Augmented Lagrangian algorithm in pure Julia

Description

Percival.jl is a pure Julia implementation of the augmented Lagrangian algorithm. Both first and second order versions of the algorithm are available.

Quick start

Given a model model and an initial solution x0, the following can be used to optimize the model using Percival.

using Nonconvex
Nonconvex.@load Percival

alg = AugLag()
options = AugLagOptions()
result = optimize(model, alg, x0, options = options)

Percival is an optional dependency of Nonconvex so you need to import it in order to use it.

Construct an instance

To construct an instance of the Ipopt algorithm, use:

alg = AugAlg()

The options keyword argument to the optimize function shown above must be an instance of the AugLagOptions struct when the algorihm is an AugLag. To specify options use keyword arguments in the constructor of AugLagOptions, e.g:

options = AugLagOptions(first_order = false, rtol = 1e-4)

The most important option is first_order which is true by default. When first_order is true, the first order augmented Lagrangian algorithm will be used. And when it is false, the second order augmented Lagrangian algorithm will be used. Other arguments include:

atol: absolute tolerance in the subproblem optimizer
rtol: relative tolerance in the subproblem optimizer
ctol: absolute feasibility tolerance
max_iter: maximum number of iterations
max_time: maximum time in seconds
max_eval: maximum number of function evaluations

When using the first order augmented Lagrangian and a block constraint (i.e. a constraint function that returns a vector), the use of reverse-mode AD will only require calling the adjoint operator of the block constraint function in order to compute the gradient of the augmented Lagrangian. This is particularly suitable for constraint functions whose Jacobians are expensive but the adjoint operator is relatively inexpensive.