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1 | | -# [Using Equality and Inequality Constraints](@id constraints) |
| 1 | +# [Using Equality and Inequality Constraints](@id constraints) |
| 2 | + |
| 3 | +Multiple optmization packages available with the MathOptInterface and Optim's `IPNewton` solver can handle non-linear constraints. |
| 4 | +Optimization.jl provides a simple interface to define the constraint as a julia function and then specify the bounds for the output |
| 5 | +in `OptimizationFunction` to indicate if it's an equality or inequality constraint. |
| 6 | + |
| 7 | +Let's define the rosenbrock function as our objective function and consider the below inequalities as our constraints. |
| 8 | + |
| 9 | +```math |
| 10 | +\begin{aligned} |
| 11 | +
|
| 12 | +x_1^2 + x_2^2 \leq 0.8 \\ |
| 13 | +
|
| 14 | +0.0 \leq x_1 * x_2 \leq 5.0 |
| 15 | +\end{aligned} |
| 16 | +``` |
| 17 | + |
| 18 | +```@example constraints |
| 19 | +using Optimization, OptimizationMOI, OptimizationOptimJL, ForwardDiff, ModelingToolkit |
| 20 | +
|
| 21 | +rosenbrock(x, p) = (p[1] - x[1])^2 + p[2] * (x[2] - x[1]^2)^2 |
| 22 | +x0 = zeros(2) |
| 23 | +_p = [1.0, 1.0] |
| 24 | +``` |
| 25 | + |
| 26 | +Next we define the sum of squares and the product of the optimization variables as our constraint functions. |
| 27 | + |
| 28 | +```@example constraints |
| 29 | +cons(res, x, p) = (res .= [x[1]^2+x[2]^2, x[1]*x[2]]) |
| 30 | +``` |
| 31 | + |
| 32 | +We'll use the `IPNewton` solver from Optim to solve the problem. |
| 33 | + |
| 34 | +```@example constraints |
| 35 | +optprob = OptimizationFunction(rosenbrock, Optimization.AutoForwardDiff(), cons = cons) |
| 36 | +prob = OptimizationProblem(optprob, x0, _p, lcons = [-Inf, -1.0], ucons = [0.8, 2.0]) |
| 37 | +sol = solve(prob, IPNewton()) |
| 38 | +``` |
| 39 | + |
| 40 | +Let's check that the constraints are satisfied and the objective is lower than at initial values to be sure. |
| 41 | + |
| 42 | +```@example constraints |
| 43 | +res = zeros(2) |
| 44 | +cons(res, sol.u, _p) |
| 45 | +res |
| 46 | +``` |
| 47 | + |
| 48 | +```@example constraints |
| 49 | +prob.f(sol.u, _p) |
| 50 | +``` |
| 51 | + |
| 52 | +We can also use the Ipopt library with the OptimizationMOI package. |
| 53 | + |
| 54 | +```@example constraints |
| 55 | +sol = solve(prob, Ipopt.Optimizer()) |
| 56 | +``` |
| 57 | + |
| 58 | +```@example constraints |
| 59 | +res = zeros(2) |
| 60 | +cons(res, sol.u, _p) |
| 61 | +res |
| 62 | +``` |
| 63 | + |
| 64 | +```@example constraints |
| 65 | +prob.f(sol.u, _p) |
| 66 | +``` |
| 67 | + |
| 68 | +We can also use ModelingToolkit as our AD backend and generate symbolic derivatives and expression graph for the objective and constraints. |
| 69 | + |
| 70 | +Let's modify the bounds to use the function as an equality constraint. The constraint now becomes - |
| 71 | + |
| 72 | +```math |
| 73 | +\begin{aligned} |
| 74 | +
|
| 75 | +x_1^2 + x_2^2 = 1.0 \\ |
| 76 | +
|
| 77 | +x_1 * x_2 = 0.5 |
| 78 | +\end{aligned} |
| 79 | +``` |
| 80 | + |
| 81 | +```@example constraints |
| 82 | +optprob = OptimizationFunction(rosenbrock, Optimization.AutoModelingToolkit(), cons = cons) |
| 83 | +prob = OptimizationProblem(optprob, x0, _p, lcons = [1.0, 0.5], ucons = [1.0, 0.5]) |
| 84 | +``` |
| 85 | + |
| 86 | +Below the AmplNLWriter.jl package is used with to use the Ipopt library to solve the problem. |
| 87 | + |
| 88 | +```@example constraints |
| 89 | +using AmplNLWriter, Ipopt_jll |
| 90 | +sol = solve(prob, AmplNLWriter.Optimizer(Ipopt_jll.amplexe)) |
| 91 | +``` |
| 92 | + |
| 93 | +The constraints evaluate to 1.0 and 0.5 respectively as expected. |
| 94 | + |
| 95 | +```@example constraints |
| 96 | +res = zeros(2) |
| 97 | +cons(res, sol.u, _p) |
| 98 | +println(res) |
| 99 | +``` |
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