ARCTR

Several ARC and TR optimization solvers.

Purpose

This package implement several Trust-Region and ARC methods to solve the unconstrained problem

min f(x)

where f is a twice continuously differentiabe function.

Trust-Region subproblem

Multiple ways are provided to solve the Trust-Region subproblem, which is to find a direction d(λ):

(∇²f(x)+λI)d(λ) = -∇f(x)

such that ||d(λ)|| ⩽ Δ, with Δ being the size of the trust region.

Installing

The optimal use of this package is through the State and Stopping packages.

] add https://github.com/Goysa2/State.jl
] add https://github.com/Goysa2/Stopping.jl

Although it is possible to use ARCTR in a self contained manner, support and update for future version of Julia will be garanteed only for usage with State and Stopping.

] add https://github.com/Goysa2/ARCTR.jl

Usage example

Let's solve a famous problem: minimize the Rosenbrock function.

function rosenbrock(x)
	n = 2; m = 2;
	f = []
	push!(f, 10 * (x[2]-x[1]^2))
	push!(f, (x[1]-1))
	return sum(f[i]^2 for i=1:m)
end

We have to use NLPModels.jl to put our problems in a structure our algorithms can understand. Our starting point will be [-1.2, 1.0]. We also have to create our State and Stopping objects for this problem.

nlp = ADNLPModel(rosenbrock, [-1.2, 1.0]);
nlpstop = NLPStopping(nlp, Stopping.unconstrained, NLPAtX([-1.2, 1.0]));

We offer multiple ways to now solve the problem:

final_state, optimal = TRLDLt(nlp, nlpstop, verbose = true)
final_state, optimal = TRLDLt_abs(nlp, nlpstop, verbose = true)
final_state, optimal = TRSpectral(nlp, nlpstop, verbose = true)
final_state, optimal = TRSpectral_abs(nlp, nlpstop, verbose = true)
final_state, optimal = ARCSpectral(nlp, nlpstop, verbose = true)
final_state, optimal = ARCLDLt(nlp, nlpstop, verbose = true)
final_state, optimal = ARCqKOp(nlp, nlpstop, verbose = true)

The final_state provide information at the last iteration and optimal is a boolean value saying if the problem has reached an optimal solution or not.

High-order correction

A limited selection of our methods offer the option to add an high-order correction if we reduce to Newton method's. The high-order correction we offer are Shamanskii, Chebyshev, Halley and SuperHalley. More documentation will be provided when those methods are more developed.

Long term goals

* Introduce dynamic precision in algorithms
* Introduce high order correction in an efficient manner.
* Provide more documentation and examples
* Making the package more self contained.