Majorization algorithms for inspecting circles, ellipses, squares, rectangles, and rhombi

In several disciplines, as diverse as shape analysis, locationtheory, quality control, archaeology, and psychometrics, it can beof interest to fit a circle through a set of points. We use theresult that it suffices to locate a center for which the varianceof the distances from the center to a set of given points isminimal. In this paper, we propose a new algorithm based oniterative majorization to locate the center. This algorithm isguaranteed to yield a series nonincreasing variances until astationary point is obtained. In all practical cases, thestationary point turns out to be a local minimum. Numericalexperiments show that the majorizing algorithm is stable and fast.In addition, we extend the method to fit other shapes, such as asquare, an ellipse, a rectangle, and a rhombus by making use ofthe class of $l_p$ distances and dimension weighting. In addition,we allow for rotations for shapes that might be rotated in theplane. We illustrate how this extended algorithm can be used as atool for shape recognition.iterative majorization;location;optimization;shape analysis

Data clustering for circle detection

This paper considers a multiple-circle detection problem on the basis of given data. The problem is solved by application of the center-based clustering method. For the purpose of searching for a locally optimal partition modeled on the well-known k-means algorithm, the k-closest circles algorithm has been constructed. The method has been illustrated by several numerical examples

Majorization algorithms for inspecting circles, ellipses, squares, rectangles, and rhombi

In several disciplines, as diverse as shape analysis, location theory, quality control, archaeology, and psychometrics, it can be of interest to fit a circle through a set of points. We use the result that it suffices to locate a center for which the variance of the distances from the center to a set of given points is minimal. In this paper, we propose a new algorithm based on iterative majorization to locate the center. This algorithm is guaranteed to yield a series nonincreasing variances until a stationary point is obtained. In all practical cases, the stationary point turns out to be a local minimum. Numerical experiments show that the majorizing algorithm is stable and fast. In addition, we extend the method to fit other shapes, such as a square, an ellipse, a rectangle, and a rhombus by making use of the class of $l_p$ distances and dimension weighting. In addition, we allow for rotations for shapes that might be rotated in the plane. We illustrate how this extended algorithm can be used as a tool for shape recognition

On location-allocation problems for dimensional facilities

This paper deals with a bilevel approach of the location-allocation problem with dimensional facilities. We present a general model that allows us to consider very general shapes of domains for the dimensional facilities and we prove the existence of optimal solutions under mild, natural assumptions. To achieve these results we borrow tools from optimal transport mass theory that allow us to give explicit solution structure of the considered lower level problem. We also provide a discretization approach that can approximate, up to any degree of accuracy, the optimal solution of the original problem. This discrete approximation can be optimally solved via a mixedinteger linear program. To address very large instance sizes we also provide a GRASP heuristic that performs rather well according to our experimental results. The paper also reports some experiments run on test data.Ministerio de Economía y Competitividad (MINECO). Españ

A Study on Efficient Algorithms for some Numerical Optimization Problems

학위논문 (박사)-- 서울대학교 대학원 : 수리과학부, 2013. 2. 신동우.This thesis is mainly divided into two parts: parameter estimation problem in linear differential equations and a minimization algorithm which is applicable to some industrial problem. In general, mathematical optimization problems are to find optimal ele- ments of a set which minimize (or maximize) the value of a given objective function. It is well known problem and arises in a various field of applications such as science, engineering, business and so on. It has a long history and there are still very much a work in progress. Optimization problems usually depend on the properties of objective func- tions involved. If functions are simple, e.g., linear, the the problem is easy to solve and moreover mathematical theories completed. If it is complex, however, it is hard to solve it theoretically and/or numerically. In this thesis, we suggest algorithms which is related to two specific opti- mization problems. These problems are both include nonlinear objective func- tions. The first part is to find a optimal parameter function of a differential equation and the second part is to find optimal solution of a facility location problem. Each parts contains theories about the solution, such as the exis- tence and the uniqueness of the optimal solution, and numerical examples are included.Abstract i I Parameter Estimation Problem 1 Chapter 1 Introduction 2 1.1 Background............................. 2 1.2 Motivation ............................. 3 1.3 Model problem ........................... 4 Chapter 2 Basic Properties of Algorithm 8 2.1 CaseStudy ............................. 8 2.1.1 The case where the subintervals are known . . . . . . . 8 2.1.2 The case where the subintervals are yet to be determined 13 2.2 An algorithm for parameter function estimation . . . . . . . . . 22 Chapter 3 Numerical Simulations 27 3.1 Data set 1.............................. 27 3.2 Data set 2.............................. 30 3.3 Details of calculation in Lemma 2.1.6. . . . . . . . . . . . . . . 33 II Facility Location Problem 39 Chapter 4 Introduction 40 Chapter 5 The nonlinear minimax problem 44 5.1 Reformulation of the minimax problem. . . . . . . . . . . . . . 44 5.1.1 Algorithm for the location of a circle . . . . . . . . . . . 51 5.1.2 Computational complexity................. 51 Chapter 6 Numerical results 53 6.1 Test case 1 ............................. 53 6.2 Test case 2 ............................. 54 6.3 Test case 3 ............................. 58 6.4 Conclusions............................. 58 국문초록 63Docto