Add tutorial with manual sparsity pattern (#209)

* Add tutorial with manual sparsity pattern

---------

Co-authored-by: Alexis Montoison <[email protected]>

Author: tmigot

docs/make.jl

@@ -20,6 +20,7 @@ makedocs(
     "Support multiple precision" => "generic.md",
     "Sparse Jacobian and Hessian" => "sparse.md",
     "Performance tips" => "performance.md",
+    "Provide sparsity pattern for sparse derivatives" => "sparsity_pattern.md",
     "Reference" => "reference.md",
   ],
 )

docs/src/sparsity_pattern.md

@@ -0,0 +1,113 @@
+# Improve sparse derivatives
+
+In this tutorial, we show a feature of ADNLPModels.jl to potentially improve the computation of sparse Jacobian and Hessian.
+
+Our test problem is an academic investment control problem:
+
+```math
+\begin{aligned}
+\min_{u,x} \quad & \int_0^1 (u(t) - 1) x(t) \\
+& \dot{x}(t) = \gamma u(t) x(t).
+\end{aligned}
+```
+
+Using a simple quadrature formula for the objective functional and a forward finite difference for the differential equation, one can obtain a finite-dimensional continuous optimization problem.
+One implementation is available in the package [`OptimizationProblems.jl`](https://github.com/JuliaSmoothOptimizers/OptimizationProblems.jl).
+
+```@example ex1
+using ADNLPModels
+using SparseArrays
+
+T = Float64
+n = 100000
+N = div(n, 2)
+h = 1 // N
+x0 = 1
+gamma = 3
+function f(y; N = N, h = h)
+  @views x, u = y[1:N], y[(N + 1):end]
+  return 1 // 2 * h * sum((u[k] - 1) * x[k] + (u[k + 1] - 1) * x[k + 1] for k = 1:(N - 1))
+end
+function c!(cx, y; N = N, h = h, gamma = gamma)
+  @views x, u = y[1:N], y[(N + 1):end]
+  for k = 1:(N - 1)
+    cx[k] = x[k + 1] - x[k] - 1 // 2 * h * gamma * (u[k] * x[k] + u[k + 1] * x[k + 1])
+  end
+  return cx
+end
+lvar = vcat(-T(Inf) * ones(T, N), zeros(T, N))
+uvar = vcat(T(Inf) * ones(T, N), ones(T, N))
+xi = vcat(ones(T, N), zeros(T, N))
+lcon = ucon = vcat(one(T), zeros(T, N - 1))
+
+@elapsed begin
+  nlp = ADNLPModel!(f, xi, lvar, uvar, [1], [1], T[1], c!, lcon, ucon; hessian_backend = ADNLPModels.EmptyADbackend)
+end
+
+```
+
+`ADNLPModel` will automatically prepare an AD backend for computing sparse Jacobian and Hessian.
+We disabled the Hessian computation here to focus the measurement on the Jacobian computation.
+The keyword argument `show_time = true` can also be passed to the problem's constructor to get more detailed information about the time used to prepare the AD backend.
+
+```@example ex1
+using NLPModels
+x = sqrt(2) * ones(n)
+jac_nln(nlp, x)
+```
+
+However, it can be rather costly to determine for a given function the sparsity pattern of the Jacobian and the Hessian of the Lagrangian.
+The good news is that determining this pattern a priori can be relatively straightforward, especially for problems like our optimal control investment problem and other problems with differential equations in the constraints.
+
+The following example instantiates the Jacobian backend while manually providing the sparsity pattern.
+
+```@example ex2
+using ADNLPModels
+using SparseArrays
+
+T = Float64
+n = 100000
+N = div(n, 2)
+h = 1 // N
+x0 = 1
+gamma = 3
+function f(y; N = N, h = h)
+  @views x, u = y[1:N], y[(N + 1):end]
+  return 1 // 2 * h * sum((u[k] - 1) * x[k] + (u[k + 1] - 1) * x[k + 1] for k = 1:(N - 1))
+end
+function c!(cx, y; N = N, h = h, gamma = gamma)
+  @views x, u = y[1:N], y[(N + 1):end]
+  for k = 1:(N - 1)
+    cx[k] = x[k + 1] - x[k] - 1 // 2 * h * gamma * (u[k] * x[k] + u[k + 1] * x[k + 1])
+  end
+  return cx
+end
+lvar = vcat(-T(Inf) * ones(T, N), zeros(T, N))
+uvar = vcat(T(Inf) * ones(T, N), ones(T, N))
+xi = vcat(ones(T, N), zeros(T, N))
+lcon = ucon = vcat(one(T), zeros(T, N - 1))
+
+@elapsed begin
+  Is = Vector{Int}(undef, 4 * (N - 1))
+  Js = Vector{Int}(undef, 4 * (N - 1))
+  Vs = ones(Bool, 4 * (N - 1))
+  for i = 1:(N - 1)
+    Is[((i - 1) * 4 + 1):(i * 4)] = [i; i; i; i]
+    Js[((i - 1) * 4 + 1):(i * 4)] = [i; i + 1; N + i; N + i + 1]
+  end
+  J = sparse(Is, Js, Vs, N - 1, n)
+
+  jac_back = ADNLPModels.SparseADJacobian(n, f, N - 1, c!, J)
+  nlp = ADNLPModel!(f, xi, lvar, uvar, [1], [1], T[1], c!, lcon, ucon; hessian_backend = ADNLPModels.EmptyADbackend, jacobian_backend = jac_back)
+end
+```
+
+We recover the same Jacobian.
+
+```@example ex2
+using NLPModels
+x = sqrt(2) * ones(n)
+jac_nln(nlp, x)
+```
+
+The same can be done for the Hessian of the Lagrangian.