Using Equality and Inequality Constraints

Multiple optmization packages available with the MathOptInterface and Optim's IPNewton solver can handle non-linear constraints. Optimization.jl provides a simple interface to define the constraint as a julia function and then specify the bounds for the output in OptimizationFunction to indicate if it's an equality or inequality constraint.

Let's define the rosenbrock function as our objective function and consider the below inequalities as our constraints.

\[\begin{aligned} x_1^2 + x_2^2 \leq 0.8 \\ 0.0 \leq x_1 * x_2 \leq 5.0 \end{aligned}\]

using Optimization, OptimizationMOI, OptimizationOptimJL, ForwardDiff, ModelingToolkit

rosenbrock(x, p) = (p[1] - x[1])^2 + p[2] * (x[2] - x[1]^2)^2
x0 = zeros(2)
_p = [1.0, 1.0]

Next we define the sum of squares and the product of the optimization variables as our constraint functions.

cons(res, x, p) = (res .= [x[1]^2+x[2]^2, x[1]*x[2]])
cons (generic function with 1 method)

We'll use the IPNewton solver from Optim to solve the problem.

optprob = OptimizationFunction(rosenbrock, Optimization.AutoForwardDiff(), cons = cons)
prob = OptimizationProblem(optprob, x0, _p, lcons = [-Inf, -1.0], ucons = [0.8, 2.0])
sol = solve(prob, IPNewton())

Let's check that the constraints are satisfied and the objective is lower than at initial values to be sure.

res = zeros(2)
cons(res, sol.u, _p)
prob.f(sol.u, _p)

We can also use the Ipopt library with the OptimizationMOI package.

sol = solve(prob, Ipopt.Optimizer())
res = zeros(2)
cons(res, sol.u, _p)
prob.f(sol.u, _p)

We can also use ModelingToolkit as our AD backend and generate symbolic derivatives and expression graph for the objective and constraints.

Let's modify the bounds to use the function as an equality constraint. The constraint now becomes -

\[\begin{aligned} x_1^2 + x_2^2 = 1.0 \\ x_1 * x_2 = 0.5 \end{aligned}\]

optprob = OptimizationFunction(rosenbrock, Optimization.AutoModelingToolkit(), cons = cons)
prob = OptimizationProblem(optprob, x0, _p, lcons = [1.0, 0.5], ucons = [1.0, 0.5])

Below the AmplNLWriter.jl package is used with to use the Ipopt library to solve the problem.

using AmplNLWriter, Ipopt_jll
sol = solve(prob, AmplNLWriter.Optimizer(Ipopt_jll.amplexe))

The constraints evaluate to 1.0 and 0.5 respectively as expected.

res = zeros(2)
cons(res, sol.u, _p)