Expected-Improvement-Based Methods for Adaptive Sampling in Multi-Objective Optimization Problems

2016 ◽  
Author(s):  
Jesper Kristensen ◽  
You Ling ◽  
Isaac Asher ◽  
Liping Wang
Keyword(s):  
Convex Hull ◽  
Design Optimization ◽  
Adaptive Sampling ◽  
Pareto Front ◽  
Performance Metrics ◽  
Sampling Methods ◽  
Optimization Problems ◽  
Expected Improvement ◽  
Multi Objective Optimization ◽  
Multi Objective

Adaptive sampling methods have been used to build accurate meta-models across large design spaces from which engineers can explore data trends, investigate optimal designs, study the sensitivity of objectives on the modeling design features, etc. For global design optimization applications, adaptive sampling methods need to be extended to sample more efficiently near the optimal domains of the design space (i.e., the Pareto front/frontier in multi-objective optimization). Expected Improvement (EI) methods have been shown to be efficient to solve design optimization problems using meta-models by incorporating prediction uncertainty. In this paper, a set of state-of-the-art methods (hypervolume EI method and centroid EI method) are presented and implemented for selecting sampling points for multi-objective optimizations. The classical hypervolume EI method uses hyperrectangles to represent the Pareto front, which shows undesirable behavior at the tails of the Pareto front. This issue is addressed utilizing the concepts from physical programming to shape the Pareto front. The modified hypervolume EI method can be extended to increase local Pareto front accuracy in any area identified by an engineer, and this method can be applied to Pareto frontiers of any shape. A novel hypervolume EI method is also developed that does not rely on the assumption of hyperrectangles, but instead assumes the Pareto frontier can be represented by a convex hull. The method exploits fast methods for convex hull construction and numerical integration, and results in a Pareto front shape that is desired in many practical applications. Various performance metrics are defined in order to quantitatively compare and discuss all methods applied to a particular 2D optimization problem from the literature. The modified hypervolume EI methods lead to dramatic resource savings while improving the predictive capabilities near the optimal objective values.

scholarly journals The dilemma between eliminating dominance-resistant solutions and preserving boundary solutions of extremely convex Pareto fronts

2021 ◽  
Author(s):  
Zhenkun Wang ◽  
Qingyan Li ◽  
Qite Yang ◽  
Hisao Ishibuchi
Keyword(s):  
Evolutionary Algorithms ◽  
Coping Strategies ◽  
Test Problem ◽  
Pareto Front ◽  
Optimization Problems ◽  
Feasible Region ◽  
Multi Objective Optimization ◽  
Multi Objective ◽  
Pareto Fronts ◽  
Boundary Solutions

AbstractIt has been acknowledged that dominance-resistant solutions (DRSs) extensively exist in the feasible region of multi-objective optimization problems. Recent studies show that DRSs can cause serious performance degradation of many multi-objective evolutionary algorithms (MOEAs). Thereafter, various strategies (e.g., the $$\epsilon $$ ϵ -dominance and the modified objective calculation) to eliminate DRSs have been proposed. However, these strategies may in turn cause algorithm inefficiency in other aspects. We argue that these coping strategies prevent the algorithm from obtaining some boundary solutions of an extremely convex Pareto front (ECPF). That is, there is a dilemma between eliminating DRSs and preserving boundary solutions of the ECPF. To illustrate such a dilemma, we propose a new multi-objective optimization test problem with the ECPF as well as DRSs. Using this test problem, we investigate the performance of six representative MOEAs in terms of boundary solutions preservation and DRS elimination. The results reveal that it is quite challenging to distinguish between DRSs and boundary solutions of the ECPF.

Toward the Use of Pareto Performance Solutions and Pareto Robustness Solutions for Multi-Objective Robust Optimization Problems

2012 ◽  
Author(s):  
Weijun Wang ◽  
Stéphane Caro ◽  
Fouad Bennis ◽  
Oscar Brito Augusto
Keyword(s):  
Robust Optimization ◽  
Pareto Front ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Multi Objective Optimization ◽  
Robust Solutions ◽  
Multi Objective ◽  
Robust Optimization Problem ◽  
Design Solutions ◽  
Further Development

For Multi-Objective Robust Optimization Problem (MOROP), it is important to obtain design solutions that are both optimal and robust. To find these solutions, usually, the designer need to set a threshold of the variation of Performance Functions (PFs) before optimization, or add the effects of uncertainties on the original PFs to generate a new Pareto robust front. In this paper, we divide a MOROP into two Multi-Objective Optimization Problems (MOOPs). One is the original MOOP, another one is that we take the Robustness Functions (RFs), robust counterparts of the original PFs, as optimization objectives. After solving these two MOOPs separately, two sets of solutions come out, namely the Pareto Performance Solutions (PP) and the Pareto Robustness Solutions (PR). Make a further development on these two sets, we can get two types of solutions, namely the Pareto Robustness Solutions among the Pareto Performance Solutions (PR(PP)), and the Pareto Performance Solutions among the Pareto Robustness Solutions (PP(PR)). Further more, the intersection of PR(PP) and PP(PR) can represent the intersection of PR and PP well. Then the designer can choose good solutions by comparing the results of PR(PP) and PP(PR). Thanks to this method, we can find out the optimal and robust solutions without setting the threshold of the variation of PFs nor losing the initial Pareto front. Finally, an illustrative example highlights the contributions of the paper.

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.

scholarly journals Industrial Applications of Intelligent Adaptive Sampling Methods for Multi-Objective Optimization

2020 ◽  
Author(s):  
Jesper Kristensen ◽  
Waad Subber ◽  
Yiming Zhang ◽  
Sayan Ghosh ◽  
Natarajan Chennimalai Kumar ◽  
...  
Keyword(s):  
Adaptive Sampling ◽  
Sampling Methods ◽  
Industrial Applications ◽  
Multi Objective Optimization ◽  
Multi Objective

scholarly journals A Multi-objective Particle Swarm Optimization Based on P System Theory

2018 ◽  
Vol 232 ◽  
pp. 03039
Author(s):  
Taowei Chen ◽  
Yiming Yu ◽  
Kun Zhao
Keyword(s):  
Particle Swarm Optimization ◽  
Pareto Front ◽  
Search Strategy ◽  
Optimization Problems ◽  
Particle Swarm ◽  
Pso Algorithm ◽  
P System ◽  
Multi Objective Optimization ◽  
Swarm Optimization ◽  
Multi Objective

Particle swarm optimization(PSO) algorithm has been widely applied in solving multi-objective optimization problems(MOPs) since it was proposed. However, PSO algorithms updated the velocity of each particle using a single search strategy, which may be difficult to obtain approximate Pareto front for complex MOPs. In this paper, inspired by the theory of P system, a multi-objective particle swarm optimization (PSO) algorithm based on the framework of membrane system(PMOPSO) is proposed to solve MOPs. According to the hierarchical structure, objects and rules of P system, the PSO approach is used in elementary membranes to execute multiple search strategy. And non-dominated sorting and crowding distance is used in skin membrane for improving speed of convergence and maintaining population diversity by evolutionary rules. Compared with other multi-objective optimization algorithm including MOPSO, dMOPSO, SMPSO, MMOPSO, MOEA/D, SPEA2, PESA2, NSGAII on a benchmark series function, the experimental results indicate that the proposed algorithm is not only feasible and effective but also have a better convergence to true Pareto front.

Population Decomposition-Based Greedy Approach Algorithm for the Multi-Objective Knapsack Problems

2017 ◽  
Vol 31 (04) ◽  
pp. 1759006 ◽  
Author(s):  
Jiawei Yuan ◽  
Hai-Lin Liu ◽  
Chaoda Peng
Keyword(s):  
Pareto Front ◽  
Optimization Problems ◽  
Experimental Studies ◽  
Weight Vector ◽  
Knapsack Problems ◽  
Multi Objective Optimization ◽  
Greedy Approach ◽  
Greedy Strategy ◽  
Multi Objective ◽  
Repair Strategy

Despite the effectiveness of the decomposition-based multi-objective evolutional algorithm (MOEA/D-M2M) in solving continuous multi-objective optimization problems (MOPs), its performance in addressing 0/1 multi-objective knapsack problems (MOKPs) has not been fully explored. In this paper, we use MOEA/D-M2M with an improved greedy repair strategy to solve MOKPs. It first decomposes an MOKP into a number of simple optimization subproblems and solves them in a collaborative way. Each subproblem has its own subpopulation, and then an improved greedy strategy is introduced to improve the performance of the proposed algorithm on MOKPs. Therein, a weight vector chosen randomly from a corresponding subpopulation is utilized to repair infeasible individuals or improve feasible individuals to have a better fitness, which improves the convergence of the population. Experimental studies on a set of test instances indicate that the MOEA/D-M2M with the improved greedy strategy is superior to MOGLS and MOEA/D in terms of finding better approximations to the Pareto front.

An Evolutionary Sequential Sampling Algorithm for Multi-Objective Optimization

2016 ◽  
Vol 33 (01) ◽  
pp. 1650006
Author(s):  
Aristotelis E. Thanos ◽  
Nurcin Celik ◽  
Juan P. Sáenz
Keyword(s):  
Pareto Front ◽  
Optimization Problems ◽  
Sequential Sampling ◽  
Extreme Points ◽  
Nonuniform Distribution ◽  
Solution Set ◽  
Multi Objective Optimization ◽  
Sampling Methodology ◽  
Multi Objective ◽  
Problem Instances

In this paper, we present a novel sequential sampling methodology for solving multi-objective optimization problems. Random sequential sampling is performed using the information from within the non-dominated solution set generated by the algorithm, while resampling is performed using the extreme points of the non-dominated solution set. The proposed approach has been benchmarked against well-known multi-objective optimization algorithms that exist in the literature through a series of problem instances. The proposed algorithm has been demonstrated to perform at least as good as the alternatives found in the literature in problems where the Pareto front presents convexity, nonconvexity, or discontinuity; while producing very promising results in problem instances where there is multi-modality or nonuniform distribution of the solutions along the Pareto front.

scholarly journals Improved NSGA-III using transfer learning and centroid distance for dynamic multi-objective optimization

2021 ◽  
Author(s):  
Haijuan Zhang ◽  
Gai-Ge Wang
Keyword(s):  
Transfer Learning ◽  
Pareto Front ◽  
Optimization Problems ◽  
Prediction Method ◽  
Reference Points ◽  
Initial Population ◽  
Multi Objective Optimization ◽  
Multi Objective ◽  
Centroid Distance ◽  
Dynamic Strategy

AbstractMulti-objective problems in real world are often contradictory and even change over time. As we know, how to find the changing Pareto front quickly and accurately is challenging during the process of solving dynamic multi-objective optimization problems (DMOPs). In addition, most solutions obey different distributions in decision space and the performance of NSGA-III when dealing with DMOPs should be further improved. In this paper, centroid distance is proposed and combined into NSGA-III with transfer learning together for DMOPs, called TC_NSGAIII. Centroid distance-based strategy is regarded as a prediction method to prevent some inappropriate individuals through measuring the distance of the population centroid and reference points. After the distance strategy, transfer learning is used for generating an initial population using the past experience. To verify the effectiveness of our proposed algorithm, NSGAIII, Tr_NSGAIII (NSGA-III combining with transfer learning only), Ce_NSGAIII (NSGA-III combining with centroid distance only), and TC_NSGAIII are compared. Seven state-of-the-art algorithms have been used for comparison on CEC 2015 benchmarks. Besides, transfer learning and centroid distance are regarded as a dynamic strategy, which is incorporated into three static algorithms, and the performance improvement is measured. What’s more, twelve benchmark functions from CEC 2015 and eight sets of parameters in each function are used in our experiments. The experimental results show that the performance of algorithms can be greatly improved through the proposed approach.

On deterministic procedures for low-cost multi-objective design optimization of miniaturized impedance matching transformers

2017 ◽  
Vol 34 (2) ◽  
pp. 403-419 ◽  
Author(s):  
Slawomir Koziel ◽  
Adrian Bekasiewicz
Keyword(s):  
Design Optimization ◽  
Pareto Front ◽  
Low Cost ◽  
Impedance Matching ◽  
Simulation Models ◽  
Multi Objective Optimization ◽  
Content Type ◽  
Multi Objective ◽  
Modeling Techniques ◽  
Em Simulation

Purpose This paper aims to investigate deterministic strategies for low-cost multi-objective design optimization of compact microwave structures, specifically, impedance matching transformers. The considered methods involve surrogate modeling techniques and variable-fidelity electromagnetic (EM) simulations. In contrary to majority of conventional approaches, they do not rely on population-based metaheuristics, which permit lowering the design cost and improve reliability. Design/methodology/approach There are two algorithmic frameworks presented, both fully deterministic. The first algorithm involves creating a path covering the Pareto front and arranged as a sequence of patches relocated in the course of optimization. Response correction techniques are used to find the Pareto front representation at the high-fidelity EM simulation level. The second algorithm exploits Pareto front exploration where subsequent Pareto-optimal designs are obtained by moving along the front by means of solving appropriately defined local constrained optimization problems. Numerical case studies are provided demonstrating feasibility of solving real-world problems involving expensive EM-simulation models of impedance transformer structures. Findings It is possible, by means of combining surrogate modeling techniques and constrained local optimization, to identify the set of alternative designs representing Pareto-optimal solutions, in a realistic time frame corresponding to a few dozen of high-fidelity EM simulations of the respective structures. Multi-objective optimization for the considered class of structures can be realized using deterministic approaches without defaulting to evolutionary methods. Research limitations/implications The present study can be considered a step toward further studies on expedited optimization of computationally expensive simulation models for miniaturized microwave components. Originality/value The proposed algorithmic solutions proved useful for expedited multi-objective design optimization of miniaturized microwave structures. The problem is extremely challenging when using conventional methods, in particular evolutionary algorithms. To the authors’ knowledge, this is one of the first attempts to investigate deterministic surrogate-assisted multi-objective optimization of compact components at the EM-simulation level.

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