Generalized Rosenbrock

This example is taken from Fig. 1 of:

A. Pal et al., "NonlinearSolve.jl: High-performance and robust solvers for systems of nonlinear equations in Julia," arXiv [math.NA], 24-Mar-2024. https://arxiv.org/abs/2403.16341

Packages

using Ariadne

Problem definition

The generalized Rosenbrock function in $N$ dimensions:

\[F(x)_1 = 1 - x_1, \quad F(x)_i = 10(x_i - x_{i-1}^2), \quad i = 2, \ldots, N\]

function generalized_rosenbrock(x, _)
    return vcat(
        1 - x[1],
        10 .* (x[2:end] .- x[1:(end - 1)] .* x[1:(end - 1)])
    )
end

generalized_rosenbrock (generic function with 1 method)

The standard starting point is $x_1 = -1.2$, $x_i = 1$ for $i \geq 2$.

N = 12
x_start = vcat(-1.2, ones(N - 1))

12-element Vector{Float64}:
 -1.2
  1.0
  1.0
  1.0
  1.0
  1.0
  1.0
  1.0
  1.0
  1.0
  1.0
  1.0

Without line search

Solving with GMRES and no line search (NoLineSearch). The number of iterations required grows quickly with $N$ and the solver fails to converge for $N \geq 9$ within the iteration budget.

_, stats = newton_krylov(
    generalized_rosenbrock,
    copy(x_start);
    algo = :gmres,
    linesearch! = NoLineSearch(),
    max_niter = 100_000
)
stats

(solved = false, stats = Ariadne.Stats(100001, 253275, 2.9955768708885603e31), t = 0.735461647)

With backtracking line search

Using BacktrackingLineSearch stabilizes convergence for larger $N$. Pal et al. report that their backtracking implementation does not converge for $N = 10$ (using abstol = 1e-8); with abstol = 1e-12 our implementation converges for all $N \leq 12$.

_, stats = newton_krylov(
    generalized_rosenbrock,
    copy(x_start);
    algo = :gmres,
    linesearch! = BacktrackingLineSearch(),
    max_niter = 100_000
)
stats

(solved = true, stats = Ariadne.Stats(2346, 26604, 4.8675897499934246e-6), t = 0.037716682)

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