, sorry I should have been clearer: you are allowed constraints of the form `f(x) <= 0` where `f` is convex. Not `f(x) >= 0`