Help rewriting Convex.jl problem in JuMP.jl



I am having a hard time trying to translate a convex optimization problem from Convex.jl to JuMP.jl:

using Convex, ECOS

# generate some data
n = 100
e = [sin(50i)*sin(50j) for i=1:n, j=1:n]

# optimization variable
h = Variable(n, n)

# objective
C = sum(pos(h[i,j] - e[i,j]) + pos(e[i,j] - h[i,j]) for i=1:n, j=1:n)

prob = minimize(C)
solve!(prob, ECOSSolver())

My attempt as a first-time user was as follows:

using JuMP, ECOS

# generate some data
n = 100
e = [sin(50i)*sin(50j) for i=1:n, j=1:n]

# define problem
m = Model(solver = ECOSSolver())

# optimization variable
@variable(m, h[1:n,1:n])

# objective
C = sum(max(0,h[i,j]-e[i,j]) + max(0,e[i,j]-h[i,j]) for i=1:n, j=1:n)

@objective(m, Min, C)

The code doesn’t compile and gives me the following error:

MethodError: no method matching isless(::JuMP.GenericAffExpr{Float64,JuMP.Variable}, ::Int64)
Closest candidates are:
isless(::Char, ::Integer) at deprecated.jl:49
isless(::AbstractFloat, ::Real) at operators.jl:42
isless(::ForwardDiff.Dual{N,T<:Real}, ::Real) at /home/juliohm/.julia/v0.5/ForwardDiff/src/dual.jl:160

  1. Can you please explain what is the issue? Are we allowed to call Julia functions like max directly in JuMP.jl?

  2. I am also wondering why the Convex.jl version takes a while (more than 5 minutes) to return even though the solver takes less than a second to solve the problem. Could someone please elaborate on what is happening with the Convex.jl approach?

`max` constraints in JuMP

No, you can’t call functions like max on JuMP variables or expressions.
You’ll have to manually apply the transformations that Convex.jl does


You mean rewrite the problem in LP form? Can you confirm that the following formulation is equivalent?

minimize sum(t)
subject to
t[i,j] >= 0
h[i,j] - e[i,j] <= t[i,j]
e[i,j] - h[i,j] <= t[i,j]

Also, I’d be happy to stick with Convex.jl if I could solve the slow parsing issue. Is there any workaround for that?


Yes, the transformation works. I like Convex.jl though, I wish I could stick with it.


PRs addressing any performance issue are welcomed.


Well, if I knew what exactly is causing the performance issue to start with…


Are you sure the two formulations are equivalent?

I restrict myself to 1 variable: I call d one of your differences h[i,j]-e[i,j]. Let us also assume for the sake of argument that there is an additional constraint d <= 0. If I don’t get you wrong, you say that

maximize max(0,d)
subject to
d <= 0

which has optimal objective value of 0, is equivalent to

maximize t
subject to
t >= 0
d <= t
d <= 0

which to me seems unbounded.


@mzaffalon max(0,d) is convex, you cannot maximize it. Or more precisely, the maximum is infinity. Now, if you make the explicit constraint d <= 0 implicit to the objective, the max(0,d) becomes indeed 0 and you can maximize it.

I believe the formulation for the minimization problem I wrote is equivalent.


You are right, I misread your post. Thank you for the explanation.