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

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.Comment: Accepted for publication in Applied Mathematics & Optimizatio

Increasing the Creativity of ENGINO Toy Sets and Generating Automatic Building Instructions

During the First Study Group with Industry which was held in Limassol, Cyprus, the ENGINO⃝R TOY SYSTEM introduced two challenging problems. The first is to get bounds on the number of possible models/toys which can be constructed using a given package of building blocks. And the second is to generate automatically the assembly instructions for a given toy. In this report we summarize our insights and provide preliminary results for the two challenges

Successive linear programing approach for solving the nonlinear split feasibility problem

The Split Feasibility Problem (SFP), which was introduced by Censor and Elfving, consists of finding a point in a set C in one space such that its image under a linear transformation belongs to another set Q in the other space. This problem was well studied both theoretically and practically as it was also used in practice in the area of Intensity-Modulated Radiation Therapy (IMRT) treatment planning. Recently Li et. al. extended the SFP to the non-linear framework. Their algorithm tries to follow the algorithm for the linear case. But, unlike the linear case, the involved proximity function is not necessarily convex. Therefore in order to use Baillon-Haddad and Dolidze Theorems, the authors assume convexity in order to prove convergence of the projected gradient method. Since convexity of the proximity function is too restrictive, we consider here a Successive Linear Programing (SLP) approach in order to obtain local optima for the non-convex case. We also aim to intro duce a non-linear version of the Split Variational Inequality Problem (SVIP)

Reformulating the Pascoletti-Serafini problem as a Bi-level optimization problem

We propose a new reformulation of the linear Pascoletti-Serafini problem as a bi-Level optimization problem. The Pascoletti-Serafini problem stands at the core of many Multi-Criteria Optimization problems, and in particular its linear version is used for navigation purposes on the Pareto frontier. The new reformulation is based on the split feasibility problem and thus enables us to apply projection methods. We show how Solodov's method can be applied to solving the equivalent bi-level optimization problem. The method is a projected gradient method, iteratively applied to a parametrized family of functions

Outer approximated projection and contraction method for solving variational inequalities

Abstract In this paper we focus on solving the classical variational inequality (VI) problem. Most common methods for solving VIs use some kind of projection onto the associated feasible set. Thus, when the involved set is not simple to project onto, then the applicability and computational effort of the proposed method could be arguable. One such scenario is when the given set is represented as a finite intersection of sublevel sets of convex functions. In this work we develop an outer approximation method that replaces the projection onto the VI’s feasible set by a simple, closed formula projection onto some “superset”. The proposed method also combines several known ideas such as the inertial technique and self-adaptive step size. Under standard assumptions, a strong minimum-norm convergence is proved and several numerical experiments validate and exhibit the performance of our scheme