How to do "for all" in sum notation in Julia/JuMP

Question

I am trying to add constraints to a linear optimization problem in Julia using JuMP. I am using the sum{} function however, I am having trouble with some of the constraints. Does anyone know how to write "for all" in JuMP (the upside down A)? Here is the code I have so far:

using JuMP
m = Model()
c= [3 5 2 ; 4 3 5 ; 4 5 3 ; 5 4 3 ; 3 5 4]
@variable(m, x[i=1:5,j=1:3] >= 0)
@objective(m,Min,sum{c[i,j]*x[i,j],i=1:5,j=1:3})
for i=1:5
    @constraint(m, sum{x[i,j],i,j=1:3} <= 480)
end

What I am trying to get is this:

I am trying to use the for loop as a substitute of "for all i from 1 to 5" however I keep getting errors. Is there another way to do this?

Answer 1

In mathematical notation, you sum across i, and do so for each j. In Julia/JuMP, you can think of "∀" as being a for loop ("for all"), and a "Σ" as being a sum{ }:

using JuMP
m = Model()
c= [3 5 2;
    4 3 5;
    4 5 3;
    5 4 3;
    3 5 4]
# x_ij >= 0  ∀ i = 1,...,5, j = 1,...,3
@variable(m, x[i=1:5,j=1:3] >= 0)
@objective(m,Min,sum{c[i,j]*x[i,j],i=1:5,j=1:3})
# ∀j = 1,...,3
for j in 1:3
    @constraint(m, sum{x[i,j],i=1:5} <= 480)
end