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CATrustRegionMethod

This package implements a trust-region method for unconstrained optimization i.e.,

$$\min_{x \in \mathbb{R}^n} f(x).$$

The method finds stationary points i.e. points with $|| \nabla f(x) || \leq \epsilon$. In particular, in our paper we show that the method acieves the best possible convergence bound up to an additive log factor, for finding an $\epsilon$-approximate stationary point, i.e., $O( \Delta_f L^{1/2} \epsilon^{-3/2}) + \tilde{O}(1)$ iterations where $L$ is the Lipschitz constant of the Hessian, $\Delta_f$ is the optimality gap, and $\epsilon$ is the termination tolerance for the gradient norm.

Consistently adaptive (CA) in the package name refers to the method achieving the best possible convergence bound without requiring knowledge of the Lipschitz constant ($L$) of the Hessian.

License

CATrustRegionMethod.jl is licensed under the MIT License.

Installation

Installing the CATrustRegionMethod solver can be done in two different ways:

Install CATrustRegionMethod as a package

Install Julia 1.10.4 or later. CATrustRegionMethod can be installed and tested through the Julia package manager:

julia> ]
pkg> add CATrustRegionMethod
pkg> test CATrustRegionMethod

One-time setup

Install Julia 1.10.4 or later. From the root directory of the repository, run:

$ julia --project=. -e 'import Pkg; Pkg.instantiate()'

Validate setup by running the unit tests:

$ julia --project=. test/runtests.jl

Running

How to use with JuMP

Here is a simple example where a JuMP model is passed to the CAT solver

using TrustCAT, JuMP
model = Model()
@variable(model, x)
@variable(model, y)
@NLobjective(model, Min, (2.0 - x)^2 + 100 * (y - x^2)^2)
set_optimizer(model, CATrustRegionMethod.Optimizer)
MOI.set(model, MOI.RawOptimizerAttribute("time_limit"), 1800.0)
MOI.set(model, MOI.RawOptimizerAttribute("algorithm_params!r_1"), 100.0)
optimize!(model)
status = MOI.get(model, MOI.TerminationStatus())
# Retrieve the solver instance
optimizer = backend(model).optimizer.model
# Algorithm stats (total function evalation, ...)
algorithm_counter = optimizer.inner.algorithm_counter

CUTEst test set

To test our solver on CUTEst test set, please use the script:

solve_cutest.jl

To see the meaning of each argument:

$ julia --project=. scripts/solve_cutest.jl --help

Here is a simple example:

$ julia --project=. scripts/solve_cutest.jl --output_dir ./scripts/benchmark/results/cutest --default_problems true

Plots for CUTEst test set

$ julia --project=. scripts/plot_CUTEst_results.jl --output_dir ./scripts/benchmark/results/cutest

Instructions for reproducing our experiments

CUTEst test set

$ julia --project=. scripts/solve_cutest.jl --output_dir ./scripts/benchmark/results/cutest --default_problems true
$ julia --project=. scripts/solve_cutest.jl --output_dir ./scripts/benchmark/results/cutest --default_problems true --θ 0.0
$ julia --project=. scripts/run_ablation_study.jl --output_dir ./scripts/benchmark/results_ablation_study/cutest --default_problems true

Examples

Examples can be found under the test directory

References

Citing

If you use our method in your research, you are kindly asked to cite the relevant papers:

@article{hamad2024simple,
  title={A simple and practical adaptive trust-region method},
  author={Hamad, Fadi and Hinder, Oliver},
  journal={arXiv preprint arXiv:2412.02079},
  year={2024}
}

@article{hamad2022consistently,
  title={A consistently adaptive trust-region method},
  author={Hamad, Fadi and Hinder, Oliver},
  journal={Advances in Neural Information Processing Systems},
  volume={35},
  pages={6640--6653},
  year={2022}
}