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

2020 ◽  
Vol 54 (3) ◽  
pp. 873-882
Author(s):  
Pedro Henrique González Silva ◽  
Ana Flávia U. S. Macambira ◽  
Renan Vicente Pinto ◽  
Luidi Simonetti ◽  
Nelson Maculan ◽  
...  
Keyword(s):  
Data Mining ◽  
Convex Optimization ◽  
Objective Function ◽  
Hybrid Method ◽  
Optimization Problem ◽  
Compact Set ◽  
Computational Results ◽  
Quadratic Constraints ◽  
Convex Optimization Problem ◽  
Finite Set

Given a solid T, represented by a compact set in ℝ3, 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.

scholarly journals An Approximate Quasi-Newton Bundle-Type Method for Nonsmooth Optimization

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Jie Shen ◽  
Li-Ping Pang ◽  
Dan Li
Keyword(s):  
Convex Optimization ◽  
Objective Function ◽  
Rate Of Convergence ◽  
Nonsmooth Optimization ◽  
Optimization Problem ◽  
Nonsmooth Convex Optimization ◽  
Convex Optimization Problem ◽  
Type Method ◽  
Yosida Regularization ◽  
Quasi Newton

An implementable algorithm for solving a nonsmooth convex optimization problem is proposed by combining Moreau-Yosida regularization and bundle and quasi-Newton ideas. In contrast with quasi-Newton bundle methods of Mifflin et al. (1998), we only assume that the values of the objective function and its subgradients are evaluated approximately, which makes the method easier to implement. Under some reasonable assumptions, the proposed method is shown to have a Q-superlinear rate of convergence.

scholarly journals Stochastic Recursive Inclusions in Two Timescales with Nonadditive Iterate-Dependent Markov Noise

2020 ◽  
Vol 45 (4) ◽  
pp. 1405-1444
Author(s):  
Vinayaka G. Yaji ◽  
Shalabh Bhatnagar
Keyword(s):  
Asymptotic Behavior ◽  
Markov Chain ◽  
Convex Optimization ◽  
Objective Function ◽  
Recent Work ◽  
Differential Inclusion ◽  
Stochastic Approximation ◽  
Optimization Problem ◽  
Approximation Scheme ◽  
Convex Optimization Problem

In this paper, we study the asymptotic behavior of a stochastic approximation scheme on two timescales with set-valued drift functions and in the presence of nonadditive iterate-dependent Markov noise. We show that the recursion on each timescale tracks the flow of a differential inclusion obtained by averaging the set-valued drift function in the recursion with respect to a set of measures accounting for both averaging with respect to the stationary distributions of the Markov noise terms and the interdependence between the two recursions on different timescales. The framework studied in this paper builds on a recent work by Ramaswamy and Bhatnagar, by allowing for the presence of nonadditive iterate-dependent Markov noise. As an application, we consider the problem of computing the optimum in a constrained convex optimization problem, where the objective function and the constraints are averaged with respect to the stationary distribution of an underlying Markov chain. Further, the proposed scheme neither requires the differentiability of the objective function nor the knowledge of the averaging measure.

scholarly journals Heliostat Field Layout Optimization with Evolutionary Algorithms

10.29007/7p6t ◽  
2018 ◽  
Author(s):  
Pascal Richter ◽  
David Laukamp ◽  
Levin Gerdes ◽  
Martin Frank ◽  
Erika Ábrahám
Keyword(s):  
Power Plant ◽  
Convex Optimization ◽  
Evolutionary Algorithms ◽  
Convergence Rate ◽  
Evolutionary Algorithm ◽  
Computer Science ◽  
Energy Supply ◽  
Optimization Problem ◽  
Convex Optimization Problem ◽  
New Genotype

The exploitation of solar power for energy supply is of increasing importance. While technical development mainly takes place in the engineering disciplines, computer science offers adequate techniques for optimization. This work addresses the problem of finding an optimal heliostat field arrangement for a solar tower power plant.We propose a solution to this global, non-convex optimization problem by using an evolutionary algorithm. We show that the convergence rate of a conventional evolutionary algorithm is too slow, such that modifications of the recombination and mutation need to be tailored to the problem. This is achieved with a new genotype representation of the individuals.Experimental results show the applicability of our approach.

scholarly journals Reconstruction of the early Universe as a convex optimization problem

2003 ◽  
Vol 346 (2) ◽  
pp. 501-524 ◽  
Author(s):  
Y. Brenier ◽  
U. Frisch ◽  
M. Hénon ◽  
G. Loeper ◽  
S. Matarrese ◽  
...  
Keyword(s):  
Convex Optimization ◽  
Early Universe ◽  
Optimization Problem ◽  
Convex Optimization Problem

Design of Low Order Controllers for Decoupled MIMO Systems With Time Response Specifications

2018 ◽  
pp. 90-128
Author(s):  
Maher Ben Hariz ◽  
Wassila Chagra ◽  
Faouzi Bouani
Keyword(s):  
Convex Optimization ◽  
Optimization Problem ◽  
Closed Loop ◽  
Mimo Systems ◽  
Design Method ◽  
Optimal Solution ◽  
Time Response ◽  
Convex Optimization Problem ◽  
Tito Systems ◽  
Low Order

The design of a low order controller for decoupled MIMO systems is proposed. The main objective of this controller is to guarantee some closed loop time response performances such as the settling time and the overshoot. The controller parameters are obtained by resolving a non-convex optimization problem. In order to obtain an optimal solution, the use of a global optimization method is suggested. In this chapter, the proposed solution is the GGP method. The principle of this method consists of transforming a non-convex optimization problem to a convex one by some mathematical transformations. So as to accomplish the fixed goal, it is imperative to decouple the coupled MIMO systems. To approve the controllers' design method, the synthesis of fixed low order controller for decoupled TITO systems is presented firstly. Then, this design method is generalized in the case of MIMO systems. Simulation results and a comparison study between the presented approach and a PI controller are given in order to show the efficiency of the proposed controller. It is remarkable that the obtained solution meets the desired closed loop time specifications for each system output. It is also noted that by considering the proposed approach the user can fix the desired closed loop performances for each output independently.

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