How to refer to a JuMP variable using a string when the model is defined in a module?

Specific Domains Optimization (Mathematical)
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3

First. Never use eval. It is always the wrong thing to do.

Either use variable_by_name, or, if the variable is registered, convert the string to a symbol and do model[Symbol(k)]:

using JuMP, Ipopt
function foo()
    fvars   = Dict{String,Float64}("c" => 5)
    m = Model(Ipopt.Optimizer)
    @variable(m, c) 
    for (k, v) in fvars
        # Option 1
        x = variable_by_name(m, k)
        fix(x, v; force = true)
        # Option 2
        x = m[Symbol(k)]
        fix(x, v; force = true)
    end
end

More generally, consider using some sort of data structure that you control:

using JuMP, Ipopt
function foo()
    fvars = Dict{String,Float64}("c" => 5)
    my_variables = Dict{String,VariableRef}()
    m = Model(Ipopt.Optimizer)
    my_variables["c"] = @variable(m, c)
    for (k, v) in fvars
        fix(my_variables[k], v; force = true)
    end
end
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