MathOptInterface.OTHER_ERROR when trying to use ISRES of NLopt through JuMP - julia

Question

I am trying to minimize a nonlinear function with nonlinear inequality constraints with NLopt and JuMP.
In my test code below, I am minimizing a function with a known global minima.
Local optimizers such as LD_MMA fails to find this global minima, so I am trying to use global optimizers of NLopt that allow nonlinear inequality constraintes.
However, when I check my termination status, it says “termination_status(model) = MathOptInterface.OTHER_ERROR”. I am not sure which part of my code to check for this error.
What could be the cause?
I am using JuMP since in the future I plan to use other solvers such as KNITRO as well, but should I rather use the NLopt syntax?
Below is my code:
# THIS IS A CODE TO SOLVE FOR THE TOYMODEL
# THE EQUILIBRIUM IS CHARACTERIZED BY A NONLINEAR SYSTEM OF ODEs OF INCREASING FUCTIONS B(x) and S(y)
# THE GOAL IS TO APPROXIMATE B(x) and S(y) WITH POLYNOMIALS
# FIND THE POLYNOMIAL COEFFICIENTS THAT MINIMIZE THE LEAST SQUARES OF THE EQUILIBRIUM EQUATIONS
# load packages
using Roots, NLopt, JuMP
# model primitives and other parameters
k = .5 # equal split
d = 1 # degree of polynomial
nparam = 2*d+2 # number of parameters to estimate
m = 10 # number of grids
m -= 1
vGrid = range(0,1,m) # discretize values
c1 = 0 # lower bound for B'() and S'()
c2 = 2 # lower and upper bounds for offers
c3 = 1 # lower and upper bounds for the parameters to be estimated
# objective function to be minimized
function obj(α::T...) where {T<:Real}
# split parameters
αb = α[1:d+1] # coefficients for B(x)
αs = α[d+2:end] # coefficients for S(y)
# define B(x), B'(x), S(y), and S'(y)
B(v) = sum([αb[i] * v .^ (i-1) for i in 1:d+1])
B1(v) = sum([αb[i] * (i-1) * v ^ (i-2) for i in 2:d+1])
S(v) = sum([αs[i] * v .^ (i-1) for i in 1:d+1])
S1(v) = sum([αs[i] * (i-1) * v ^ (i-2) for i in 2:d+1])
# the equilibrium is characterized by the following first order conditions
#FOCb(y) = B(k * y * S1(y) + S(y)) - S(y)
#FOCs(x) = S(- (1-k) * (1-x) * B1(x) + B(x)) - B(x)
function FOCb(y)
sy = S(y)
binv = find_zero(q -> B(q) - sy, (-c2, c2))
return k * y * S1(y) + sy - binv
end
function FOCs(x)
bx = B(x)
sinv = find_zero(q -> S(q) - bx, (-c2, c2))
return (1-k) * (1-x) * B1(x) - B(x) + sinv
end
# evaluate the FOCs at each grid point and return the sum of squares
Eb = [FOCb(y) for y in vGrid]
Es = [FOCs(x) for x in vGrid]
E = [Eb; Es]
return E' * E
end
# this is the actual global minimum
αa = [1/12, 2/3, 1/4, 2/3]
obj(αa...)
# do optimization
model = Model(NLopt.Optimizer)
set_optimizer_attribute(model, "algorithm", :GN_ISRES)
#variable(model, -c3 <= α[1:nparam] <= c3)
#NLconstraint(model, [j = 1:m], sum(α[i] * (i-1) * vGrid[j] ^ (i-2) for i in 2:d+1) >= c1) # B should be increasing
#NLconstraint(model, [j = 1:m], sum(α[d+1+i] * (i-1) * vGrid[j] ^ (i-2) for i in 2:d+1) >= c1) # S should be increasing
register(model, :obj, nparam, obj, autodiff=true)
#NLobjective(model, Min, obj(α...))
println("")
println("Initial values:")
for i in 1:nparam
set_start_value(α[i], αa[i]+rand()*.1)
println(start_value(α[i]))
end
JuMP.optimize!(model)
println("")
#show termination_status(model)
#show objective_value(model)
println("")
println("Solution:")
sol = [value(α[i]) for i in 1:nparam]
My output:
Initial values:
0.11233072522513032
0.7631843020124309
0.3331559403539963
0.7161240026812674
termination_status(model) = MathOptInterface.OTHER_ERROR
objective_value(model) = 0.19116585196576466
Solution:
4-element Vector{Float64}:
0.11233072522513032
0.7631843020124309
0.3331559403539963
0.7161240026812674

Answer 1

I answered on the Julia forum: https://discourse.julialang.org/t/mathoptinterface-other-error-when-trying-to-use-isres-of-nlopt-through-jump/87420/2.
Posting my answer for posterity:
You have multiple issues:
range(0,1,m) should be range(0,1; length = m) (how did this work otherwise?) This is true for Julia 1.6. The range(start, stop, length) method was added for Julia v1.8
Sometimes your objective function errors because the root doesn't exist. If I run with Ipopt, I get
ERROR: ArgumentError: The interval [a,b] is not a bracketing interval.
You need f(a) and f(b) to have different signs (f(a) * f(b) < 0).
Consider a different bracket or try fzero(f, c) with an initial guess c.
Here's what I would do:
using JuMP
import Ipopt
import Roots
function main()
k, d, c1, c2, c3, m = 0.5, 1, 0, 2, 1, 10
nparam = 2 * d + 2
m -= 1
vGrid = range(0, 1; length = m)
function obj(α::T...) where {T<:Real}
αb, αs = α[1:d+1], α[d+2:end]
B(v) = sum(αb[i] * v^(i-1) for i in 1:d+1)
B1(v) = sum(αb[i] * (i-1) * v^(i-2) for i in 2:d+1)
S(v) = sum(αs[i] * v^(i-1) for i in 1:d+1)
S1(v) = sum(αs[i] * (i-1) * v^(i-2) for i in 2:d+1)
function FOCb(y)
sy = S(y)
binv = Roots.fzero(q -> B(q) - sy, zero(T))
return k * y * S1(y) + sy - binv
end
function FOCs(x)
bx = B(x)
sinv = Roots.fzero(q -> S(q) - bx, zero(T))
return (1-k) * (1-x) * B1(x) - B(x) + sinv
end
return sum(FOCb(x)^2 + FOCs(x)^2 for x in vGrid)
end
αa = [1/12, 2/3, 1/4, 2/3]
model = Model(Ipopt.Optimizer)
#variable(model, -c3 <= α[i=1:nparam] <= c3, start = αa[i]+ 0.1 * rand())
#constraints(model, begin
[j = 1:m], sum(α[i] * (i-1) * vGrid[j]^(i-2) for i in 2:d+1) >= c1
[j = 1:m], sum(α[d+1+i] * (i-1) * vGrid[j]^(i-2) for i in 2:d+1) >= c1
end)
register(model, :obj, nparam, obj; autodiff = true)
#NLobjective(model, Min, obj(α...))
optimize!(model)
print(solution_summary(model))
return value.(α)
end
main()