Efficient algorithms for nonconvex optimization with applications in proton therapy and distributionally robust optimization

Abstract

Efficient algorithms in nonconvex optimization can significantly improve accuracy, speed, and scalability in various fields. This thesis presents our development of such algorithms, which have the potential to greatly influence many applications. The first part of this thesis focuses on the nonconvex Minimum-monitor-unit (MMU) constraint in pencil beam scanning proton therapy. This constraint plays a crucial role in potential cancer treatment modalities such as efficient IMPT, FLASH RT, and proton arc (ARC) delivery, where a large MMU threshold is required. However, existing methods can only handle the MMU problem with relatively small thresholds. To address this issue, we propose the stochastic coordinate descent (SCD) method in this part. The SCD method ensures the optimization objective’s monotonic decrease and provides explicit updates for each coordinate. Numerical experiments conducted on three clinical cases validate that the SCD method can offer better treatment plan quality with relatively rigorous spot reduction (RSR) method, motivated by the improved delivery efficiency and plan quality achieved through spot reduction. In the RSR method, we mathematically formulate the constrained IMPT problem with MMU constraint and ℓ₀ norm constraint. By obtaining a closed-form solution for the proximal problem involving these two constraints, we efficiently solve the problem using the ADMM method. Numerical experiments demonstrate that the RSR method can achieve comparable plan quality to the SCD method in a faster manner. In the second part, we study the Bures-Wasserstein (BW) gradient descent method. This method has gained considerable attention in various domains, including Gaussian barycenter, matrix recovery and variational inference problems, due to its alignment with the Wasserstein geometry of normal distributions. Despite its popularity, existing convergence analysis are often contingent upon specific loss functions, and the exploration of constrained settings within this framework remains limited. In this part, we make an attempt to bridge this gap by providing a general convergence rate guarantee for BW gradient descent when the Euclidean strongly convexity of the loss and the constraints is assumed. In an effort to advance practical implementations, we also derive a closed-form solution for the projection onto BW distance-constrained sets, which enables the fast implementation of projected BW gradient descent for problems that arise in the constrained barycenter and distributionally robust optimization literature. Experimental results demonstrate significant improvements in computational efficiency and convergence speed, underscoring the efficacy of our method in practical scenarios.

Date of Award	2024
Original language	English
Awarding Institution	The Hong Kong University of Science and Technology
Supervisor	Jianfeng CAI (Supervisor)