Accelerating Two Projection Methods via Perturbations with Application to Intensity-Modulated Radiation Therapy

Abstract

Constrained convex optimization problems arise naturally in many realworld 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.

Nesterov perturbations and projection methods applied to IMRT

“…These perturbations are then plugged into the level set scheme for solving (1.1) with the simultaneous and cyclic subgradient projection methods as operator T . In contrast to the perturbations used in [3], the Nesterov perturbations do not generate a new non-zigzagging direction. Instead they enlarge the steps into the directions generated by T itself and thereby amplify the movement towards the feasible set, which is already contained in those directions.…”

“…where 18) and {b k } ∞ k=0 is bounded. Proposition 2 in [3] implies that the inner and the outer perturbation scheme with β k chosen as in (2.14) can be used interchangeably with regard to weak and strong convergence of the resulting sequences of iterates {y k } ∞ k=0 and {z k } ∞ k=0 when y 0 = z 0 .…”

“…In what follows, we first compare results on the Nesterov perturbation for different threshold values ε max , next we keep ε max fixed and illustrate the results of choosing different step sizes λ NE and finally we compare the results produced by the Nesterov perturbation to those achieved by the heavy ball and surrogate constraint perturbation, which were introduced in [3]. By Φ * method , we denote the lowest objective function value, for which the level set scheme using the specified unperturbed method is able to find a solution within the given maximum number n max of iterations per CFP P s as described in (2.4).…”

“…Our goal is to use perturbations to address this behavior and accelerate the progress of the projection methods. In [3], we did this by constructing new non-zigzagging directions and using those directions as perturbations; see also [37]. Clearly there exist many optimization techniques such as [18,21,22,31], but here we mainly focus on first order methods with perturbations.…”

“…Following the recent results in [3], in this work, we introduce new inner and outer perturbations, called Nesterov perturbations, in order to address the zigzagging behavior in a new way. These perturbations are then plugged into the level set scheme for solving (1.1) with the simultaneous and cyclic subgradient projection methods as operator T .…”