CaNNOLeS.jl tutorial

CaNNOLeS is a solver for equality-constrained nonlinear least-squares problems, i.e., optimization problems of the form

\[ \min \frac{1}{2}\|F(x)\|^2_2 \quad \text{s.to} \quad c(x) = 0. \]

It uses other JuliaSmoothOptimizers packages for development. In particular, NLPModels.jl is used for defining the problem, and SolverCore for the output. It also uses HSL.jl's MA57 as main solver, but you can pass linsolve=:ldlfactorizations to use LDLFactorizations.jl.

Fine-tune CaNNOLeS

CaNNOLeS.jl exports the function cannoles:

cannoles(nlp :: AbstractNLPModel; kwargs...)

Find below a list of the main options of cannoles.

Tolerances on the problem

Parameters	Type	Default	Description
atol	AbstractFloat	√eps(eltype(x))	absolute tolerance
rtol	AbstractFloat	√eps(eltype(x))	relative tolerance: the algorithm uses ϵtol := atol + rtol * ‖∇F(x⁰)ᵀF(x⁰) - ∇c(x⁰)ᵀλ⁰‖
Fatol	AbstractFloat	√eps(eltype(x))	absolute tolerance on the residual
Frtol	AbstractFloat	√eps(eltype(x))	relative tolerance on the residual, the algorithm stops when ‖F(xᵏ)‖ ≤ Fatol + Frtol * ‖F(x⁰)‖ and ‖c(xᵏ)‖∞ ≤ √ϵtol
unbounded_threshold	AbstractFloat	-1e5	below this threshold the problem is unbounded
max_eval	Integer	100000	maximum number of evaluations computed by neval_residual(nls) + neval_cons(nls)
max_iter	Integer	-1	maximum number of iterations
max_time	AbstractFloat	30.0	maximum number of seconds
max_inner	Integer	10000	maximum number of inner iterations (return stalled if this limit is reached)
verbose	Integer	0	if > 0, display iteration details every verbose iteration

Algorithmic parameters

Parameters	Type	Default	Description
x	AbstractVector	nls.meta.x0	initial guess
λ	AbstractVector	nls.meta.y0	initial guess for the Lagrange mutlipliers
useinitialmultiplier	Bool	false	if true use λ for the initial stopping tests
method	Symbol	:Newton	method to compute direction, :Newton, :LM, :Newton_noFHess, or :Newton_vanishing
linsolve	Symbol	:ma57	solver use to compute the factorization: :ma57, :ma97, :ldlfactorizations
verbose	Int	0	if > 0, display iteration details every verbose iteration
checksmallresidual	Bool	true	if true, stop whenever \( \|F(x)\|^2_2 \leq \) ϵtol and \( \|c(x)\|_\infty \leq \) √ϵtol
alwaysacceptextrapolation	Bool	false	if true, run even if the extrapolation step fails
δdec	Real	eltype(x)(0.1)	reducing factor on the parameter δ

Examples

using CaNNOLeS, ADNLPModels

# Rosenbrock
nls = ADNLSModel(x -> [x[1] - 1; 10 * (x[2] - x[1]^2)], [-1.2; 1.0], 2)
stats = cannoles(nls, rtol = 1e-5, x = ones(2))

"Execution stats: first-order stationary"

# Constrained
nls = ADNLSModel(
  x -> [x[1] - 1; 10 * (x[2] - x[1]^2)],
  [-1.2; 1.0],
  2,
  x -> [x[1] * x[2] - 1],
  [0.0],
  [0.0],
)
stats = cannoles(nls, max_time = 10., linsolve = :ldlfactorizations)

"Execution stats: first-order stationary"