scholarly journals Optimization In A Finite-Dimensional Euclide Space

2020 ◽  
Vol 7 (1) ◽  
pp. 20-28
Author(s):  
A.I. Kosolap ◽  
Keyword(s):  
Euclidean Space ◽  
Objective Function ◽  
Optimization Model ◽  
Optimization Problems ◽  
Complexity Class ◽  
Feasible Region ◽  
Complexity Classes ◽  
The Third ◽  
Primal Dual ◽  
Multimodal Problems

In this paper, optimization models in Euclidean space are divided into four complexity classes. Ef-fective algorithms have been developed to solve the problems of the first two classes of complexity. These are the primal-dual interior-point methods. Discrete and combinatorial optimization problems of the third complexity class are recommended to be converted to the fourth complexity class with continuous change of variables. Effective algorithms have not been developed for problems of the third and fourth complexity classes, with the exception of a narrow class of problems that are unimodal. The general optimization problem is formulated as a minimum (maximum) objective function in the presence of constraints. The complexity of the problem depends on the structure of the objective function and its feasible region. If the functions that determine the optimization model are quadratic or polynomial, then semidefinite programming can be used to obtain estimates of so-lutions in such problems. Effective methods have been developed for semidefinite optimization problems. Sometimes it’s enough to develop an algorithm without building a mathematical model. We see such an example when sorting an array of numbers. Effective algorithms have been devel-oped to solve this problem. In the work for sorting problems, an optimization model is constructed, and it coincides with the model of the assignment problem. It follows from this that the sorting problem is unimodal. Effective algorithms have not been developed to solve multimodal problems. The paper proposes a simple and effective algorithm for the optimal allocation of resources in mul-tiprocessor systems. This problem is multimodal. In the general case, for solving multimodal prob-lems, a method of exact quadratic regularization is proposed. This method has proven its compara-tive effectiveness in solving many test problems of various dimensions. Keywords: Euclidean space, optimization, unimodal problems, multimodal problems, complexity classes, numerical methods.

Research on the Properties of Coaxality Error Objective Function

2009 ◽  
Vol 16-19 ◽  
pp. 1164-1168 ◽  
Author(s):  
Ping Liu ◽  
San Yang Liu
Keyword(s):  
Euclidean Space ◽  
Objective Function ◽  
Optimization Model ◽  
Dimensional Euclidean Space ◽  
Modern Theory ◽  
Minimal Point ◽  
Zone Method ◽  
Deviation Measurement ◽  
Minimum Zone ◽  
Theoretical Results

The unconstrained optimization model applying to radial deviation measurement is established for assessing coaxality errors by the positioned minimum zone method. The properties of the objective function in the optimization model are thoroughly researched. On the basis of the modern theory of convex functions, it is strictly proved that the objective function is a continuous and non-differentiable and convex function defined on the four-dimensional Euclidean space R4. Therefore, the global minimal value of the objective function is unique and any of its minimal point must be its global minimal point. Thus, any existing optimization algorithm, as long as it is convergent, can be used to solve the objective function to get the wanted values of coaxality errors by the positioned minimum zone assessment. An example is given to verify the theoretical results presented.

Some Properties of Mathematical Model for Cylindricity Errors

2010 ◽  
Vol 37-38 ◽  
pp. 1214-1218 ◽  
Author(s):  
Ping Liu ◽  
Hui Yi Miao
Keyword(s):  
Mathematical Model ◽  
Euclidean Space ◽  
Convex Function ◽  
Objective Function ◽  
Optimization Model ◽  
Dimensional Euclidean Space ◽  
Modern Theory ◽  
Minimal Point ◽  
Deviation Measurement ◽  
Theoretical Results

An unconstrained optimization model, applicable to radial deviation measurement, is established for assessing cylindricity errors by the minimum circumscribed cylinder evaluation. The properties of the objective function in the optimization model are thoroughly investigated. On the basis of the modern theory of convex functions, it is strictly proved that the objective function is a continuous and non-differentiable and convex function defined on the four-dimensional Euclidean space R4. Therefore, the minimal value of the objective function is unique and any of its minimal point must be its global minimal point. Thus, any existing optimization algorithm, so long as it is convergent, can be used to solve the objective function to get the wanted values of cylindricity errors by the minimum circumscribed cylinder assessment. An example is given to verify the theoretical results presented.

An Efficient Kriging-Based Constrained Optimization Algorithm by Global and Local Sampling in Feasible Region

2019 ◽  
Vol 142 (5) ◽  
Author(s):  
Tianzeng Tao ◽  
Guozhong Zhao ◽  
Shanhong Ren
Keyword(s):  
Phase I ◽  
Constrained Optimization ◽  
Objective Function ◽  
Optimization Design ◽  
Optimization Problems ◽  
Phase Iii ◽  
Benchmark Problems ◽  
Feasible Region ◽  
Sample Points ◽  
Local Sampling

Abstract To solve challenging optimization problems with time-consuming objective and constraints, a novel efficient Kriging-based constrained optimization (EKCO) algorithm is proposed in this paper. The EKCO mainly consists of three sampling phases. In phase I of EKCO, considering the significance of constraints, feasible region is constructed via employing a feasible region sampling (FRS) criterion. The FRS criterion can avoid the local clustering phenomenon of sample points. Therefore, phase I is also a global sampling process for the objective function in the feasible region. However, the objective function may be higher-order nonlinear than constraints. In phase II, by maximizing the prediction variance of the surrogate objective, more accurate objective function in the feasible region can be obtained. After global sampling, to accelerate the convergence of EKCO, an objective local sampling criterion is introduced in phase III. The verification of the EKCO algorithm is examined on 18 benchmark problems by several recently published surrogate-based optimization algorithms. The results indicate that the sampling efficiency of EKCO is higher than or comparable with that of the recently published algorithms while maintaining the high accuracy of the optimal solution, and the adaptive ability of the proposed algorithm also be validated. To verify the ability of EKCO to solve practical engineering problems, an optimization design problem of aeronautical structure is presented. The result indicates EKCO can find a better feasible design than the initial design with limited sample points, which demonstrates practicality of EKCO.

scholarly journals An enumerative algorithm for non-linear multi-level integer programming problem

2009 ◽  
Vol 19 (2) ◽  
pp. 263-279 ◽  
Author(s):  
Ritu Narang ◽  
S.R. Arora
Keyword(s):  
Programming Problem ◽  
Optimization Problems ◽  
Feasible Region ◽  
Integer Programming Problem ◽  
The Third ◽  
Level Problem ◽  
Enumerative Algorithm ◽  
Indefinite Quadratic ◽  
Feasible Solutions ◽  
Optimum Solution

In this paper a multilevel programming problem, that is, three level programming problem is considered. It involves three optimization problems where the constraint region of the first level problem is implicitly determined by two other optimization problems. The objective function of the first level is indefinite quadratic, the second one is linear and the third one is linear fractional. The feasible region is a convex polyhedron. Considering the relationship between feasible solutions to the problem and bases of the coefficient sub-matrix associated to the variables of the third level, an enumerative algorithm is proposed, which finds an optimum solution to the given problem. It is illustrated with the help of an example. .

Mathematical Model for Evaluating Cylindricity Errors by Maximum Inscribed Cylinder

2011 ◽  
Vol 474-476 ◽  
pp. 1418-1422
Author(s):  
Ping Liu ◽  
Hui Yi Miao
Keyword(s):  
Mathematical Model ◽  
Euclidean Space ◽  
Convex Function ◽  
Objective Function ◽  
Optimization Model ◽  
Dimensional Euclidean Space ◽  
Modern Theory ◽  
Minimal Point ◽  
Deviation Measurement ◽  
Theoretical Results

An unconstrained optimization model applicable to radial deviation measurement is established for assessing cylindricity errors by the maximum inscribed cylinder evaluation. The properties of the objective function in the optimization model are thoroughly researched. On the basis of the modern theory of convex functions, it is strictly proved that the objective function is a continuous and non-differentiable and convex function defined on a subset of the four-dimensional Euclidean space. The minimal value of the objective function is unique and any of its minimal point must be its global minimal point. Any existing optimization algorithm, so long as it is convergent, can be used to solve the objective function to get the wanted values of cylindricity errors by the maximum inscribed cylinder assessment. An example is given to verify the theoretical results presented.

An inner-point modification of PSO for constrained optimization

2015 ◽  
Vol 32 (7) ◽  
pp. 2005-2019 ◽  
Author(s):  
Daniele Peri
Keyword(s):  
Objective Function ◽  
Optimization Problems ◽  
Real Life ◽  
Optimal Solution ◽  
Pso Algorithm ◽  
Parallel Structure ◽  
Feasible Region ◽  
Content Type ◽  
Design Variables ◽  
Particle Level

Purpose – The purpose of this paper is to propose a modification of the original PSO algorithm in order to avoid the evaluation of the objective function outside the feasible set, improving the parallel performances of the algorithm in the view of its application on parallel architectures. Design/methodology/approach – Classical PSO iteration is repeated for each particle until a feasible point is found: the global search is performed by a set of independent sub-iteration, at the particle level, and the evaluation of the objective function is performed only once the full swarm is feasible. After that, the main attractors are updated and a new sub-iteration is initiated. Findings – While the main qualities of PSO are preserved, a great advantage in terms of identification of the feasible region and detection of the best feasible solution is obtained. Furthermore, the parallel structure of the algorithm is preserved, and the load balance improved. The results of the application to real-life optimization problems, where constraint satisfaction sometime represents a problem itself, gives the measure of this advantage: an improvement of about 10 percent of the optimal solution is obtained by using the modified version of the algorithm, with a more precise identification of the optimal design variables. Originality/value – Differently from the standard approach, utilizing a penalty function in order to discharge unfeasible points, here only feasible points are produced, improving the exploration of the feasible region and preserving the parallel structure of the algorithm.

Spreadsheet Modeling to Determine Optimum Ship Main Dimensions and Power Requirements at Basic Design Stage

2003 ◽  
Vol 40 (01) ◽  
pp. 61-70
Author(s):  
Ketut Buda Artana ◽  
Kenji Ishida
Keyword(s):  
Objective Function ◽  
Optimization Model ◽  
Optimization Problems ◽  
Economic Cost ◽  
Design Stage ◽  
Basic Design ◽  
Power Requirement ◽  
Cost Of Transport ◽  
Decision Variables

The objective of this paper is to describe and evaluate a scheme of engineering-economic analysis in determining optimum ship main dimensions and power requirements at the basic design stage. An optimization model designs the problem and is arranged into five main parts, namely, Input, Equation, Constraint, Output and Objective Function. The constraints, which are the considerations to be fulfilled, become the director of this process and a minimum and a maximum value are set on each constraint so as to give the working area of the optimization process. The outputs (decision variables) are optimized in favor of minimizing the objective function. Microsoft Excel-Premium Solver Platform (PSP), a spreadsheet modeling tool is utilized to model the optimization problem. The first part of this paper contains a description on general optimization problems, followed by model construction of the optimization program. A case study on the determination of ship main dimensions and its power requirements for a tanker is introduced with the main objective being to minimize the economic cost of transport (ECT). After simulating the model and verifying the results, it is observed that this method yields considerably comparable results with the main dimensions and power requirement database of the real operated ships (tanker). It is also believed that this process needs no painful and exhaustive efforts to produce the programming code, if the problem and optimization model have been well defined.

scholarly journals A new method for global optimization

2021 ◽  
Vol 71 ◽  
pp. 121-130
Author(s):  
Anatolii Kosolap
Keyword(s):  
Global Optimization ◽  
Interior Point Method ◽  
Numerical Experiments ◽  
Convex Set ◽  
New Method ◽  
Feasible Region ◽  
Global Extremum ◽  
Positive Orthant ◽  
Primal Dual ◽  
Multimodal Problems

This paper presents a new method for global optimization. We use exact quadratic regularization for the transformation of the multimodal problems to a problem of a maximum norm vector on a convex set. Quadratic regularization often allows you to convert a multimodal problem into a unimodal problem. For this, we use the shift of the feasible region along the bisector of the positive orthant. We use only local search (primal-dual interior point method) and a dichotomy method for search of a global extremum in the multimodal problems. The comparative numerical experiments have shown that this method is very efficient and promising.

scholarly journals MATHEMATICAL MODELING OF OPTIMAL HEIGHTS OF EXTERNAL GEODESIC SIGNS

2021 ◽  
Vol 1 (161) ◽  
pp. 109-115
Author(s):  
O. Voronkov ◽  
O. Baistryk ◽  
A. Danylyuk
Keyword(s):  
Quadratic Programming ◽  
Objective Function ◽  
Optimization Model ◽  
Optimization Problems ◽  
Geodetic Network ◽  
Local Maximum ◽  
Quadratic Model ◽  
Design Stage ◽  
Mathematical Apparatus ◽  
Financial Costs

Due to the great importance of geodetic networks for the formation of a unified coordinate system on the territory of Ukraine, external geodetic signs have been established, which need to be restored and further developed. At the design stage, the calculation of the heights of geodetic signs is performed on topographic maps. The cost of erection of geodetic signs on average is 50 - 60% of the total cost of creating a geodetic network, so there is a need to pay close attention to the choice of places to build signs that provide their optimal height. The article presents a methodical approach to determining the heights of external geodetic signs, based on the mathematical apparatus used for modeling and solving optimization problems. The principle of construction of the optimization model of the problem during the design of external geodetic signs in the conditions when their direct visibility should be provided is considered. The article considers in detail the types and structures of external geodetic signs, identifies the features of their location and construction. The resulting optimization model includes objective function, which is a quadratic form, and line restriction. This model is a model of quadratic programming, that belongs to a class of nonlinear programming models, but have their particular case and the simplest of nonlinear. This is because property quadratic model, which consists in the fact that since the problem of quadratic programming set of feasible solutions is convex, then, if the objective function is concave, any local maximum is global, and if the objective function is convex, then any local minimum is also global. The necessity of solving the problem of optimizing the heights of geodetic signs is substantiated, which is still connected with the financial costs for their construction and reconstruction. It is concluded that the approach to determining the heights of external geodetic signs presented in the article, which uses a mathematical apparatus for solving optimization problems, is an effective and efficient approach, and allows to numerically justify the minimum required and sufficient height of external geodetic signs. Using the present approach to the determination of geodetic heights external signs to optimize the financial costs of their construction, which is essential.

COMPARISON OF TWO OPTIMIZATION MODELS FOR CONSTRUCTING PATTERNS IN THE METHOD OF LOGICAL ANALYSIS OF DATA

2020 ◽  
pp. 10-13
Author(s):  
Р.И. Кузьмич ◽  
А.А. Ступина ◽  
С.Н. Ежеманская ◽  
А.П. Шугалей
Keyword(s):  
Objective Function ◽  
Optimization Model ◽  
Logical Analysis ◽  
Optimization Models ◽  
Logical Analysis Of Data

Предлагаются две оптимизационные модели для построения информативных закономерностей. Приводится эмпирическое подтверждение целесообразности использования критерия бустинга в качестве целевой функции оптимизационной модели для получения информативных закономерностей. Информативность, закономерность, критерий бустинга, оптимизационная модель Comparison of two optimization models for constructing patterns in the method of logical analysis of data Two optimization models for constructing informative patterns are proposed. An empirical confirmation of the expediency of using the boosting criterion as an objective function of the optimization model for obtaining informative patterns is given.

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