Bethany Caldwell

Research Associate at the University of New South Wales

Research Keywords

Optimisation, convex optimisation, numerical algorithms, numerical analysis, optimal control, projection methods/operator splitting methods, robust optimisation

About Me

I am a mathematician with expertise in optimisation. I have recently completed a 12-month contract at the University of New South Wales on robust optimisation as a Research Associate with Jeya Jeyakumar (UNSW) and Guoyin Li (UNSW).I recieved my PhD in 2025 from the University of South Australia (UniSA) under the supervision of Regina Burachik (UniSA), Yalçın Kaya (UniSA) and Walaa Moursi (U. Waterloo) for my thesis titled Douglas–Rachford Algorithm for Optimal Control.

Past/Upcoming Talks

  • A primal-dual splitting method for robust convex optimisation @ SigmaOpt Workshop, Australian National University, Canberra, February 2026

  • A Primal-Dual Splitting Method for Robust Convex Optimisation @ 69th Annual Meeting of the Australian Mathematical Society (AustMS), La Trobe, Sydney, December 2025

  • Optimal Control Duality and the Douglas–Rachford Algorithm @ Strategies for Handling Applications with Nonconvexity (SHAWN) workshop, Banff International Research Station, Banff, May 2025

Research

B. I. Caldwell, N. D. Dizon, V. Jeyakumar, G. Li, A Duality-Guided Proximal Splitting Method for Robust Constrained Best Approximation via Convex Semi-Definite Program Reformulations, Set-Valued Var. Anal., 34(7) (2026), https://doi.org/10.1007/s11228-026-00793-7R. S. Burachik, B. I. Caldwell, C. Y. Kaya, W. M. Moursi, Optimal control duality and the Douglas–Rachford algorithm, SIAM J. Control Optim., 62(1), pp. 680-698 (2024),https://doi.org/10.1137/23M1558549R. S. Burachik, B. I. Caldwell, C. Y. Kaya, Douglas–Rachford Algorithm for Control- and State-constrained Optimal Control Problems, AIMS Mathematics, 9(6), pp. 13874-13893 (2024), https://doi.org/10.3934/math.2024675R. S. Burachik, B. I. Caldwell, C. Y. Kaya, Douglas–Rachford algorithm for control-constrained minimum-energy control problems, ESAIM Control Optim. Calc. Var., 30(18) (2024), https://doi.org/10.1051/cocv/2024004

Maths Artwork

Research

(Note that authors are listed alphabetically, reflecting the traditional practice in mathematics)Journal Articles
B. I. Caldwell, N. D. Dizon, V. Jeyakumar, G. Li, A Duality-Guided Proximal Splitting Method for Robust Constrained Best Approximation via Convex Semi-Definite Program Reformulations, Set-Valued Var. Anal., 34(7) (2026), https://doi.org/10.1007/s11228-026-00793-7.R. S. Burachik, B. I. Caldwell, C. Y. Kaya, W. M. Moursi, Optimal control duality and the Douglas–Rachford algorithm, SIAM J. Control Optim., 62(1), pp. 680-698 (2024),https://doi.org/10.1137/23M1558549.R. S. Burachik, B. I. Caldwell, C. Y. Kaya, Douglas–Rachford Algorithm for Control- and State-constrained Optimal Control Problems, AIMS Mathematics, 9(6), pp. 13874-13893 (2024), https://doi.org/10.3934/math.2024675.R. S. Burachik, B. I. Caldwell, C. Y. Kaya, Douglas–Rachford algorithm for control-constrained minimum-energy control problems, ESAIM Control Optim. Calc. Var., 30(18) (2024), https://doi.org/10.1051/cocv/2024004.R. S. Burachik, B. I. Caldwell, C. Y. Kaya, A generalized multivariable Newton method, Fixed Point Theory and Algorithms Sci. Eng., 15 (2021), https://doi.org/10.1186/s13663-021-00700-9.Theses
B. I. Caldwell, Douglas–Rachford Algorithm for Optimal Control Problems, (2024), https://hdl.handle.net/11541.2/42180.Under review
B. I. Caldwell, N. D. Dizon, V. Jeyakumar, G. Li, Interwoven SDP in Primal-Dual Proximal Splitting Methods for Adjustable Robust Convex Optimisation with SOS-Convex Polynomial Constraints, (2026), https://arxiv.org/abs/2602.14624.R. S. Burachik, B. I. Caldwell, C. Y. Kaya, W. M. Moursi, Best Approximation Optimal Control for Infeasible Double Integrator and Douglas–Rachford algorithm, (2026), https://arxiv.org/abs/2602.07851.R. S. Burachik, B. I. Caldwell, C. Y. Kaya, W. M. Moursi, M. Saurette, On the Douglas–Rachford and Peaceman–Rachford algorithms in the presence of uniform monotonicity and the absence of minimizers, (2024), https://arxiv.org/abs/2201.06661.

Past/Upcoming Talks

2026

  • A primal-dual splitting method for robust convex optimisation @ SigmaOpt Workshop, Australian National University, Canberra, February 2026

2025

  • A Primal-Dual Splitting Method for Robust Convex Optimisation @ 69th Annual Meeting of the Australian Mathematical Society (AustMS), La Trobe University, Sydney, December 2025

  • Optimal Control Duality and the Douglas–Rachford Algorithm @ Strategies for Handling Applications with Nonconvexity (SHAWN) workshop, Banff International Research Station, Banff, May 2025

  • Optimal Control Duality and the Douglas–Rachford Algorithm @ Splitting Algorithms – Advances, Challenges and Opportunities, MATRIX institute, Creswick, February 2025

2024

  • The Douglas–Rachford Algorithm for Inconsistent Problems @ Joint Meeting of the New Zealand, Australian and American Mathematical Societies, University of Auckland, Auckland, December 2024

  • Douglas–Rachford Algorithm for Optimal Control Problems @ Lund University, Lund, September 2024

  • Douglas–Rachford Algorithm for Optimal Control Problems @ UniSA Industrial AI Seminar, Mawson Lakes, August 2024

  • The Douglas–Rachford Algorithm for Inconsistent Problems @ International Symposium on Mathematical Programming, Montréal, July 2024

2023

  • Douglas–Rachford Algorithm for Control- and State-constrained Optimal Control Problems @ Workshop on Optimisation, Metric Bounds, Approximation and Transversality (WOMBAT), University of Sydney, Sydney, December 2023

  • Douglas–Rachford Algorithm for Control- and State-constrained Optimal Control Problems @ Undergraduate Research Assistantships Seminar, University of Waterloo, Waterloo, July 2023

  • Douglas–Rachford Algorithm for Control- and State-constrained Optimal Control Problems @ Continuous Optimisation Seminar, University of Waterloo, Waterloo, June 2023

  • Douglas–Rachford Algorithm for Control- and State-constrained Optimal Control Problems @ SIAM Conference on Optimisation, The Sheraton Grand Seattle, Seattle, May 2023

2022

  • Splitting and Projection Methods for Control-constrained Linear-quadratic Optimal Control Problems @ AustMS, University of New South Wales, December 2022

  • Splitting and Projection Methods for Control-constrained Linear-quadratic Optimal Control Problems @ WOMBAT, Curtin University, Perth, December 2022

  • Splitting and Projection Methods for Control-constrained Linear-quadratic Optimal Control Problems @ EUROPT Workshop on Advance in Continuous Optimisation, NOVA University Lisbon, Lisbon, July 2022

  • Splitting and Projection Methods for Control-constrained Linear-quadratic Optimal Control Problems @ ORCOS Viennese Conference on Optimal Control and Dynamic Games, Vienna University of Technology, Vienna, July 2022

2020-2021

  • Splitting and Projection Methods for Control-constrained Linear-quadratic Optimal Control Problems @ AustMS, University of Newcastle, online, December 2021

  • Visualisations of a Generalised Newton Method @ AustMS, University of Newcastle, online, December 2021

  • Multivariable Extensions of a Generalized Newton Method @ AustMS, University of New England, online, December 2020

Maths Artwork

Nonlinear equations arise in disciplines such as mathematics, engineering, economics, finance and biology. These equations can almost never be solved analytically; therefore, numerical methods are used to find an approximation of a solution. The Newton method is one such iterative method that is locally convergent. Another is the generalised Newton method from my work with Regina Burachik and Yalçın Kaya in this paper. Unlike the classical Newton method, this generalised method requires an auxiliary function which, depending on the choice, may increase the region of convergence. The images below was generated by applying the generalised Newton method to find zeros of systems of nonlinear equations in ℝ2. Each image is a grid in ℝ2 where each point was used as a starting point for the algorithm and the assigned colour of that point corresponds to the number of iterations the algorithm took to converge from that point. For example, the first image (yellow and blue) was generated when applying the generalised Newton method with the auxiliary function as the hyperbolic sine function to find a zero of a nonlinear equation from a signal processing problem.These works have been featured in the CARMA Art/Poster Competition (second place in 2021), the UniSA Images of Research and Teaching (finalist in 2021) and the Intercultural Science-Art Exhibition during the 11th Heidelberg Laureate Forum.