Internal functions

projected_newton!(solver, x, H, g, Δ, cgtol, ℓ, u, s, Hs; max_time = Inf, max_cgiter = 50, subsolver_verbose = 0)

Compute an approximate solution d for

min q(d) = ¹/₂dᵀHs + dᵀg s.t. ℓ ≦ x + d ≦ u, ‖d‖ ≦ Δ

starting from s. The steps are computed using the conjugate gradient method projected on the active bounds.

s = projected_line_search!(x, H, g, d, ℓ, u, Hs; μ₀ = 1e-2)

Performs a projected line search, searching for a step size t such that

0.5sᵀHs + sᵀg ≦ μ₀sᵀg,

where s = P(x + t * d) - x, while remaining on the same face as x + d. Backtracking is performed from t = 1.0. x is updated in place.

α, s = cauchy!(x, H, g, Δ, ℓ, u, s, Hs; μ₀ = 1e-2, μ₁ = 1.0, σ=10.0)

Computes a Cauchy step s = P(x - α g) - x for

min  q(s) = ¹/₂sᵀHs + gᵀs     s.t.    ‖s‖ ≦ μ₁Δ,  ℓ ≦ x + s ≦ u,

with the sufficient decrease condition

q(s) ≦ μ₀sᵀg.

projected_gauss_newton!(solver, x, A, Fx, Δ, gctol, s, max_cgiter, ℓ, u; max_cgiter = 50, max_time = Inf, subsolver_verbose = 0)

Compute an approximate solution d for

min q(d) = ½‖Ad + Fx‖² - ½‖Fx‖²     s.t.    ℓ ≦ x + d ≦ u,  ‖d‖ ≦ Δ

starting from s. The steps are computed using the conjugate gradient method projected on the active bounds.

s = projected_line_search_ls!(x, A, g, d, ℓ, u, As, s; μ₀ = 1e-2)

Performs a projected line search, searching for a step size t such that

½‖As + Fx‖² ≤ ½‖Fx‖² + μ₀FxᵀAs

where s = P(x + t * d) - x, while remaining on the same face as x + d. Backtracking is performed from t = 1.0. x is updated in place.

α, s = cauchy_ls!(x, A, Fx, g, Δ, ℓ, u, s, As; μ₀ = 1e-2, μ₁ = 1.0, σ=10.0)

Computes a Cauchy step s = P(x - α g) - x for

min  q(s) = ½‖As + Fx‖² - ½‖Fx‖²     s.t.    ‖s‖ ≦ μ₁Δ,  ℓ ≦ x + s ≦ u,

with the sufficient decrease condition

q(s) ≦ μ₀gᵀs,

where g = AᵀFx.