Hello,
im struggling to get one of the julia optimization packages (optim, NLopt, Jump etc…) to work to minimize a constrained function.

In particular, im interested to find the minimum distance between a point and a conic section. Here is a python code using scipy.minimize that works.

from scipy.optimize import minimize
def distance_to_conic(point, a, b, c, d, e, f):
x0, y0 = point
def objective(xy):
x, y = xy
return (x - x0)**2 + (y - y0)**2
def constraint(xy):
x, y = xy
return a*x**2 + b*x*y + c*y**2 + d*x + e*y + f
initial_guess = np.array([x0, y0])
cons = {'type': 'eq', 'fun': constraint}
result = minimize(objective, initial_guess, constraints=cons)
closest_point = result.x
min_distance = np.sqrt(objective(closest_point))
return min_distance, closest_point

However, the python implementation is rather slow, so i want to speed things up in julia. Has anyone proper easy to follow documentation or examples for a similar problem in julia? I kinda lost in the package documentations …

Hi there!
If I’m not mistaken this is a quadratically constrained quadratic program. This kind of problem is rather well-suited to JuMP, see e.g. the tutorial

Are you sure that your constraint is an equality and not an inequality?

using JuMP, Ipopt
function distance_to_conic(point, a, b, c, d, e, f)
x0, y0 = point
model = Model(Ipopt.Optimizer)
set_silent(model)
@variable(model, x, start = x0)
@variable(model, y, start = y0)
@objective(model, Min, (x - x0)^2 + (y - y0)^2)
@constraint(model, a * x^2 + b * x * y + c * y^2 + d * x + e * y + f == 0)
optimize!(model)
@assert is_solved_and_feasible(model)
closest_point = value(x), value(y)
min_distance = sqrt(objective_value(model))
return min_distance, closest_point
end

I didn’t test this, so there might be a typo, etc, but it should point you in the right direction.