Convexity and Optimality Conditions for Constrained Least-Squares Fitting of Planes and Parallel Planes to Establish Datums

2018 ◽  
Vol 19 (1) ◽  
Author(s):  
Craig M. Shakarji ◽  
Vijay Srinivasan
Keyword(s):  
Least Squares ◽  
Optimality Conditions ◽  
Global Optimum ◽  
Objective Functions ◽  
Constrained Least Squares ◽  
Least Squares Fitting ◽  
Optimum Solution ◽  
Set Of Points ◽  
Theoretical Issues ◽  
Global Optimum Solution

This paper addresses some important theoretical issues for constrained least-squares fitting of planes and parallel planes to a set of points. In particular, it addresses the convexity of the objective function and the combinatorial characterizations of the optimality conditions. These problems arise in establishing planar datums and systems of planar datums in digital manufacturing. It is shown that even when the set of points (i.e., the input points) are in general position, (1) a primary planar datum can contact 1, 2, or 3 input points, (2) a secondary planar datum can contact 1 or 2 input points, and (3) two parallel planes can each contact 1, 2, or 3 input points, but there are some constraints to these combinatorial counts. In addition, it is shown that the objective functions are convex over the domains of interest. The optimality conditions and convexity of objective functions proved in this paper will enable one to verify whether a given solution is a feasible solution, and to design efficient algorithms to find the global optimum solution.

Convexity and Optimality Conditions for Constrained Least-Squares Fitting of Planes and Parallel Planes to Establish Datums

2017 ◽  
Author(s):  
Craig M. Shakarji ◽  
Vijay Srinivasan
Keyword(s):  
Least Squares ◽  
Optimality Conditions ◽  
Feasible Solution ◽  
Global Optimum ◽  
Objective Functions ◽  
Constrained Least Squares ◽  
Least Squares Fitting ◽  
Optimum Solution ◽  
Theoretical Issues ◽  
Global Optimum Solution

This paper addresses some important theoretical issues for constrained least-squares fitting of planes and parallel planes to a set of input points. In particular, it addresses the convexity of the objective function and the combinatorial characterizations of the optimality conditions. These problems arise in establishing planar datums and systems of planar datums in digital manufacturing. It is shown that even when the input points are in general position: (1) a primary planar datum can contact 1, 2, or 3 input points, (2) a secondary planar datum can contact 1 or 2 input points, and (3) two parallel planes can each contact 1, 2, or 3 input points, but there are some constraints to these combinatorial counts. In addition, it is shown that the objective functions are convex over the domains of interest. The optimality conditions and convexity of objective functions proved in this paper will enable one to verify whether a given solution is a feasible solution, and to design efficient algorithms to find the global optimum solution.

Optimality Conditions for Constrained Least-Squares Fitting of Circles, Cylinders, and Spheres to Establish Datums

2018 ◽  
Vol 18 (3) ◽  
Author(s):  
Craig M. Shakarji ◽  
Vijay Srinivasan
Keyword(s):  
Least Squares ◽  
Optimality Conditions ◽  
Necessary Conditions ◽  
Three Dimensional ◽  
Necessary Condition ◽  
Constrained Least Squares ◽  
Least Squares Fitting ◽  
Candidate Solution ◽  
Cylinders And Spheres ◽  
New Algorithms

This paper addresses the combinatorial characterizations of the optimality conditions for constrained least-squares fitting of circles, cylinders, and spheres to a set of input points. It is shown that the necessary condition for optimization requires contacting at least two input points. It is also shown that there exist cases where the optimal condition is achieved while contacting only two input points. These problems arise in digital manufacturing, where one is confronted with the task of processing a (potentially large) number of points with three-dimensional coordinates to establish datums on manufactured parts. The optimality conditions reported in this paper provide the necessary conditions to verify if a candidate solution is feasible, and to design new algorithms to compute globally optimal solutions.

Optimality Conditions for Constrained Least-Squares Fitting of Circles, Cylinders, and Spheres to Establish Datums

2017 ◽  
Author(s):  
Craig M. Shakarji ◽  
Vijay Srinivasan
Keyword(s):  
Least Squares ◽  
Optimality Conditions ◽  
Necessary Conditions ◽  
Three Dimensional ◽  
Necessary Condition ◽  
Constrained Least Squares ◽  
Least Squares Fitting ◽  
Candidate Solution ◽  
Cylinders And Spheres ◽  
New Algorithms

This paper addresses the combinatorial characterizations of the optimality conditions for constrained least-squares fitting of circles, cylinders, and spheres to a set of input points. It is shown that the necessary condition for optimization requires contacting at least two input points. It is also shown that there exist cases where the optimal condition is achieved while contacting only two input points. These problems arise in digital manufacturing, where one is confronted with the task of processing a (potentially large) number of points with three-dimensional coordinates to establish datums on manufactured parts. The optimality conditions reported in this paper provide the necessary conditions to verify if a candidate solution is feasible, and to design new algorithms to compute globally optimal solutions.

On Algorithms and Heuristics for Constrained Least-Squares Fitting of Circles and Spheres to Support Standards

2018 ◽  
Author(s):  
Craig M. Shakarji ◽  
Vijay Srinivasan
Keyword(s):  
Least Squares ◽  
Theoretical Work ◽  
Two Dimensions ◽  
International Standards ◽  
Three Dimensions ◽  
Constrained Least Squares ◽  
Least Squares Fitting ◽  
Practical Algorithms ◽  
Coordinate Metrology ◽  
Set Of Points

Constrained least-squares fitting has gained considerable popularity among national and international standards committees as the default method for establishing datums on manufactured parts. This has resulted in the emergence of several interesting and urgent problems in computational coordinate metrology. Among them is the problem of fitting inscribing and circumscribing circles (in two-dimensions) and spheres (in three-dimensions) using constrained least-squares criterion to a set of points that are usually described as a ‘point-cloud.’ This paper builds on earlier theoretical work, and provides practical algorithms and heuristics to compute such circles and spheres. Representative codes that implement these algorithms and heuristics are also given to encourage industrial use and rapid adoption of the emerging standards.

On Algorithms and Heuristics for Constrained Least-Squares Fitting of Circles and Spheres to Support Standards

2019 ◽  
Vol 19 (3) ◽  
Author(s):  
Craig M. Shakarji ◽  
Vijay Srinivasan
Keyword(s):  
Least Squares ◽  
Theoretical Work ◽  
Two Dimensions ◽  
International Standards ◽  
Three Dimensions ◽  
Constrained Least Squares ◽  
Least Squares Fitting ◽  
Practical Algorithms ◽  
Coordinate Metrology ◽  
Set Of Points

Constrained least-squares fitting has gained considerable popularity among national and international standards committees as the default method for establishing datums on manufactured parts. This has resulted in the emergence of several interesting and urgent problems in computational coordinate metrology. Among them is the problem of fitting inscribing and circumscribing circles (in two dimensions) and spheres (in three dimensions) using constrained least-squares criterion to a set of points that are usually described as a “point-cloud.” This paper builds on earlier theoretical work, and provides practical algorithms and heuristics to compute such circles and spheres. Representative codes that implement these algorithms and heuristics are also given to encourage industrial use and rapid adoption of the emerging standards.

Hookes-Jeeves-Based Variant of Memetic Algorithm

2016 ◽  
pp. 450-475
Author(s):  
Dipti Singh ◽  
Kusum Deep
Keyword(s):  
Genetic Algorithms ◽  
Memetic Algorithm ◽  
Optimization Problems ◽  
Optimal Solution ◽  
Memetic Algorithms ◽  
Global Optimum ◽  
Benchmark Problems ◽  
New Variant ◽  
Optimum Solution ◽  
Global Optimum Solution

Due to their wide applicability and easy implementation, Genetic algorithms (GAs) are preferred to solve many optimization problems over other techniques. When a local search (LS) has been included in Genetic algorithms, it is known as Memetic algorithms. In this chapter, a new variant of single-meme Memetic Algorithm is proposed to improve the efficiency of GA. Though GAs are efficient at finding the global optimum solution of nonlinear optimization problems but usually converge slow and sometimes arrive at premature convergence. On the other hand, LS algorithms are fast but are poor global searchers. To exploit the good qualities of both techniques, they are combined in a way that maximum benefits of both the approaches are reaped. It lets the population of individuals evolve using GA and then applies LS to get the optimal solution. To validate our claims, it is tested on five benchmark problems of dimension 10, 30 and 50 and a comparison between GA and MA has been made.

Genetic Range Genetic Algorithms to Obtain Quasi-Optimum Solutions

2003 ◽  
Author(s):  
Masao Arakawa ◽  
Tomoyuki Miyashita ◽  
Hiroshi Ishikawa
Keyword(s):  
Genetic Algorithms ◽  
Fitness Function ◽  
New Product ◽  
Global Optimum ◽  
Test Problems ◽  
Local Optimum ◽  
Optimum Solution ◽  
Product Response ◽  
Peaked Function ◽  
Global Optimum Solution

In some cases of developing a new product, response surface of an objective function is not always single peaked function, and it is often multi-peaked function. In that case, designers would like to have not oniy global optimum solution but also as many local optimum solutions and/or quasi-optimum solutions as possible, so that he or she can select one out of them considering the other conditions that are not taken into account priori to optimization. Although this information is quite useful, it is not that easy to obtain with a single trial of optimization. In this study, we will propose a screening of fitness function in genetic algorithms (GA). Which change fitness function during searching. Therefore, GA needs to have higher flexibility in searching. Genetic Range Genetic Algorithms include a number of searching range in a single generation. Just like there are a number of species in wild life. Therefore, it can arrange to have both global searching range and also local searching range with different fitness function. In this paper, we demonstrate the effectiveness of the proposed method through a simple benchmark test problems.

A Composite Framework of Cuckoo Search and PSO Algorithm

2015 ◽  
Vol 713-715 ◽  
pp. 1491-1494 ◽  
Author(s):  
Zhi Qiang Gao ◽  
Li Xia Liu ◽  
Wei Wei Kong ◽  
Xiao Hong Wang
Keyword(s):  
Cloud Model ◽  
Cuckoo Search ◽  
Pso Algorithm ◽  
Global Optimum ◽  
High Dimensional ◽  
Test Results ◽  
Swarm Optimization ◽  
Local Solutions ◽  
Optimum Solution ◽  
Global Optimum Solution

A novel composite framework of Cuckoo Search (CS) and Particle Swarm Optimization (PSO) algorithm called CS-PSO is proposed in this paper. In CS-PSO, initialization is substituted by chaotic system, and then Cuckoo shares optimums in the global best solutions pool with particles in PSO to improve parallel cooperation and social interaction. Furthermore, Cloud Model, famous for its outstanding characteristics of the process of transforming qualitative concepts to a set of quantitative numerical values, is adopted to exploit the surrounding of the local solutions obtained from the global best solution pool. Benchmark test results show that, CS-PSO can converge to the global optimum solution rapidly and accurately, compared with other algorithms, especially in high dimensional problems.

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