Supplementary code for the paper: Empirical risk minimization for risk-neutral composite optimal control with applications to bang-bang control

This repository contains supplementary code for the paper

J. Milz and D. Walter: Empirical risk minimization for risk-neutral composite optimal control with applications to bang-bang control, https://arxiv.org/abs/2408.10384, 2024.

Abstract

Nonsmooth composite optimization problems under uncertainty are prevalent in various scientific and engineering applications. We consider risk-neutral composite optimal control problems, where the objective function is the sum of a potentially nonconvex expectation function and a nonsmooth convex function. To approximate the risk-neutral optimization problems, we use a Monte Carlo sample-based approach, study its asymptotic consistency, and derive nonasymptotic sample size estimates. Our analyses leverage problem structure commonly encountered in PDE-constrained optimization problems, including compact embeddings and growth conditions. We apply our findings to bang-bang-type optimal control problems and propose the use of a conditional gradient method to solve them effectively. We present numerical illustrations.

Getting started

Docker

We provide a pre-build Docker image which can be used to run the the code in this repository. First thing you need to do is to ensure that you have docker installed.

To start an interactive docker container you can execute the following command

docker run --rm -it ghcr.io/milzj/mcrnnoc:latest

The Dockerfile can be built and run using

docker build -t mcrnnoc . --no-cache --network=host
docker run -it mcrnnoc

Installation without using Docker

conda env create -f environment.yml
conda activate MCRNNOC

Running the simulations

Linear Problem

Nominal Problem

cd src/mcrnnoc/examples/linear/nominal

Execute

./simulate_nominal.sh

to start the simulation.

Reference problems

cd src/mcrnnoc/examples/linear/risk_neutral

Execute

sbatch simulate_reference.sbatch

to start the simulation.

The following shell script solves a reference problem for small mesh width and sample size

./test_simulate_reference.sh

SAA experiments

cd src/mcrnnoc/examples/linear/risk_neutral

Execute

sbatch simulate_experiment.sbatch

to start the simulation.

The following shell script simulates SAA experiments for small mesh width and sample sizes

./test_simulate_experiment.sh

Postprocessing: Generating convergence plots

./plot_experiment.sh

Bilinear Problem

Nominal Problem

cd src/mcrnnoc/examples/bilinear/nominal

Execute

./simulate_nominal.sh

to start the simulation.

Reference problems

cd src/mcrnnoc/examples/bilinear/risk_neutral

Execute

sbatch simulate_reference.sbatch

to start the simulation.

The following shell script solves a reference problem for small mesh width and sample size

./test_simulate_reference.sh

SAA experiments

cd src/mcrnnoc/examples/bilinear/risk_neutral

Execute

sbatch simulate_experiment.sbatch

to start the simulation.

The following shell script simulates SAA experiments for small mesh width and sample sizes

./test_simulate_experiment.sh

Postprocessing: Generating convergence plots

./plot_experiment.sh

References

• Partnership for an Advanced Computing Environment (PACE) at the Georgia Institute of Technology, Atlanta, Georgia, USA, http://www.pace.gatech.edu

• The implementation of the random field is adapted from poisson-risk-neutral-lognormal. It implements the KKL expansion defined in Example 7.56 ("separable exponential $d=2$") with $a = a_1 = a_2 = 1/2$ in

  G. J. Lord, C. E. Powell, and T. Shardlow. An Introduction to Computational Stochastic PDEs. Cambridge Texts Appl. Math. 50. Cambridge University Press, Cambridge, 2014. doi:10.1017/CBO9781139017329.

• We used line_profiler and kernproof.

Having issues

If you have any troubles please file an issue in the GitHub repository.

License

See LICENSE

Acknowledgement

We thank the developers of https://github.com/scientificcomputing/example-paper.