A Circle Intersection Method for Bi-Objective Optimization

2021 ◽  
pp. 1-36
Author(s):  
Jianhua Zhou ◽  
Mian Li ◽  
Xiaojin Fu
Keyword(s):  
Pareto Front ◽  
Computational Cost ◽  
Multi Objective Optimization ◽  
Nsga Ii ◽  
New Approach ◽  
Pareto Point ◽  
Objective Space ◽  
Anchor Points ◽  
Intersection Method

Abstract Multi-Objective Optimization (MOO) problems are encountered in many applications, of which bi-objective problems are frequently met. Despite the computational efforts, the quality of the Pareto front is also a considerable issue. An evenly distributed Pareto front is desirable in certain cases when a continuous representation of the Pareto front is needed. In this paper, a new approach called Circle Intersection (CI) is proposed. Firstly, the anchor points are computed. Then in the normalized objective space, a circle with a proper radius of r centering at one of the anchor points or the latest obtained Pareto point is drawn. Interestingly, the intersection of the circle and the feasible boundary can be determined whether it is a Pareto point or not. For a convex or concave feasible boundary, the intersection is exactly the Pareto point, while for other cases the intersection can provide useful information for searching the true Pareto point even if it is not a Pareto point. A novel MOO formulation is proposed for CI correspondingly. Sixteen examples are used to demonstrate the applicability of the proposed method and results are compared to those of NNC, MOGOA, and NSGA-II. Computational results show that the proposed CI method is able to obtain a well-distributed Pareto front with a better quality or with less computational cost.

A Normalized Circle Intersection Method for Bi-Objective Optimization Programming

2017 ◽  
Author(s):  
Jianhua Zhou ◽  
Tingting Xia ◽  
Mian Li ◽  
Min Xu
Keyword(s):  
Pareto Front ◽  
Method Comparison ◽  
Multi Objective Optimization ◽  
New Approach ◽  
Pareto Point ◽  
Objective Space ◽  
Anchor Points ◽  
Intersection Method ◽  
Analytical Expressions

Multi-objective optimization (MOO) problems are encountered in many applications and a number of approaches have been proposed to deal with this kind of problems. Despite the computational efforts, the quality of the Pareto front is also a considerable issue. An evenly distributed Pareto front is desirable for developing analytical expressions. In this paper, a brand new approach called Normalized Circle Intersection (NCI) is proposed, which is able to efficiently generate a Pareto front with evenly-distributed Pareto points for bi-objective problems, no matter the feasible boundary is convex or not. Firstly, the anchor points are computed using an existing sequential MOO (SMOO) approach. Then in the normalized objective space, a circle with a radius of r centering at one of the anchor points or the latest obtained Pareto point is drawn. The intersection of the circle and the feasible boundary, which exists for sure, can be determined whether it is a Pareto point or not. For a convex or concave feasible boundary, the intersection is exactly the Pareto point to be found, while for a non-convex boundary the intersection can provide useful information for searching the true Pareto point even if it self is not a Pareto point. A novel MOO formulation is proposed for NCI correspondingly. Four examples, including two numerical and two engineering examples, are provided to demonstrate the applicability of the proposed method. Comparison of the computational results with WS, NNC and SMOO shows the effectiveness of the proposed method.

scholarly journals An Effective Couple Method for Reliability-Based Multi-Objective Optimization of Truss Structures with Static and Dynamic Constraints

2019 ◽  
Vol 17 (06) ◽  
pp. 1950016 ◽  
Author(s):  
T. Vo-Duy ◽  
D. Duong-Gia ◽  
V. Ho-Huu ◽  
T. Nguyen-Thoi
Keyword(s):  
Cross Section ◽  
Optimization Problems ◽  
Computational Cost ◽  
Probabilistic Constraints ◽  
Truss Structures ◽  
Multi Objective Optimization ◽  
Nsga Ii ◽  
Dynamic Constraints ◽  
Multi Objective ◽  
Couple Method

This paper proposes an effective couple method for solving reliability-based multi-objective optimization problems of truss structures with static and dynamic constraints. The proposed coupling method integrates a single-loop deterministic method (SLDM) into the nondominated sorting genetic algorithm II (NSGA-II) algorithm to give the so-called SLDM-NSGA-II. Thanks to the advantage of SLDM, the probabilistic constraints are treated as approximating deterministic constraints. And therefore the reliability-based multi-objective optimization problems can be transformed into the deterministic multi-objective optimization problems of which the computational cost is reduced significantly. In these reliability-based multi-objective optimization problems, the conflicting objective functions are to minimize the weight and the displacements of the truss. The design variables are cross-section areas of the bars and contraints include static and dynamic constraints. For reliability analysis, the effect of uncertainty of parameters such as force, added mass in the nodes, material properties and cross-section areas of the bars are taken into account. The effectiveness and reliability of the proposed method are demonstrated through three benchmark-type truss structures including a 10-bar planar truss, a 72-bar spatial truss and a 200-bar planar truss. Moreover, the influence of parameters on the reliability-based Pareto optimal fronts is also carried out.

Optimization of Vehicle-Borne Radar Antenna Pedestal Based on Modified NSGA-II

2014 ◽  
Vol 945-949 ◽  
pp. 2241-2247
Author(s):  
De Gao Zhao ◽  
Qiang Li
Keyword(s):  
Pareto Front ◽  
Optimization Problems ◽  
Population Diversity ◽  
Multi Objective Optimization ◽  
Nsga Ii ◽  
Multi Objective ◽  
Sorting Process ◽  
Definition Of ◽  
Radar Antenna ◽  
Modified Algorithm

This paper deals with application of Non-dominated Sorting Genetic Algorithm with elitism (NSGA-II) to solve multi-objective optimization problems of designing a vehicle-borne radar antenna pedestal. Five technical improvements are proposed due to the disadvantages of NSGA-II. They are as follow: (1) presenting a new method to calculate the fitness of individuals in population; (2) renewing the definition of crowding distance; (3) introducing a threshold for choosing elitist; (4) reducing some redundant sorting process; (5) developing a self-adaptive arithmetic cross and mutation probability. The modified algorithm can lead to better population diversity than the original NSGA-II. Simulation results prove rationality and validity of the modified NSGA-II. A uniformly distributed Pareto front can be obtained by using the modified NSGA-II. Finally, a multi-objective problem of designing a vehicle-borne radar antenna pedestal is settled with the modified algorithm.

On Benchmark Problems and Metrics for Decision Space Performance Analysis in Multi-Objective Optimization

2017 ◽  
Vol 16 (01) ◽  
pp. 1750006 ◽  
Author(s):  
Bin Zhang ◽  
Kamran Shafi ◽  
Hussein Abbass
Keyword(s):  
Performance Analysis ◽  
Fitness Landscape ◽  
Benchmark Problems ◽  
Test Problems ◽  
Multi Objective Optimization ◽  
Decision Space ◽  
Nsga Ii ◽  
Multi Objective ◽  
Objective Space ◽  
Problem Instances

A number of benchmark problems exist for evaluating multi-objective evolutionary algorithms (MOEAs) in the objective space. However, the decision space performance analysis is a recent and relatively less explored topic in evolutionary multi-objective optimization research. Among other implications, such analysis can lead to designing more realistic test problems, gaining better understanding about optimal and robust design areas, and design and evaluation of knowledge-based optimization algorithms. This paper complements the existing research in this area and proposes a new method to generate multi-objective optimization test problems with clustered Pareto sets in hyper-rectangular defined areas of decision space. The test problem is parametrized to control number of decision variables, number and position of optimal areas in the decision space and modality of fitness landscape. Three leading MOEAs, including NSGA-II, NSGA-III, and MOEA/D, are evaluated on a number of problem instances with varying characteristics. A new metric is proposed that measures the performance of algorithms in terms of their coverage of the optimal areas in the decision space. The empirical analysis presented in this research shows that the decision space performance may not necessarily be reflective of the objective space performance and that all algorithms are sensitive to population size parameter for the new test problems.

Development of New Approach in Reliability Analysis for Excellent Predictive Quality of the Approximation Using Adaptive Kriging

2019 ◽  
Vol 44 ◽  
pp. 44-63
Author(s):  
Nassim Kernou ◽  
Youcef Bouafia
Keyword(s):  
Structural Reliability ◽  
Computational Cost ◽  
Computation Time ◽  
Point Method ◽  
Global Approximation ◽  
New Approach ◽  
Predictive Quality ◽  
Kriging Approximation ◽  
Adaptive Kriging

This study presents the results of a new approach for structural reliability analyses using adaptive kriging, confirmation simulation, and the pilot point method. Its main objective is to develop an efficient and accurate global approximation while controlling the computational cost and accuracy of prediction. The main contribution of research is to reduce computation time and successfully analyze complex problems with accurate results while ensuring excellent predictive quality of the approximation. For an excellent predictability of the kriging approximation, pilot point method and confirmation simulation are proposed. Simply, the predictive quality of the initial kriging approximation is improved by adding adaptive information, and the points are referred to as “pilot points” in areas where the kriging variance is maximized. Outcomes are confirmed with numerical simulations. The purpose is to select the minimum number of design experiments to ensure a good relative accuracy of the predictors with respect to the original models. Numerical examples show the efficiency of the proposed method compared to other structural reliability approaches.

scholarly journals A Generalized SDP Multi-Objective Optimization Method for EM-Based Microwave Device Design

Sensors ◽  
2019 ◽  
Vol 19 (14) ◽  
pp. 3065 ◽  
Author(s):  
Ying Liu ◽  
Qingsha S. Cheng ◽  
Slawomir Koziel
Keyword(s):  
Computational Cost ◽  
Optimization Method ◽  
Multi Objective Optimization ◽  
Uwb Antenna ◽  
Nsga Ii ◽  
Multi Objective ◽  
Iot Applications ◽  
Comparison Results ◽  
Pareto Fronts ◽  
Better Than

In this article, a generalized sequential domain patching (GSDP) method for efficient multi-objective optimization based on electromagnetics (EM) simulation is proposed. The GSDP method allowing fast searching for Pareto fronts for two and three objectives is elaborated in detail in this paper. The GSDP method is compared with the NSGA-II method using multi-objective problems in the DTLZ series, and the results show the GSDP method saved computational cost by more than 85% compared to NSGA-II method. A diversity comparison indicator (DCI) is used to evaluate approximate Pareto fronts. The comparison results show the diversity performance of GSDP is better than that of NSGA-II in most cases. We demonstrate the proposed GSDP method using a practical multi-objective design example of EM-based UWB antenna for IoT applications.

Multi-objective optimization of a hybrid electromagnetic suspension system for ride comfort, road holding and regenerated power

2016 ◽  
Vol 23 (5) ◽  
pp. 782-793 ◽  
Author(s):  
Mansour Ataei ◽  
Ehsan Asadi ◽  
Avesta Goodarzi ◽  
Amir Khajepour ◽  
Mir Behrad Khamesee
Keyword(s):  
Pareto Front ◽  
Ride Comfort ◽  
Optimization Method ◽  
Suspension System ◽  
Multi Objective Optimization ◽  
Damping Force ◽  
Nsga Ii ◽  
Electromagnetic Suspension ◽  
Multi Objective ◽  
Electromagnetic Component

This paper reports work on the optimization and performance evaluation of a hybrid electromagnetic suspension system equipped with a hybrid electromagnetic damper. The hybrid damper is configured to operate with hydraulic and electromagnetic components. The hydraulic component produces a large fail-safe baseline damping force, while the electromagnetic component adds energy regeneration and adaptability to the suspension. For analyzing the system, the electromagnetic component was modeled and integrated into a 2DOF quarter-car model. Three criteria were considered for evaluating the performance of the suspension system: ride comfort, road holding and regenerated power. Using the genetic algorithm multi-objective optimization (NSGA-II), the suspension design was optimized to improve the performance of the vehicle with respect to the selected criteria. The multi-objective optimization method provided a set of solutions called Pareto front in which all solutions are equally good and the selection of each one depends on conditions and needs. Among the given solutions in the Pareto front, a small number of cases, with different design purposes, were selected. The performances of the selected designs were compared with two reference systems: a conventional and a nonoptimized hybrid suspension system. The results show that the ride comfort and road holding qualities of the optimized hybrid system are improved, and the regenerated power is considerably increased.

The Multi-Objective Optimization Design of a Forklift Steering Mechanism

2014 ◽  
Vol 1049-1050 ◽  
pp. 884-887
Author(s):  
Qin Man Fan ◽  
Yong Hai Wu
Keyword(s):  
Hydraulic System ◽  
Optimization Design ◽  
Multi Objective Optimization ◽  
Nsga Ii ◽  
Multi Objective ◽  
Oil Pump ◽  
System Pressure ◽  
Steering Mechanism ◽  
Steering Torque

The design and quality of steering mechanism is directly related to forklift traction, mobility, steering stability and safe operation. A multi-objective optimization model of the forklift steering mechanism is established in this paper. The objective function is minimum oil cylinder stroke difference and the minimum power oil pump. Steering torque, geometrical angles, geometry size and the hydraulic system pressure are used as constraint conditions. We use non dominated sorting genetic algorithm (NSGA II) based on the Pareto optimal concept to optimize and calculate model and get the optimal design of steering mechanism.

scholarly journals Linear Geometric ICA: Fundamentals and Algorithms

2003 ◽  
Vol 15 (2) ◽  
pp. 419-439 ◽  
Author(s):  
Fabian J. Theis ◽  
Andreas Jung ◽  
Carlos G. Puntonet ◽  
Elmar W. Lang
Keyword(s):  
Computational Cost ◽  
Geometric Approach ◽  
Convergence Condition ◽  
Geometric Algorithms ◽  
Data Sets ◽  
New Approach ◽  
Geometric Convergence ◽  
Mixing Matrix ◽  
Unit Vectors

Geometric algorithms for linear independent component analysis (ICA) have recently received some attention due to their pictorial description and their relative ease of implementation. The geometric approach to ICA was proposed first by Puntonet and Prieto (1995). We will reconsider geometric ICA in a theoretic framework showing that fixed points of geometric ICA fulfill a geometric convergence condition (GCC), which the mixed images of the unit vectors satisfy too. This leads to a conjecture claiming that in the nongaussian unimodal symmetric case, there is only one stable fixed point, implying the uniqueness of geometric ICA after convergence. Guided by the principles of ordinary geometric ICA, we then present a new approach to linear geometric ICA based on histograms observing a considerable improvement in separation quality of different distributions and a sizable reduction in computational cost, by a factor of 100, compared to the ordinary geometric approach. Furthermore, we explore the accuracy of the algorithm depending on the number of samples and the choice of the mixing matrix, and compare geometric algorithms with classical ICA algorithms, namely, Extended Infomax and FastICA. Finally, we discuss the problem of high-dimensional data sets within the realm of geometrical ICA algorithms.

A Population Size Reduction Approach for Nondominated Sorting-Based Optimization Algorithms

2017 ◽  
Vol 16 (01) ◽  
pp. 1750005
Author(s):  
O. Tolga Altinoz ◽  
A. Egemen Yilmaz
Keyword(s):  
Population Size ◽  
Pareto Front ◽  
Size Reduction ◽  
Benchmark Problems ◽  
Nsga Ii ◽  
General Belief ◽  
Objective Space ◽  
Overall Performance ◽  
Approximation Set ◽  
Nondominated Sorting

The solution set of any multi-objective optimization problem can be expressed as an approximation set of Pareto front. The number of solution candidates in this set could be large enough to cover the entire Pareto front as a general belief. However, among the sufficiently close points on the objective space, almost same accurate solutions can obtain. Hence, in this set, it is possible to eliminate some of the solutions without detriment to the overall performance. Therefore, in this research, the authors propose a population size reduction method which systematically reduced the population size based on the distance and angle relations between any consecutive solutions. The results are evaluated based on two-objective benchmark problems and compared with the results of NSGA-II algorithm with respect to three different performance evaluation metrics.

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