Covariance Matrix Adaptation for Multi-objective Optimization

2007 ◽  
Vol 15 (1) ◽  
pp. 1-28 ◽  
Author(s):  
Christian Igel ◽  
Nikolaus Hansen ◽  
Stefan Roth
Keyword(s):  
Differential Evolution Algorithm ◽  
Size Control ◽  
Search Space ◽  
Multi Objective Optimization ◽  
Step Size ◽  
Nsga Ii ◽  
Multi Objective ◽  
Invariance Properties ◽  
Covariance Matrix Adaptation ◽  
Single Objective

The covariancematrix adaptation evolution strategy (CMA-ES) is one of themost powerful evolutionary algorithms for real-valued single-objective optimization. In this paper, we develop a variant of the CMA-ES for multi-objective optimization (MOO). We first introduce a single-objective, elitist CMA-ES using plus-selection and step size control based on a success rule. This algorithm is compared to the standard CMA-ES. The elitist CMA-ES turns out to be slightly faster on unimodal functions, but is more prone to getting stuck in sub-optimal local minima. In the new multi-objective CMAES (MO-CMA-ES) a population of individuals that adapt their search strategy as in the elitist CMA-ES is maintained. These are subject to multi-objective selection. The selection is based on non-dominated sorting using either the crowding-distance or the contributing hypervolume as second sorting criterion. Both the elitist single-objective CMA-ES and the MO-CMA-ES inherit important invariance properties, in particular invariance against rotation of the search space, from the original CMA-ES. The benefits of the new MO-CMA-ES in comparison to the well-known NSGA-II and to NSDE, a multi-objective differential evolution algorithm, are experimentally shown.

scholarly journals Multi-agent collaborative search: an agent-based memetic multi-objective optimization algorithm applied to space trajectory design

2011 ◽  
Vol 225 (11) ◽  
pp. 1211-1227 ◽  
Author(s):  
M Vasile ◽  
F Zuiani
Keyword(s):  
Selection Criterion ◽  
Search Space ◽  
Trajectory Design ◽  
Multi Objective Optimization ◽  
Nsga Ii ◽  
Agent Based ◽  
Multi Objective ◽  
Collaborative Search ◽  
Multi Agent ◽  
Space Trajectory Design

This article presents an algorithm for multi-objective optimization that blends together a number of heuristics. A population of agents combines heuristics that aim at exploring the search space both globally and in a neighbourhood of each agent. These heuristics are complemented with a combination of a local and global archive. The novel agent-based algorithm is tested at first on a set of standard problems and then on three specific problems in space trajectory design. Its performance is compared against a number of state-of-the-art multi-objective optimization algorithms that use the Pareto dominance as selection criterion: non-dominated sorting genetic algorithm (NSGA-II), Pareto archived evolution strategy (PAES), multiple objective particle swarm optimization (MOPSO), and multiple trajectory search (MTS). The results demonstrate that the agent-based search can identify parts of the Pareto set that the other algorithms were not able to capture. Furthermore, convergence is statistically better although the variance of the results is in some cases higher.

scholarly journals Multi-Objective Optimal Operations Based on Improved NSGA-II for Hanjiang to Wei River Water Diversion Project, China

Water ◽  
2019 ◽  
Vol 11 (6) ◽  
pp. 1159 ◽  
Author(s):  
Lianzhou Wu ◽  
Tao Bai ◽  
Qiang Huang ◽  
Jian Wei ◽  
Xia Liu
Keyword(s):  
Power Generation ◽  
Search Space ◽  
Water Transfer ◽  
Water Diversion ◽  
Multi Objective Optimization ◽  
Nsga Ii ◽  
Multi Objective ◽  
Water Diversion Project ◽  
Basin Water ◽  
Wei River

It is important to investigate the laws of reservoir multi-objective optimization operations, because it can obtain the best benefits from inter-basin water transfer projects to mitigate water shortage in intake areas. Given the multifaceted demands of the Hanjiang to Wei River Water Diversion Project, China (referred hereafter as “the Project”), an easy-to-operate multi-objective optimal model based on simulation is built and applied to search the multi-objective optimization operation rules between power generation and energy consumption. The Project includes two reservoirs connected by a water transfer tunnel. One is Huangjinxia, located in the mainstream of Hanjiang with abundant inflow but no regulation ability, and the other is Sanhekou, located in the tributary of Hanjiang with multi-year regulation ability but less water. The layout of the Project increases the difficulty of reservoir joint optimization operations. Therefore, an improved Non-dominated Sorting Genetic Algorithm-II (I-NSGA-II) with a feasible search space is proposed to solve the model based on long-term series data. The results show that: (1) The validated simulation model is helpful to obtain Pareto front curves to reveal the rules between power generation and energy consumption. (2) Choosing a reasonable search step size to build a feasible search space based on simulation results for the I-NSGA-II can help find more optimized solutions. Considering the influence of the initial populations of the algorithm and limited computing ability of computers, the qualified rate of Pareto points solved by I-NSGA-II are superior to NSGA-II. (3) According to the characteristics of the Project, water transfer ratio threshold value of two reservoirs are quantified for maximize economic benefits. Moreover, the flood season is a critical operation period for the Project, in which both reservoirs should supply more water to intake areas to ensure the energy balanced of the entire system. The findings provide an easy-to-operate multi-objective operation model with the I-NSGA-II that can easily be applied in optimal management of inter-basin water transfer projects by relevant authorities.

Product form evolutionary design system construction based on neural network model and multi-objective optimization

2020 ◽  
Vol 39 (5) ◽  
pp. 7977-7991
Author(s):  
Yixiang Wu
Keyword(s):  
Functional Relation ◽  
Product Form ◽  
Design System ◽  
Evolutionary Design ◽  
Multi Objective Optimization ◽  
Nsga Ii ◽  
Multi Objective ◽  
Form Design ◽  
Product Form Design ◽  
Single Objective

The product form evolutionary design based on multi-objective optimization can satisfy the complex emotional needs of consumers for product form, but most relevant literatures mainly focus on single-objective optimization or convert multiple-objective optimization into the single objective by weighting method. In order to explore the optimal product form design, we propose a hybrid product form design method based on back propagation neural networks (BP-NN) and non-dominated sorting genetic algorithm-II (NSGA-II) algorithms from the perspective of multi-objective optimization. First, the product form is deconstructed and encoded by morphological analysis method, and then the semantic difference method is used to enable consumers to evaluate product samples under a series of perceptual image vocabularies. Then, the nonlinear complex functional relation between the consumers’ perceptual image and the morphological elements is fitted with the BP-NN. Finally, the trained BP-NN is embedded into the NSGA-II multi-objective evolutionary algorithm to derive the Pareto optimal solution. Based on the hybrid BP-NN and NSGA-II algorithms, a multi-objective optimization based product form evolutionary design system is developed with the electric motorcycle as a case. The system is proved to be feasible and effective, providing theoretical reference and method guidance for the multi-image product form design.

ROTATED PROBLEMS AND ROTATIONALLY INVARIANT CROSSOVER IN EVOLUTIONARY MULTI-OBJECTIVE OPTIMIZATION

2008 ◽  
Vol 07 (02) ◽  
pp. 149-186 ◽  
Author(s):  
ANTONY IORIO ◽  
XIAODONG LI
Keyword(s):  
Evolutionary Algorithms ◽  
Coordinate System ◽  
Test Problems ◽  
Multi Objective Optimization ◽  
Problem Domain ◽  
Nsga Ii ◽  
Multi Objective ◽  
Parameter Interactions ◽  
Rotationally Invariant ◽  
Single Objective

Problems that are not aligned with the coordinate system can present difficulties to many optimization algorithms, including evolutionary algorithms, by trapping the search on a ridge. The ridge problem in single-objective optimization is understood, but until now little work has been done on understanding this issue in the multi-objective domain. Multi-objective problems with parameter interactions present difficulties to an optimization algorithm, which are not present in the single-objective domain. In this work, we have explained the nature of these difficulties, and investigated the behavior of the NSGA-II, which has difficulties with problems not aligned with the principle coordinate system. This study has investigated Simplex Crossover (SPX), Unimodal Normally Distributed Crossover (UNDX), Parent-Centric Crossover (PCX), and Differential Evolution (DE), as possible alternatives to the Simulated Binary Crossover (SBX) operator within the NSGA-II, on problems exhibiting parameter interactions through a rotation of the coordinate system. An analysis of these operators on three rotated bi-objective test problems, and a four-and eight-objective problem is provided. New observations on the behavior of rotationally invariant crossover operators in the multi-objective problem domain have been reported.

Progressive-Stepping-Based Non-Dominated Sorting Genetic Algorithm for Multi-Objective Optimization

2016 ◽  
Vol 7 (3) ◽  
pp. 17-49 ◽  
Author(s):  
Akshay Baviskar ◽  
Shankar Krishnapillai
Keyword(s):  
Genetic Algorithm ◽  
Optimization Problems ◽  
Mechanical Design ◽  
Multi Objective Optimization ◽  
Step Size ◽  
Nsga Ii ◽  
Multi Objective ◽  
Performance Metric ◽  
Mechanical Design Problem ◽  
Utopia Point

This paper demonstrates two approaches to achieve faster convergence and a better spread of Pareto solutions in fewer numbers of generations, compared to a few existing algorithms, including NSGA-II and SPEA2 to solve multi-objective optimization problems (MOP's). Two algorithms are proposed based on progressive stepping mechanism, which is obtained by the hybridization of existing Non-dominated Sorting Genetic Algorithm II (NSGA-II) with novel guided search schemes, and modified chromosome selection and replacement mechanisms. Progressive Stepping Non-dominated Sorting based on Local search (PSNS-L) controls the step size, and Progressive Stepping Non-dominated Sorting based on Utopia point (PSNS-U) method controls the number of divisions to generate better chromosomes in each generation to achieve faster convergence. Four multi-objective evolutionary algorithms (EA's) are compared for different benchmark functions and PSNS outperforms them in most cases based on various performance metric values. Finally a mechanical design problem has been solved with PSNS algorithms.

Covariance matrix adaptive strategy for a multi-objective evolutionary algorithm based on reference point

2020 ◽  
Vol 39 (5) ◽  
pp. 7315-7332
Author(s):  
Lixin Wei ◽  
JinLu Zhang ◽  
Rui Fan ◽  
Xin Li ◽  
Hao Sun
Keyword(s):  
Covariance Matrix ◽  
State Of The Art ◽  
Optimal Solution ◽  
Search Space ◽  
Continuous Functions ◽  
User Preferences ◽  
Reference Points ◽  
Multi Objective Optimization ◽  
Multi Objective ◽  
Single Objective

In this article, an effective method, called an adaptive covariance strategy based on reference points (RPCMA-ES) is proposed for multi-objective optimization. In the proposed algorithm, search space is divided into independent sub-regions by calculating the angle between the objective vector and the reference vector. The reference vectors can be used not only to decompose the original multi-objective optimization problem into a number of single-objective subproblems, but also to elucidate user preferences to target a preferred subset of the whole Pareto front (PF). In this respect, any single objective optimizers can be easily used in this algorithm framework. Inspired by the multi-objective estimation of distribution algorithms, covariance matrix adaptation evolution strategy (CMA-ES) is involved in RPCMA-ES. A state-of-the-art optimizer for single-objective continuous functions is the CMA-ES, which has proven to be able to strike a good balance between the exploration and the exploitation of search space. Furthermore, in order to avoid falling into local optimality and make the new mean closer to the optimal solution, chaos operator is added based on CMA-ES. By comparing it with four state-of-the-art multi-objective optimization algorithms, the simulation results show that the proposed algorithm is competitive and effective in terms of convergence and distribution.

scholarly journals Silhouetting the Cost-Time Front: Multi-objective Resource Optimization in Business Processes

2021 ◽  
pp. 92-108
Author(s):  
Orlenys López-Pintado ◽  
Marlon Dumas ◽  
Maksym Yerokhin ◽  
Fabrizio Maria Maggi
Keyword(s):  
Resource Utilization ◽  
Cycle Time ◽  
Search Space ◽  
Hill Climbing ◽  
Optimization Approach ◽  
Multi Objective Optimization ◽  
Resource Allocations ◽  
Nsga Ii ◽  
Multi Objective ◽  
The Cost

AbstractThe allocation of resources in a business process determines the trade-off between cycle time and resource cost. A higher resource utilization leads to lower cost and higher cycle time, while a lower resource utilization leads to higher cost and lower waiting time. In this setting, this paper presents a multi-objective optimization approach to compute a set of Pareto-optimal resource allocations for a given process concerning cost and cycle time. The approach heuristically searches through the space of possible resource allocations using a simulation model to evaluate each allocation. Given the high number of possible allocations, it is imperative to prune the search space. Accordingly, the approach incorporates a method that selectively perturbs a resource utilization to derive new candidates that are likely to Pareto-dominate the already explored ones. The perturbation method relies on two indicators: resource utilization and resource impact, the latter being the contribution of a resource to the cost or cycle time of the process. Additionally, the approach incorporates a ranking method to accelerate convergence by guiding the search towards the resource allocations closer to the current Pareto front. The perturbation and ranking methods are embedded into two search meta-heuristics, namely hill-climbing and tabu-search. Experiments show that the proposed approach explores fewer resource allocations to compute Pareto fronts comparable to those produced by a well-known genetic algorithm for multi-objective optimization, namely NSGA-II.

scholarly journals Multi-Objective Lightweight Optimization of Parameterized Suspension Components Based on NSGA-II Algorithm Coupling with Surrogate Model

Machines ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 107
Author(s):  
Rongchao Jiang ◽  
Zhenchao Jin ◽  
Dawei Liu ◽  
Dengfeng Wang
Keyword(s):  
Ride Comfort ◽  
Lightweight Design ◽  
Dynamic Performance ◽  
Vehicle Dynamic ◽  
Multi Objective Optimization ◽  
Original Design ◽  
Dynamic Simulations ◽  
Nsga Ii ◽  
Torsion Beam ◽  
Multi Objective

In order to reduce the negative effect of lightweighting of suspension components on vehicle dynamic performance, the control arm and torsion beam widely used in front and rear suspensions were taken as research objects for studying the lightweight design method of suspension components. Mesh morphing technology was employed to define design variables. Meanwhile, the rigid–flexible coupling vehicle model with flexible control arm and torsion beam was built for vehicle dynamic simulations. The total weight of control arm and torsion beam was taken as optimization objective, as well as ride comfort and handling stability performance indexes. In addition, the fatigue life, stiffness, and modal frequency of control arm and torsion beam were taken as the constraints. Then, Kriging model and NSGA-II were adopted to perform the multi-objective optimization of control arm and torsion beam for determining the lightweight scheme. By comparing the optimized and original design, it indicates that the weight of the optimized control arm and torsion beam are reduced 0.505 kg and 1.189 kg, respectively, while structural performance and vehicle performance satisfy the design requirement. The proposed multi-objective optimization method achieves a remarkable mass reduction, and proves to be feasible and effective for lightweight design of suspension components.

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