# Accelerating two projection methods via perturbations with application to Intensity-Modulated Radiation Therapy

## TL;DR

This paper introduces two novel perturbation techniques to accelerate projection methods in solving convex feasibility problems, demonstrating significant speed-ups and improved solutions in applications like intensity-modulated radiation therapy.

## Contribution

The paper develops and applies two new perturbation strategies leveraging bounded perturbation resilience to enhance projection methods for convex optimization, especially in ill-conditioned problems.

## Key findings

- Perturbed methods accelerate convergence up to 4 times faster.
- Perturbed methods achieve lower objective function values by 0.5% to 5.1%.
- Effective in applications like IMRT treatment planning.

## Abstract

Constrained convex optimization problems arise naturally in many real-world applications. One strategy to solve them in an approximate way is to translate them into a sequence of convex feasibility problems via the recently developed level set scheme and then solve each feasibility problem using projection methods. However, if the problem is ill-conditioned, projection methods often show zigzagging behavior and therefore converge slowly. To address this issue, we exploit the bounded perturbation resilience of the projection methods and introduce two new perturbations which avoid zigzagging behavior. The first perturbation is in the spirit of $k$-step methods and uses gradient information from previous iterates. The second uses the approach of surrogate constraint methods combined with relaxed, averaged projections. We apply two different projection methods in the unperturbed version, as well as the two perturbed versions, to linear feasibility problems along with nonlinear optimization problems arising from intensity-modulated radiation therapy (IMRT) treatment planning. We demonstrate that for all the considered problems the perturbations can significantly accelerate the convergence of the projection methods and hence the overall procedure of the level set scheme. For the IMRT optimization problems the perturbed projection methods found an approximate solution up to 4 times faster than the unperturbed methods while at the same time achieving objective function values which were 0.5 to 5.1% lower.