Nelder-Mead Simplex Optimization Algorithm

template<int N>
struct NelderMeadOptions

Configuration for Nelder-Mead simplex algorithm.

Controls convergence criteria and simplex transformation coefficients. Default values work well for most problems.

Template Parameters:

N – Number of optimization variables

Public Members

Kokkos::View<type_real[N], Kokkos::LayoutRight, Kokkos::HostSpace> x0

Initial guess.

int max_iterations = -1

Maximum iterations (-1 = 200*N)

type_real tol_f = 1.0e-4

Tolerance on function value spread.

type_real tol_x = 1.0e-4

Tolerance on simplex diameter.

type_real reflection = 1.0

Reflection coefficient.

type_real expansion = 2.0

Expansion coefficient.

type_real contraction = 0.5

Contraction coefficient.

type_real shrink = 0.5

Shrink coefficient.

template<int N, typename Func>
OptimizationResult<N> specfem::optimization::optimize(NelderMeadSimplex, Func &&objective, NelderMeadOptions<N> options)

Nelder-Mead downhill simplex minimization.

Derivative-free optimization using \(N+1\) simplex vertices in \(N\)-dimensional space. Iteratively applies reflection, expansion, contraction, and shrink operations to locate a local minimum.

Convergence occurs when both:

  • Function value spread < tol_f

  • Simplex diameter < tol_x

// Minimize (x-1)^2 + (y-2)^2
auto f = [](auto x) { return (x(0)-1)*(x(0)-1) + (x(1)-2)*(x(1)-2); };
Kokkos::View<type_real[2], Kokkos::LayoutRight, Kokkos::HostSpace> x0("x0");
x0(0) = 0.0; x0(1) = 0.0;
NelderMeadOptions<2> opts{x0};
auto result = optimize(NelderMeadSimplex{}, f, opts);
// result.x(0) ≈ 1.0, result.x(1) ≈ 2.0
Template Parameters:

  • N – Problem dimension

  • Func – Callable with signature: type_real(View<type_real[N]>)

Parameters:

  • tag – Algorithm selector (NelderMeadSimplex{})

  • objective – Function to minimize

  • options – Configuration with initial guess and tolerances

Returns:

OptimizationResult with solution, value, iterations, and convergence flag