Optimal configuration of gamma ray machine radiosurgery units: the sphere covering subproblem

Liberti, Leo; Maculan, Nelson; Zhang, Yue

doi:10.1007/s11590-008-0095-4

Cited by 13 publications

(21 citation statements)

References 16 publications

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“…Roughly speaking, the covering problem asks for the point configuration that minimizes the radius of overlapping spheres circumscribed around each of the points required to cover R d . The covering problem has applications in wireless communication network layouts [25], the search of high-dimensional data parameter spaces (e.g., search templates for gravitational waves) [26], and stereotactic radiation therapy [27]. The quantizer problem is concerned with finding the point configuration in R d that minimizes a "distance error" associated with a randomly placed point and the nearest point of the point process.…”

Section: Introductionmentioning

confidence: 99%

Reformulation of the covering and quantizer problems as ground states of interacting particles

Torquato

2010

Phys. Rev. E

152

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It is known that the sphere packing problem and the number variance problem (closely related to an optimization problem in number theory) can be posed as energy minimizations associated with an infinite number of point particles in d-dimensional Euclidean space R d interacting via certain repulsive pair potentials. We reformulate the covering and quantizer problems as the determination of the ground states of interacting particles in R d that generally involve singlebody, two-body, three-body, and higher-body interactions. This is done by linking the covering and quantizer problems to certain optimization problems involving the "void" nearest-neighbor functions that arise in the theory of random media and statistical mechanics. These reformulations, which again exemplifies the deep interplay between geometry and physics, allow one now to employ theoretical and numerical optimization techniques to analyze and solve these energy minimization problems. The covering and quantizer problems have relevance in numerous applications, including wireless communication network layouts, the search of high-dimensional data parameter spaces, stereotactic radiation therapy, data compression, digital communications, meshing of space for numerical analysis, and coding and cryptography, among other examples. In the first three space dimensions, the best known solutions of the sphere packing and number variance problems (or their "dual" solutions) are directly related to those of the covering and quantizer problems, but such relationships may or may not exist for d ≥ 4, depending on the peculiarities of the dimensions involved. Our reformulation sheds light on the reasons for these similarities and differences. We also show that disordered saturated sphere packings provide relatively thin (economical) coverings and may yield thinner coverings than the best known lattice coverings in sufficiently large dimensions.In the case of the quantizer problem, we derive improved upper bounds on the quantizer error using sphere-packing solutions, which are generally substantially sharper than an existing upper bound in low to moderately large dimensions. We also demonstrate that disordered saturated sphere packings yield relatively good quantizers. Finally, we remark on possible applications of our results for the detection of gravitational waves.

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Section: Introductionmentioning

confidence: 99%

Reformulation of the covering and quantizer problems as ground states of interacting particles

Torquato

2010

Phys. Rev. E

152

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“…On the topic of linearizing the Euclidean norm in the context of mathematical optimization, the examples are rare but can be found in the applications overviewed of section 1. In the context of radiotherapy equipment configuration, the authors of [16] propose to linearize the quadratic terms of the convex proximity constraints with extra variables and a notion of approximation points, but without really discussing the error made in the end on the approximated Euclidean distances. Another way of linearizing the Euclidean distances is to discretize the possible positions of the originally continuous variables allowing then to pre-compute all the possible pointto-point distances, as done both in [16] and [5].…”

Section: Linearization Of Euclidean Norm Dependent Constraints In Rmentioning

confidence: 99%

“…Most importantly, such models are adaptable to variants of the problem that are characterized by extra sets of constraints and variables. As mentioned in Section 2, the authors of [16] propose linearization and discretization techniques for a special sphere covering problem occuring in a gamma ray machine radiosurgery application. Their technique is based on the linearization of the squared distance.…”

Section: A Simple Case With Convex Distance Constraints: the Continuomentioning

confidence: 99%

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Linearization of Euclidean norm dependent inequalities applied to multibeam satellites design

et al. 2019

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Euclidean norm computations over continuous variables appear naturally in the constraints or in the objective of many problems in the optimization literature, possibly defining non-convex feasible regions or cost functions. When some other variables have discrete domains, it positions the problem in the challenging Mixed Integer Nonlinear Programming (MINLP) class. For any MINLP where the nonlinearity is only present in the form of inequality constraints involving the Euclidean norm, we propose in this article an efficient methodology for linearizing the optimization problem at the cost of entirely controllable approximations even for non convex constraints. They make it possible to rely fully on Mixed Integer Linear Programming and all its strengths. We first empirically compare this linearization approach with a previously proposed linearization approach of the literature on the continuous k−center problem. This methodology is then successfully applied to a critical problem in the telecommunication satellite industry: the optimization of the beam layouts in multibeam satellite systems. We provide a proof of the NP-hardness of this very problem along with experiments on a realistic reference scenario.

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“…In [1], besides a heuristic to find a good starting point to the problem, the authors used an inexact penalization method to remodel the constraints and the least squares method was used to estimate parameters. In [5], at first, a nonlinear non-convex mixed integer model was proposed. The solid was discretized and five reformulations of the first model were proposed.…”

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confidence: 99%

New proposals for modelling and solving the problem of covering solids using spheres of different radii

et al. 2020

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Given a solid T , represented by a compact set in R 3 , the aim of this work is to find a covering of T by a finite set of spheres of different radii. Some level of intersection between the spheres is necessary to cover the solid. Moreover, the volume occupied by the spheres on the outside of T is limited. This problem has an application in the planning of a radio-surgery treatment known by Gamma Knife and can be formulated as a non-convex optimization problem with quadratic constraints and linear objective function. In this work, two new linear mathematical formulations with binary variables and a hybrid method are proposed. The hybrid method combines heuristic, data mining and an exact method. Computational results show that the proposed methods outperform the ones presented in the literature.Mathematics Subject Classification. 90C10.

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Optimal configuration of gamma ray machine radiosurgery units: the sphere covering subproblem

Abstract: We use reformulation techniques to model and solve a complex sphere covering problem occurring in the configuration of a gamma ray machine radiotherapy equipment unit.

Cited by 13 publications

References 16 publications

Reformulation of the covering and quantizer problems as ground states of interacting particles

Reformulation of the covering and quantizer problems as ground states of interacting particles

Linearization of Euclidean norm dependent inequalities applied to multibeam satellites design

New proposals for modelling and solving the problem of covering solids using spheres of different radii

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