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Multi-objective Genetic Algorithms: Problem Difficulties and Construction of Test Problems
Abstract
In this paper, we study the problem features that may cause a multi-objective genetic algorithm (GA) difficulty in converging to the true Pareto-optimal front. Identification of such features helps us develop difficult test problems for multi-objective optimization. Multi-objective test problems are constructed from single-objective optimization problems, thereby allowing known difficult features of single-objective problems (such as multi-modality, isolation, or deception) to be directly transferred to the corresponding multi-objective problem. In addition, test problems having features specific to multi-objective optimization are also constructed. More importantly, these difficult test problems will enable researchers to test their algorithms for specific aspects of multi-objective optimization.
... These typically involve formulating the problem as the heuristic optimization algorithms [10]. At present, there are many multi-objective algorithms are available, most of which are developed based on recent meta-heuristic optimization techniques [11][12][13][14]. These meta-heuristic optimizers can provide the decision maker with a various set of best tradeoff scheduling plans, termed as Pareto optimal solutions. ...
... The concepts of optimization can be extended including reliability, Plastic analysis and design [20][21][22][23]. In order to compare with available meta-heuristic algorithms, all of examples are solved also using the multi-objective particle swarm optimization (MOPSO) [13] and the NSGA-II algorithm [14] in order to provide some comparison. Table 1 shows the parameters of these algorithms in the utilized examples. ...
The obnoxious p-median problem consists of locating p facilities among a set of sites such that the sum distance from any demand to its nearest facility and the dispersion among facilities are maximized. In this paper, the multi-objective colliding bodies optimization algorithm (MOCBO) is utilized to obtain the trade-off curve of the obnoxious p-median problems. The performance of the developed optimization method is investigated for locating obnoxious facilities through two case studies to maximize the two conflicting objectives. The performance of the MOCBO algorithm is further compared with those of the MPSO and NSGA-II algorithms representative of the state of the art in the field of multi-objective optimization. In this study, the MOCBO algorithm showed suitable convergence performance and generalization abilities compared to the MPSO and NSGA-II algorithms.
... Subsequently, Rudolph [57] introduces the SYM-PART test problems, which have a larger number of PSs. Deb [58] proposes the omni-test problems, which allow for adjustable decision variables and an adjustable number of PSs. Additionally, the SSUF test problem [25] and the HPS test problem [59] are proposed to evaluate the performance of algorithms in terms of maintaining diversity. ...
Multimodal multi-objective problems refer to situations where more than one Pareto set in the decision space corresponds to the same Pareto front in the objective space. In this paper, we propose a multimodal multi-objective coati optimization algorithm based on spectral clustering (MMOCOA-SC) for use in multimodal multi-objective problems. The algorithm starts by creating stable subpopulations via a spectral clustering method, grouping similar individuals within each subpopulation. Then, an improved coati optimization algorithm is proposed in order to increase the diversity of the population and continuously search for near-Pareto optimal solutions. Finally, we use an improved crowding distance method combined with a non-dominated sorting method to maintain and retain multiple near-Pareto optimal solutions. The MMOCOA-SC is evaluated alongside five state-of-the-art algorithms using the 2020 CEC test suite, IDMP test problems, and twelve classic engineering application problems. In comparison to the other five state-of-the-art algorithms, the experimental results show that MMOCOA-SC exhibits a superior performance when applied to the problems. This study explores multimodal multi-objective optimization algorithms with a focus on the concept of symmetry, which is crucial for multimodal multi-objective optimization problems in terms of achieving a balanced decision space and a diversity of solutions in the objective space.
... We first conduct experiments on Deb-Thiele-Laummans-Zitzler (DTLZ) 56 and Zitzler-Deb-Thiele (ZDT) 57 multi-objective benchmark functions that are commonly used in evolutionary computation community, including functions with shifted, non-separable, concave, disconnected and multi-modal characteristics. ...
Multi-objective optimization has burgeoned as a potent methodology for informed decision-making in enhanced geothermal systems, aiming to concurrently maximize economic yield, ensure enduring geothermal energy provision, and curtail carbon emissions. However, addressing a multitude of design parameters inherent in computationally intensive physics-driven simulations constitutes a formidable impediment for geothermal design optimization, as well as across a broad range of scientific and engineering domains. Here we report an Active Learning enhanced Evolutionary Multi-objective Optimization algorithm, integrated with hydrothermal simulations in fractured media, to enable efficient optimization of fractured geothermal systems using few model evaluations. We introduce probabilistic neural network as classifier to learns to predict the Pareto dominance relationship between candidate samples and reference samples, thereby facilitating the identification of promising but uncertain offspring solutions. We then use active learning strategy to conduct hypervolume based attention subspace search with surrogate model by iteratively infilling informative samples within local promising parameter subspace. We demonstrate its effectiveness by conducting extensive experimental tests of the integrated framework, including multi-objective benchmark functions, a fractured geothermal model and a large-scale enhanced geothermal system. Results demonstrate that the ALEMO approach achieves a remarkable reduction in required simulations, with a speed-up of 1-2 orders of magnitude (10-100 times faster) than traditional evolutionary methods, thereby enabling accelerated decision-making. Our method is poised to advance the state-of-the-art of renewable geothermal energy system and enable widespread application to accelerate the discovery of optimal designs for complex systems.
... This makes it difficult to optimize one objective without worsening others. Therefore, instead of obtaining a single optimal solution, the MOOPs require the retrieval of trade-off solutions, which are referred to as the Pareto optimal front [60,61]. These solutions that lie on the Pareto optimal front are non-dominated by other solutions -meaning no other solution is better in all objectives. ...
Soft robotic devices are designed for applications such as exploration, manipulation, search and rescue, medical surgery, rehabilitation, and assistance. Due to their complex kinematics, various and often hard-to-define degrees of freedom, and nonlinear properties of their material, designing and operating these devices can be quite challenging. Using tools such as optimization methods can improve the efficiency of these devices and help roboticists manufacture the robots they need. In this work, we present an extensive and systematic literature search on the optimization methods used for the mechanical design of soft robots, particularly focusing on literature exploiting evolutionary computation (EC). We completed the search in the IEEE, ACM, Springer, SAGE, Elsevier, MDPI, Scholar, and Scopus databases between 2009 and 2024 using the keywords “soft robot,” “design,” and “optimization.” We categorized our findings in terms of the type of soft robot (i.e., bio-inspired, cable-driven, continuum, fluid-driven, gripper, manipulator, modular), its application (exploration, manipulation, surgery), the optimization metrics (topology, force, locomotion, kinematics, sensors, and energy), and the optimization method (categorized as EC or non-EC methods). After providing a road map of our findings in the state of the art, we offer our observations concerning the implementation of the optimization methods and their advantages. We then conclude our paper with suggestions for future research.
... Remarkable progress has been made in building well-defined test problems to assess the efficiency and effectiveness of multiobjective evolutionary algorithms (MOEAs). These test problems, typically comprising a benchmark suite, are designed to reflect challenging problem characteristics, including but not limited to multimodality, deception, isolated optima, non-convexity, discreteness, non-uniformity, non-separability and scalability in the number of decision variables and/or objectives (Deb 1999). ...
Despite progress in multiobjective evolutionary algorithms (MOEAs) research, their efficacy in real-world scenarios remains unclear. This article introduces a diagnostic benchmarking framework to evaluate MOEAs, comprising (1) flexible MOEA construction software, (2) performance evaluation metrics and (3) real-world applications for benchmarking, reflecting diverse mathematical challenges. Utilizing this framework, NSGA-II, NSGA-III, RVEA, MOEA/D and Borg MOEA were evaluated across four applications with three to ten objectives. Collectively, the four applications capture challenges such as stochastic objectives, severe constraints, nonlinearity and complex Pareto frontiers. The study demonstrates how MOEAs that have shown strong performance on standard test problems can struggle on real-world applications. The benchmarking framework and results have value for enhancing the design and use of MOEAs in real-world applications. Further, the results highlight the need to improve the adaptability and ease-of-use of MOEAs given the often ill-defined nature of real-world problem-solving. ARTICLE HISTORY
... Statistical study of performance values on Hypervolume for x 2 ≈ 0.2 and the local minimum for x 2 ≈ 0.6 with g(0.2) = 0.7057 and g(0.6) = 1.2[14]. In this work, we use the case x 2 ≈ 0.2. ...
We propose a method for solving multiobjective optimization problems under the constraints of inequality. In this method, the initial problem is transformed into a single-objective optimization without constraints using an augmented Lagrangian function and an ϵ-constraint approach. Indeed, the augmented Lagrangian function is used to convert a given problem with multiple objective functions into a single objective function. The ϵ-constraint approach allows for the transformation of constrained optimization problems into unconstrained optimization problems. To demonstrate the admissibility and Pareto optimality of the obtained solutions, we have provided two propositions with proofs. In addition, a comparison study is made with two other well-known and widely used methods, such as NSGA-II and BoostDMS, on convergence and distribution of obtained solutions using numerical results for fifty test problems taking in the literature. Based on all these theoretical and numerical results, we can say that the proposed method is the best way to solve multiobjective optimization problems.
... Several analytical MOPs are adopted to test the proposed method, including zdt1, zdt2, zdt3, and zdt4 problems, as given in Table I. The detailed mathematical formulas can be found in Ref. 37. ...
Modern aerodynamic design optimization aims to discover optimal configurations using computational fluid dynamics under complex flow conditions, which is a typical expensive multi-objective optimization problem. The multi-objective evolutionary algorithm based on decomposition (MOEA/D) combined with efficient global optimization is a promising method but requires enhanced efficiency and faces limitations in its application to multi-objective aerodynamic design optimization (MOADO). To address the issues, an efficient parallel MOEA/D assisted with variable-fidelity optimization (VFO) is proposed for solving MOADO, called the MOEA/D-VFO algorithm. Variable-fidelity surrogates are built for objectives and constraints, achieving higher accuracy using fewer high-fidelity samples and a great number of low-fidelity samples. By retaining more good candidates, the sub-optimization problems defined by decomposing original objectives are capable of discovering more favorable samples using MOEA/D, which prompts optimization convergence. A constraint-handling strategy is developed by incorporating the probability of feasibility functions in the sub-optimizations. The selection of new samples for parallel evaluation is improved by filtering out poor candidates and selecting effective promising samples, which improves the feasibility and diversity of solved Pareto solutions. A Pareto front (PF) can be efficiently found in a single optimization run. The proposed approach is demonstrated by four analytical test functions and verified by two aerodynamic design optimizations of airfoils with and without constraints, respectively. The results indicate that the MOEA/D-VFO approach can greatly improve optimization efficiency and obtain the PF satisfying constraints within an affordable computational budget.
... Step 2: Make the population start out. The settings of genetic algorithm parameters directly impact its performance, and a reasonable choice and control of these parameters enable the genetic algorithm to search for the optimal solution along the most favorable path 57 . The parameter settings for genetic algorithms are typically determined based on the characteristics of the problem and the search space for solutions 58,59 . ...
As a main carrier mode for the sustainable development of the construction industry in China, prefabricated building may lead to problems such as cost overruns, project delays, and waste of resources due to unreasonable selection of prefabricated components. Therefore, we quantitatively analyze the contribution rate of quality optimization of prefabricated components using QFD-SEM. Under the constraints of prefabrication rate, quality optimization contribution rate, and expected values of various sub-goals, we propose a multi-objective optimization method for prefabricated component combinations based on cost, duration, and carbon emissions. By using NSGA-II to solve the model, we can obtain a set of optimal Pareto solutions for prefabricated component combinations. Based on the optimal Pareto solution set, we establish a multi-objective evaluation model using simulated annealing optimization projection tracing method, and select the optimal prefabricated component combination solution according to the projected eigenvalues of the solutions. An empirical study is conducted using an eleven-story framed building in Shenzhen, Guangdong Province, China as a case study. The results show that: (1) Using this method, optimal solutions can be obtained in an unbounded solution space, with the optimal solution having advantages over both fully cast-in-place and fully prefabricated solutions. Compared to the fully cast-in-place solution, the duration and carbon emissions are reduced by 36.62% and 12.74% respectively, while compared to the fully prefabricated solution, costs are reduced by 4.15%. (2) There is a certain negative correlation between the cost of prefabricated component combinations and duration, carbon emissions, and quality optimization, while there is a certain positive correlation with the prefabrication rate. (3) The size of the optimal projection direction vector based on the optimization objectives indicates that carbon emissions have the greatest impact on the evaluation results of the solutions.
... x L x U Reference BK1 2 2 (-5,-5) (10,10) (Huband et al., 2006) DD1 5 2 (-20,...,-20) (20,...,20) (Das & Dennis, 1998) Deb 2 2 (0.1,0.1) (1,1) (Deb, 1999) FF1 2 2 (-1,-1) (1,1) (Huband et al., 2006) Hil1 2 2 (0,0) (1,1) (Hillermeier, 2001) Imbalance1 2 2 (-2,-2) (2,2) (Chen et al., 2023a) JOS1a 50 2 (-2,...,-2) (2,...,2) (Jin et al., 2001) LE1 2 2 (-5,-5) (10,10) (Huband et al., 2006) PNR 2 2 (-2,-2) (2,2) (Preuss et al., 2006) WIT1 2 2 (-2,-2) (2,2) (Witting, 2012) For the tested problems, the partial order are induced by polyhedral cones R 2 + , K 1 , and K 2 , respectively, where ...
In this paper, we develop a unified framework and convergence analysis of descent methods for vector optimization problems (VOPs) from a majorization-minimization perspective. By choosing different surrogate functions, the generic method reduces to some existing descent methods with and without line search, respectively. The unified convergence analysis shows that the slow convergence of the steepest descent method is mainly due to the large gap between surrogate and objective functions. As a result, the performance of descent methods can be improved by narrowing the surrogate gap. Interestingly, we observe that selecting a tighter surrogate function is equivalent to using an appropriate base of the dual cone in the direction-finding subproblem. Furthermore, we use Barzilai-Borwein method to narrow the surrogate gap and devise a Barzilai-Borwein descent method for VOPs with polyhedral cone. By reformulating the subproblem, we provide a new insight into the Barzilai-Borwein descent method and bridge it to the steepest descent method. Finally, several numerical experiments confirm the efficiency of the proposed method.
... For example, unlike vector enabled genetic algorithm (VEGA), NSGA distributes the solutions across all Pareto-optimal regions, avoiding bias as a major limitation of VEGA (Richardson et al. 1989). Deb (1999) demonstrated that NSGA converges towards the true Pareto front and evenly distributes solutions. Also, NSGA is not subject to the limitation of a multi-objective genetic algorithm (MOGA) that scalarizes the objectives into a single objective with weight vectors, simplifying the optimization process but making the solution heavily dependent on the chosen weight vectors (Konak et al. 2006). ...
There is limited existing research that identifies the optimum rehabilitation strategy, taking utility decision-makers’ risk attitudes into account. The objective of this study is to detect the critical pipes of a water distribution network (WDN) for rehabilitation, maximizing the post-earthquake serviceability of the WDN while minimizing the risk of choosing a specific rehabilitation strategy. For that purpose, a multi-objective optimization framework is formulated. System Serviceability Index (SSI) is quantified to represent the serviceability of a WDN after an earthquake. One of the two objective functions in the optimization problem maximizes the expected SSI value. The second objective function minimizes the value at risk (VaR) or conditional value at risk (CVaR) of the decision-making. The solution methodology comprises five steps: pipe seismic repair rate calculation, hydraulic modeling, and analysis, Monte Carlo simulation, nondominated sorting genetic algorithm (NSGA) for optimization, and nondominated or Pareto-optimal rehabilitation strategies identification. The proposed approach is applied to a WDN to demonstrate its effectiveness. The proposed approach offers a range of nondominated or Pareto-optimal rehabilitation strategies facilitating the decision-making based on tradeoffs between post-earthquake serviceability and risk within a specific budget limit. The proposed approach outperforms existing methods by providing risk-averse decision-makers with a set of optimal rehabilitation strategies with known risk levels.
... The fitness function is a measure for evaluating the efficiency of the chromosomes. The fitness function definition is a key component that affects GA performance [34]. The process of calculating the fitness function will store the best response for future replications. ...
The price of a company's stock, which can increase in lockstep with the price of a single share, is the one of the indicator to measure its performance. Clients or stockholding companies find it challenging to make long-term projections regarding the value of specific stocks due to the unpredictable nature of stock prices. Consequently, there is no business-related subject more talked about than stock market predictions. It is crucial to resolve this issue in a way that benefits buyers and investors because they frequently experience investment losses. Machine learning is useful in developing model for stock value predictions. We are utilizing Python and Linear Regression, one of the Machine Learning statistical techniques for predictive analysis, to create a stock price prediction website in order to address this issue. Our study primarily focuses on the NIFTY50 index's performance in distributed lag with the purpose of predicting stock prices in the Indian stock market. Several useful characteristics of the NIFTY50 lag index were extracted by means of a genetic algorithm. After that, we uncovered hidden correlations between the stock index and a given stock's trend by using the linear regression classifier. For the purpose of testing our approach, we used it to forecast the future of three distinct equities. In comparison to state-of-the-art forecasting methodologies, our experimental results demonstrated an accuracy of 82.55%. For the purpose of predicting daily changes in stock prices, the NIFTY50 stock index proved to be useful.
... Ensuring a broad diversity of solutions across all objectives is crucial, providing decision-makers with a comprehensive spectrum of choices. Scholarly works abound with algorithms developed from this approach can be found in literature [13][14][15]. ...
This research introduces a novel multi-objective adaptation of the Geometric Mean Optimizer (GMO), termed the Multi-Objective Geometric Mean Optimizer (MOGMO). MOGMO melds the traditional GMO with an elite non-dominated sorting approach, allowing it to pinpoint Pareto optimal solutions through offspring creation and selection. A Crowding Distance (CD) coupled with an Information Feedback Mechanism (IFM) selection strategy is employed to maintain and amplify the convergence and diversity of potential solutions. MOGMO efficacy and capabilities are assessed using thirty notable case studies. This encompasses nineteen multi-objective benchmark problems without constraints, six with constraints and five multi-objective engineering design challenges. Based on the optimization results, the proposed MOGMO is better 54.83% in terms of GD, 64.51% in terms of IGD, 67.74% in terms of SP, 70.96% in terms of SD, 64.51% in terms of HV and 77.41% in terms of RT. Therefore, MOGMO has a better convergence and diversity for solving un-constraint, constraint and real-world application. Statistical outcomes from MOGMO are compared with those from Multi-Objective Equilibrium Optimizer (MOEO), Decomposition-Based Multi-Objective Symbiotic Organism Search (MOSOS/D), Non-dominated Sorting Genetic Algorithm (NSGA-II), Multi-Objective Multi-Verse Optimization (MOMVO) and Multi-Objective Plasma Generation Opti-mizer (MOPGO) algorithms, utilizing identical performance measures. This comparison reveals that MOGMO consistently exhibits robustness and excels in addressing an array of multi-objective challenges. The MOGMO source code is available at https:// github. com/ kanak 02/ MOGMO.
... Innovative inventions in several key technologies have driven the IoV environment forward. Prominent factors that have contributed to the IoV environment encompass the development of high-speed wireless fidelity (Wi-Fi) networks, the advance of cost-effective and energy-efficient sensors, the change in on-demand cloud computing setup, massive data sets, and the implementation of Artificial Intelligence (AI) technologies [11][12][13][14]. Access control is required to overcome significant challenges and ensure a safe and efficient IoV environment. ...
The Internet of Vehicles (IoV) is currently experiencing significant development, which has involved the introduction of an efficient Access Control Mechanism (ACM). Reliable access control is evolving into mandatory in order to provide security and efficient transmission within the IoV environment as the volume of vehicles equipped with connectivity continues to expand and they become more incorporated into any number of applications. The primary objective of this research is to develop an ACM for the IoV system based on the use of a Genetically Optimized Twin-Delayed Delayed Deep Deterministic Policy Gradient (TD3) algorithm. The TD3 model modifies access policies to be in line with the current scenario using deep reinforcement learning (Deep RL) techniques. This allows vehicles to make access decisions that are intelligent about the environment in which they are performing. To prevent energy loss while the vehicle is in transit into the client system, the model also emphasizes access based on the vehicle's energy consumption (EC). Finally, with the support of the genetic algorithm (GA), the accuracy of the access control model can be improved by optimizing the high-level parameters in a manner in which they improves efficiency. In order to further enhance the model's environmental sustainability and reliability, the recommended model provides an approach that is both profound and efficient for access control in the constantly changing setting of the IoV.
... The FDA benchmark [7] is understood to be the most popular DMOP. It is based on ZDT [99] and DTLZ [100] functions. The FDA consists of five easily constructed functions and a scalable number of decision variables. ...
Evolutionary algorithms have been widely explored and applied in optimization problems. The introduction of multi-objective evolutionary algorithms (MOEAs) has facilitated the adaptation and creation of new methods to handle more complex and realistic optimizations, such as dynamic multi-objective optimization problems (DMOPs). A dynamic MOEA (DMOEA) can be constructed by changing the MOEA structure and variation operators used to solve DMOPs. Furthermore, DMOEAs can implement change-detection strategies and mechanisms to handle the dynamics of the environment. DMOEAs are often designed to solve unconstrained DMOPs. However, several studies have been conducted to solve dynamic constrained MOPs and more recently, to solve dynamic many-objective optimization problems. There are many real-world DMOPs; however, few DMOEAs have been evaluated with respect to those problems. Evaluation of algorithm performance is an essential aspect of using DMOEAs. Several measures used to validate MOEAs have been adapted to assess DMOEAs and quantitatively compare them. This paper presents a broad review of DMOEAs. The information is summarized and organized according to current research branches. Furthermore, a more general classification scheme for DMOEAs is proposed, based on the method of incorporating diversity to solve DMOPs.
... More multi-objective optimization is necessary since the mechanism mass m was defined by the aforementioned optimization model as a constraint variable. When compared to a traditional genetic algorithm, a multi-objective optimization genetic algorithm solves the objective function using various weight coefficients, improving the objectivity of the optimization process and reducing the complexity of multi-objective problems to single-objective problems [17][18][19]. In addition, the population can be converged towards a Pareto optimum solution using multi-objective genetic algorithms [20]. ...
Underwater unmanned robots are an essential tool for human underwater exploration and detection and are widely employed in a variety of underwater operational settings. One of the hottest issues in this field is applying bionic notions to the creation of underwater unmanned robots by simulating fish swimming or cephalopod crawling. Using the tentacle suction cup adsorption technique during octopus’ predation as a model, underwater magnetic adsorption robots with the opening and closing claws were studied in this paper. First, the robot’s general structural design is presented. The claw mechanism is demonstrated by mimicking the octopus’s tentacle action during feeding, which primarily consists of an opening and closing claw that replicates the octopus’s tentacle and a magnetic adsorption unit that replicates the octopus’s suction cup adsorption. Then, the Kriging response surface optimization method is used to optimize the design of the claw mechanism to obtain excellent mechanical properties, and simulation software is used to verify. Finally, a robot prototype was built and its pool tests were conducted, with some experimental results presented. The experimental results show that after the robot reaches the predetermined position through pneumatic ejection and secondary propulsion launch, it can quickly open its claws within 0.11 s and apply 462.42 N adsorption force to complete the adsorption of the target.
... Since the hypervolume (HV) indicator reveals how close proximity between the non-dominated solutions and true Pareto front (Zitzler and Thiele 1999), and the inverted generational distance (IGD) embodies the distribution and convergence of non-dominated solutions, they are commonly used to evaluate the performance of compared algorithms (Agrawal and Deb 1995). Herein, ZDT instances (ZDTs) (Deb 1999) and Nowacki beam problem (Nowacki 1980) are conducted to validate the feasibility of proposed method. Furthermore, the comparisons of related HV and IGD values are carried out to illustrate the capability of HCSIS-SMOO. ...
High-quality Pareto front is always pursued when solving multi-objective optimization problems with surrogate-assisted multi-objective optimization (SMOO). For this purpose, a hybrid criterion-based sample infilling strategy is proposed to improve the predicted performance of surrogate model. Two infill criteria are integrated in proposed strategy, one is the expected improvement matrix-based infill criterion, by which a new sample point is generated through maximizing the expected improvement function of the non-dominated solutions in objective space. The other one is the Euclidean distance-based infill criterion, by which a newly different sample point is generated through perturbing the non-dominated solutions in design space. At each iteration in the SMOO process, two promising infilling sample points are obtained by the proposed strategy with the filter strategy, and the corresponding responses are further evaluated with the real simulation models, then the sample set and surrogate models will be updated sequentially. According to the results of the numerical validations, the SMOO algorithm with the proposed strategy has the competitive capabilities in terms of high-quality Pareto front and computational stability. Finally, the hybrid criterion-based sample infilling strategy for SMOO is applied to solve a high-dimension multi-objective optimization problem related to lightweight design of an electric bus body structure.
... Golberg [32] explains that GA were originally based on natural genetics and, as pointed out by Deb [33], they currently have various applications in optimization processes and search activities for minimizing resources, given their broad applicability, ease of use and the integration of a global perspective. ...
This paper applies Fuzzy Cognitive Maps (FCMs) to understand the diverse behavior of municipal governments in Ecuador to find common elements that influence the well-being of citizens in the short and long term. Information gathering was conducted in two stages: in the first one, a group of 16 national experts was consulted to develop the initial FCM; in the second stage, local experts from 220 municipalities were interviewed to collect information on the general validity of initial FCMs and specific values given to concepts and relationships in their municipalities. Results show the importance of certain concepts for long-term municipal performance, such as the need for a competitive entrepreneurial sector, improving human resources in the municipality, and, particularly, having a competent mayor with leadership skills and a forward-looking vision that enables the development of municipal projects required to reach an efficient and equitable coverage of goods and services throughout the city. Through the application of genetic algorithms, the FCM was calibrated to ascertain the long-term dynamics of municipal development and the optimal values of the concepts that would optimize the attainment of the set objectives. The derived outcomes suggest the desirability of the maintenance of, in principle, unwanted structures like financial transfers from the central government and the need to exploit natural resources to attain urban development.
... First, the search should be guided toward the global Pareto-optimal region. Second, In the current non-dominated front, population diversity should be maintained [ 33]. ...
This chapter will discuss the recent and notable advances in using the Genetic Algorithm for autonomous vehicles and traffic control applications. The interest of research communities, industries, and urban organizations in traffic control and autonomous driving, the high complexity of the context, and the significance of these fields’ reliability and safety caused many optimization problems to arise from the context. So, numerous optimization algorithms have been employed to address these problems. As one of the most powerful and diverse optimization methods, the Genetic Algorithm has been used to improve and stabilize the performances of autonomous vehicles and traffic control tasks. This chapter will be divided into two sections. The first section will explore the genetic algorithm’s advances in urban traffic control problems. The advances in autonomous driving tasks using the genetic algorithm will be discussed in the second section.
... This represents an invalid relocation process. To address this issue, a proposed approach incorporates the initial stacking in the chromosome encoding while utilizing a heuristic algorithm [8] for determining the target stack. The heuristic algorithm follows a set of defined steps [9], which are outlined as follows: ...
This paper is based on the research background of block relocation in the automated warehouse and proposes a solution utilizing AGV to relocate goods within the warehouse. Building upon the automated warehouse layout and block relocation process, we formulate a mathematical model to minimize the the number of AGV moving goods and the total travel distance of the AGV. To solve the model, we develop an enhanced genetic algorithm, which zones the block relocation process into multiple stages. Each stage integrates conventional heuristic algorithms to enhance the efficiency of the solution process. The improved genetic algorithm can effectively solve the problem, significantly enhancing the automation level of the warehouse.
Over the past two decades, descent methods have received substantial attention within the multiobjective optimization field. Nonetheless, both theoretical analyses and empirical evidence reveal that existing first-order methods for multiobjective optimization converge slowly, even for well-conditioned problems, due to the objective imbalances. To address this limitation, we incorporate curvature information to scale each objective within the direction-finding subprob-lem, introducing a scaled proximal gradient method for multiobjective optimization (SPGMO). We demonstrate that the proposed method achieves improved linear convergence, exhibiting rapid convergence in well-conditioned scenarios. Furthermore, by applying small scaling to linear objectives, we prove that the SPGMO attains improved linear convergence for problems with multiple linear objectives. Additionally, integrating Nesterov's acceleration technique further enhances the linear convergence of SPGMO. To the best of our knowledge, this advancement in linear convergence is the first theoretical result that directly addresses objective imbalances in multiobjective first-order methods. Finally, we provide numerical experiments to validate the efficiency of the proposed methods and confirm the theoretical findings.
Water distribution networks (WDNs) are critical infrastructures playing crucial role for the development of cities and population health by delivering water to end-users. Ensuring an efficient water supply system represents an important task for the water utilities as compromise solution between system reliability and cost (investment/maintenance) for appropriate pipe diameter sizing and isolation valve placement. Generally, these two tasks are addressed separately, and valve positioning is defined on an already designed WDN. In this work, a novel coupled multi-objective optimization approach is proposed for simultaneously define optimal pipe diameter and optimal valve placement. To solve this problem, the evolutionary genetic algorithm was combined with the hydraulic simulation software EPANET in the Matlab environment. Two objective functions were adopted, the average demand shortfall related to segmental isolation (as a surrogate for WDN reliability) and the total investment cost (pipe and valve costs). The proposed approach is applied to the WDN of the city of Goro (Italy). Design solutions are compared with those obtained by applying traditional approach for the valve placement. Results show the effectiveness of the proposed method in defining more beneficial solutions in terms of total cost, reliability and hydraulic performance.
The article proposes an optimization algorithm using a hierarchical environment selection strategyto solve the deficiencies of current multimodal multi-objective optimization algorithms in obtaining the completeness and convergence of Pareto optimal Sets (PSs). Firstly, the algorithm in this article is framed by a differential evolutionary algorithm (DE) and uses a special crowding distance to design a neighborhood-based individual variation strategy, which also ensures the diversity, and then special crowding distance is used to help populations with non-dominated sorting. In the stage of environmental selection, a strategy of hierarchical selection of individuals was designed, which selects sorted non-dominant ranked individual layer by layer according to the ratio, which allows potential individuals tobe explored. Finally, in the stage of evolution of individuals, the convergence and diversity of populations were investigated, anddifferent mutation strategies were selectedaccording to the characteristics of individuals. DE reproduction strategies are used for iteration, preventing individuals from avoiding premature convergence and ensuring the algorithm’s searchability. These strategies help the algorithm to obtain more diverse and uniformly distributed PSs and Pareto Front (PF). The algorithm of this article compares with several other excellent algorithms on 13 test problems, and the test results show that all the algorithms of this article exhibit superior performance.
In multimodal multiobjective optimization problems (MMOPs), there are several Pareto optimal solutions corresponding to the identical objective vector. This paper proposes a new differential evolution algorithm to solve MMOPs with higher-dimensional decision variables. Due to the increase in the dimensions of decision variables in real-world MMOPs, it is difficult for current multimodal multiobjective optimization evolutionary algorithms (MMOEAs) to find multiple Pareto optimal solutions. The proposed algorithm adopts a dual-population framework and an improved environmental selection method. It utilizes a convergence archive to help the first population improve the quality of solutions. The improved environmental selection method enables the other population to search the remaining decision space and reserve more Pareto optimal solutions through the information of the first population. The combination of these two strategies helps to effectively balance and enhance convergence and diversity performance. In addition, to study the performance of the proposed algorithm, a novel set of multimodal multiobjective optimization test functions with extensible decision variables is designed. The proposed MMOEA is certified to be effective through comparison with six state-of-the-art MMOEAs on the test functions.
The current global energy crisis necessitates a shift to renewable energy sources to mitigate climate change impacts. Wave energy emerges as a promising renewable resource to fulfill increasing energy demands. This energy can be extracted using wave energy converters (WECs), with multi-axis WECs (MA-WECs) being more effective than single-axis versions due to their capacity to harness energy from waves in various directions. The challenge lies in determining the ideal geometric design for MA-WECs, that can be tackled through multi-objective optimization (MOO) techniques. This research focuses on evaluating different MOO algorithms for the optimal geometric design of MA-WECs. To assess the structural response of different geometries and sizes, the study utilized the NEMOH boundary element method solver, aiming to maximize power output, lower the levelized cost of energy (LCOE), and optimize the geometry configuration. Findings indicate that the choice of optimization algorithm considerably influences the MA-WEC's optimal design, enhancing power efficiency, reducing device volume, and cutting costs more effectively than the initial design.
The correspondence between the decision space and the objective space is often many-to-one in multi-objective optimization problems. Therefore, a class of problems with such mapping relationships is defined as a MMOPs. For these problems, how to ensure the final solution converges to each Pareto solution set and guarantees the diversity of the algorithm is an urgent problem. The method of the paper with opposition-based strategy, a multimodal multi-objective optimization algorithm, is proposed. The algorithm proposed is called MMODE_OP, which is framed by a differential evolutionary algorithm, and opposition-based learning is applied to the initialization phase and generation-hopping phase to filter out the more promising individuals in the population for iteration to enhance the global search capability and the diversity of population. In addition, different Gaussian perturbation strategies are adopted with iteration to achieve the search of the neighborhood, which can further not only improve the quality of the Pareto solution set but also enable the convergence of the Pareto solution set quickly. This method improves the algorithm’s local and global search ability, and enables multiple the Pareto solution set and improving the convergence. In the meantime, adaptive scaling factors and crossover factors are designed in this paper to enhance the improved search capability. Finally, the experiment results of MMODE_OP and other excellent algorithms on 13 test problems corroborate the proposed methods have superior performance.
Nonlinear conjugate gradient methods have recently garnered significant attention within the multiobjective optimization community. These methods aim to maintain consistency in conjugate parameters with their single-objective optimization counterparts. However, the preservation of the attractive conjugate property of search directions remains uncertain, even for quadratic cases, in multiobjective conjugate gradient methods. This loss of interpretabil-ity of the last search direction significantly limits the applicability of these methods. To shed light on the role of the last search direction, we introduce a novel approach called the subspace minimization Barzilai-Borwein method for multiobjective optimization problems (SMBBMO). In SMBBMO, each search direction is derived by optimizing a preconditioned Barzilai-Borwein subproblem within a two-dimensional subspace generated by the last search direction and the current Barzilai-Borwein descent direction. Furthermore, to ensure the global convergence of SMBBMO, we employ a modified Cholesky factorization on a transformed scale matrix, capturing the local curvature information of the problem within the two-dimensional subspace. Under mild assumptions, we establish both global and Q-linear convergence of the proposed method. Finally , comparative numerical experiments confirm the efficacy of SMBBMO, even when tackling large-scale and ill-conditioned problems.
In this article, we propose an adaptation of the Hyperbolic Augmented Lagrangian algorithm to solve multiobjective optimization problems. After conducting a literature review, we provide a definition of the Hyperbolic Augmented Lagrange method. Under clearly defined assumptions, we demonstrate that any limit point of a sequence generated by the proposed approach is feasible and constitutes a Pareto-optimal solution. To assess the effectiveness of the proposed algorithm, we incorporate two variants of the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method, thereby offering two approaches for solving well-known tests problems from the literature. We employ metrics to evaluate the performance of these two approaches.
In recent years, computational design methods have attracted much attention due to their capability and convenience in rapidly generating potential solutions to a design problem. Among various computational design techniques, topology optimisation, a performance-based design tool, has become increasingly popular, as it enables designers to create efficient and innovative structural forms. However, in the field of structural optimisation, a premium has often been placed on achieving the global optimal solution, and the obtained optimal structural layout may not satisfy all design requirements (e.g., aesthetic and functional considerations). Moreover, conventional optimisation methods usually operate like a "black box", in which designers are not allowed to exert their personal preferences on the evolving design. In practice, it is highly desirable to obtain diverse design solutions that possess high structural performance.
This research aims to develop a systematic methodology under the computational framework of the bi-directional evolutionary structural optimisation (BESO) method to create diverse and competitive structural designs. This thesis provides three interrelated research studies focused on the development of different approaches to generating alternative design options. The developed approach from each study is readily applicable to different design stages.
The first study proposes three stochastic approaches to generating diverse and competitive designs. The approaches include (1) penalising elemental sensitivities, (2) changing initial designs, and (3) integrating the genetic algorithm into the BESO method. Each approach can randomly change the search path of the standard BESO algorithm, thereby influencing the design outcome in a stochastic way. By using any of these approaches, the designer can obtain a new structural shape when the optimisation process is restarted every time. The numerical results from several 3D optimisation problems demonstrate that the proposed approaches are capable of producing a variety of efficient structural designs with distinctly different structural forms.
The second study proposes a thinning algorithm based approach to controlling the structural complexity (i.e., the number of cavities and tunnels) for both 2D and 3D optimisation problems. This approach is developed by integrating the topology-preserving feature of the simple points into the BESO method. In this work, the structural complexity can be directly controlled by preserving the topological properties of the prescribed initial design. Designers can alter the topology of the initial design to achieve multiple design solutions with different topologies. The developed methodology has been successfully applied in the computational morphogenesis of various structures. The results demonstrate that the control of design diversity and structural performance can be well balanced.
In the third study, a hole-filling method is integrated into the BESO method to control the number and size of existing cavities and tunnels during the form-finding process. The hole-filling method is implemented to fill in extra cavities with solid material and cover extra tunnels by building sheet-like patches. Compared with the second study, the structural complexity control can start from an initial full design. Thus, it requires less human intervention to impose the topological constraints. Moreover, the numerical results demonstrated that the generation of sheet-like components could considerably increase the stiffness of the optimised structure. Thus, the developed approach has great potential to help engineers and architects search for efficient structural designs with controllable topologies.
The novel approaches proposed in this PhD project can be applied in all stages of a design process. The stochastic approaches are capable of generating a large number of unexpected designs with high performance. Thus, the obtained solutions can be a rich source of inspiration for the early-stage exploration of a design process. In the detailed design stage, the structural complexity control approaches allow designers to directly control the topology of the evolving designs. In this respect, the designers can integrate aesthetic preferences and other requirements into the optimisation process. The research outcome holds great potential for practical applications in architecture and engineering, where diverse and competitive solutions are in high demand.
The investigation of seepage situation of a reservoir under complex geological conditions is greatly significant and is necessary to be determined by geologic survey or numerical analysis. Based on previous geological survey data, a 3-dimensional seepage back analysis was conducted. In this study, two hydraulic conditions of unsteady and steady seepage were considered, and the corresponding back analysis came down to a multi-objective decision-making problem. The GRNN model was trained by PSO algorithm for obtaining the relationship between permeability coefficients and monitoring data, and by combining the NSGA-II algorithm, the best unbiased solution of permeability coefficients for the multi-objective function established by monitoring data of seepage discharge and pressure head was searched via iteration calculation. On this basis, the seepage safety of the reservoir was evaluated. Through taking the seepage discharge of the whole reservoir into account, the problem in the traditional seepage field back analysis that the foundation strata seepage parameters are insensitive to the pressure head inside the dam was solved. And the inversion results can provide a supplement for the geological survey under complex geological conditions. The method adopted in this paper may provide significant references for the anti-seepage design and reinforcement of hydraulic structures with complex geological conditions.
Preconditioning is a powerful approach for solving ill-conditioned problems in optimization , where a preconditioning matrix is used to reduce the condition number and speed up the convergence of first-order method. Unfortunately, it is impossible to capture the curvature of all objective functions with a single preconditioning matrix in multiobjective optimization. Instead, second-order methods for multiobjective optimization problems (MOPs) use different matrices for objectives in direction-finding subproblems, leading to a prohibitive per-iteration cost. To balance per-iteration cost and better curvature exploration, we propose a precondi-tioned Barzilai-Borwein descent method for MOPs (PBBMO). In the direction-finding subprob-lems, we employ a scale matrix to explore the curvature of an implicit scalarization function. The Barzilai-Borwein method is then applied to the matrix metric to tune the gradients of the objective functions, which can also be considered as an extra diagonal preconditioner based on the scale matrix for each objective, and mitigates the effect of imbalances among objectives. From a preconditioning perspective, we use BFGS update formula to approximate a trade-off of Hessian matrices. Under mild assumption, we give a simple convergence analysis for the Barzilai-Borwein quasi-Newton method. Finally, comparative numerical results confirm the efficiency of the proposed method, even when applied to large-scale and ill-conditioned problems.
This paper uses an augmented Lagrangian method based on an inexact exponential penalty function to solve constrained multiobjective optimization problems. Two algorithms have been proposed in this study. The first algorithm uses a projected gradient, while the second uses the steepest descent method. By these algorithms, we have been able to generate a set of nondominated points that approximate the Pareto optimal solutions of the initial problem. Some proofs of theoretical convergence are also proposed for two different criteria for the set of generated stationary Pareto points. In addition, we compared our method with the NSGA-II and augmented the Lagrangian cone method on some test problems from the literature. A numerical analysis of the obtained solutions indicates that our method is competitive with regard to the test problems used for the comparison.
This dissertation describes experiments conducted to explore the efficacy of using vector-valued feedback with a class of adaptive procedures called genetic algorithms. The software system developed was called VEGA for Vector Evaluated Genetic Algorithm and was first used on multiple objective optimization problems. The main conclusion of these experiments was that VEGA provided a powerful and robust search technique for complex multiobjective optimization problems of high order when little or no a priori knowledge was available to guide the search. These results were similar to those found by previous researchers using scalar genetic algorithms for scalar optimization problems. The VEGA technique was then applied to multiclass pattern discrimination tasks. The resulting software system was called LS-2 for Learning System-Two, since it followed closely the lead of a scalar-valued learning system called LS-1 developed by Stephen Smith. These experiments revealed that LS-2 was able to evolve high performance production system programs to perform the pattern discrimination tasks it was given. The VEGA approach demonstrates the efficacy of extending the previously demonstrated power of genetic algorithms to vector-valued problems and, thereby, provides a new approach to machine learning.
. This paper presents a novel evolutionary approach of approximating the shape of the Pareto-optimal set of multi-objective optimization problems. The evolutionary algorithm (EA) uses the predator-prey model from ecology. The prey are the usual individuals of an EA that represent possible solutions to the optimization task. They are placed at vertices of a graph, remain stationary, reproduce, and are chased by predators that traverse the graph. The predators chase the prey only within its current neighborhood and according to one of the optimization criteria. Because there are several predators with different selection criteria, those prey individuals, which perform best with respect to all objectives, are able to produce more descendants than inferior ones. As soon as a vertex for the prey becomes free, it is refilled by descendants from alive parents in the usual way of EA, i.e., by inheriting slightly altered attributes. After a while, the prey concentrate at Pareto-opti...
A different crossover operator, uniform crossover, is presented. It is compared theoretically and empirically with one-point and two-point crossover and is shown to be superior in most cases.
Although there are many versions of evolutionary algorithms that
are tailored to multi-criteria optimization, theoretical results are
apparently not yet available. In this paper, it is shown that results
known from the theory of evolutionary algorithms in case of
single-criterion optimization do not carry over to the multi-criterion
case. At first, three different step size rules are investigated
numerically for a selected problem with two conflicting objectives. The
empirical results obtained by these experiments lead to the observation
that only one of these step size rules may have the property to ensure
convergence to the Pareto set. A theoretical analysis finally shows that
a special version of an evolutionary algorithm with this step size rule
converges with probability one to the Pareto set for the test problem
under consideration
Many, if not most, optimization problems have multiple objectives.
Historically, multiple objectives have been combined ad hoc to form a
scalar objective function, usually through a linear combination
(weighted sum) of the multiple attributes, or by turning objectives into
constraints. The genetic algorithm (GA), however, is readily modified to
deal with multiple objectives by incorporating the concept of Pareto
domination in its selection operator, and applying a niching pressure to
spread its population out along the Pareto optimal tradeoff surface. We
introduce the Niched Pareto GA as an algorithm for finding the Pareto
optimal set. We demonstrate its ability to find and maintain a diverse
“Pareto optimal population” on two artificial problems and
an open problem in hydrosystems
Learning Gene Linkage to Efficiently Solve Problems of Bounded Difficulty Using Genetic Algorithms by Georges Raif Harik Co-Chairs: Keki B. Irani and David E. Goldberg The complicated nature of modern scientific endeavors often times requires the employment of black-box optimization. For the past twenty years, the simple genetic algorithm (sGA) has proven to be a fertile inspiration for such techniques. Yet, many attempts to improve or adapt the sGA remain disconnected with its prevailing theory. This theory suggests that the sGA works by propagating building blocks---highly fit similarities in the structure of its solutions---and that it can fail by not recombining these building blocks in one optimal solution. The most successful of previous attempts to facilitate building block recombination have strayed far from the operation of the sGA, resulting in techniques that are difficult to use and implement. This dissertation presents an approach to solving the recombination problem witho...
Parallel genetic algorithms are often very different from the "traditional" genetic algorithm proposed by Holland, especially with regards to population structure and selection mechanisms. In this paper we compare several parallel genetic algorithms across a wide range of optimization functions in an attempt to determine whether these changes have positive or negative impact on their problemsolving capabilities. The findings indicate that the parallel structures perform as well as or better than standard versions, even without taking parallel hardware into account.
This paper presents a static analysis of deception in multimodal functions. Deception in a bipolar function of unitation (a function with two global optima and a number of deceptive attractors) is defined, and a set of sufficient conditions relating function values is obtained. A bipolar deceptive function is also constructed from low-order Walsh coefficients. Multimodal functions of bounded deception are formed by concatenating several bipolar deceptive functions. These functions offer a great challenge to global optimization algorithms (including genetic algorithms) because they are deceptive and have a large number of attractors, of which only a few are global optima. These functions also open doors for generalizing the notion of deception, and allow us to better understand the importance of deception in the study of genetic algorithms.
This paper considers the effect of stochasticity on the quality of convergence of genetic algorithms (GAs). In many problems, the variance of building-block fitness or so-called collateral noise is the major source of variance, and a population-sizing equation is derived to ensure that average signal-to-collateral-noise ratios are favorable to the discrimination of the best building blocks required to solve a problem of bounded deception. The sizing relation is modified to permit the inclusion of other sources of stochasticity, such as the noise of selection, the noise of genetic operators, and the explicit noise or nondeterminism of the objective function. In a test suite of five functions, the sizing relation proves to be a conservative predictor of average correct convergence, as long as all major sources of noise are considered in the sizing calculation. These results suggest how the sizing equation may be viewed as a coarse delineation of a boundary between what a physicist might call two distinct phases of GA behavior. At low population sizes the GA makes many errors of decision, and the quality of convergence is largely left to the vagaries of chance or the serial fixup of flawed results through mutation or other serial injection of diversity. At large population sizes, GAs can reliably discriminate between good and bad building blocks, and parallel processing and recombination of building blocks lead to quick solution of even difficult deceptive problems. Additionally, the paper outlines a number of extensions to this work, including the development of more refined models of the relation between generational average error and ultimate convergence quality, the development of online methods for sizing populations via the estimation of population-s...
In trying to solve multiobjective optimization problems, many traditional methods scalarize the objective vector into a single objective. In those cases, the obtained solution is highly sensitive to the weight vector used in the scalarization process and demands that the user have knowledge about the underlying problem. Moreover, in solving multiobjective problems, designers may be interested in a set of Pareto-optimal points, instead of a single point. Since genetic algorithms (GAs) work with a population of points, it seems natural to use GAs in multiobjective optimization problems to capture a number of solutions simultaneously. Although a vector evaluated GA (VEGA) has been implemented by Schaffer and has been tried to solve a number of multiobjective problems, the algorithm seems to have bias toward some regions. In this paper, we investigate Goldberg's notion of nondominated sorting in GAs along with a niche and speciation method to find multiple Pareto-optimal points simultaneously. The proof-of-principle results obtained on three problems used by Schaffer and others suggest that the proposed method can be extended to higher dimensional and more difficult multiobjective problems. A number of suggestions for extension and application of the algorithm are also discussed.
The nondominated sorting genetic algorithm (NSGA) is adapted and used to obtain multiobjective Pareto optimal solutions for three grades of nylon 6 being produced in an industrial semibatch reactor. The total reaction time and the concentra- tion of an undesirable cyclic dimer in the product are taken as two individual objectives for minimization, while simultaneously requiring the attainment of design values of the final monomer conversion and for the number-average chain length. Substantial improvements in the operation of the nylon 6 reactor are indicated by this study. The technique used is very general in nature and can be used for multiobjective optimization of other reactors. Good mathematical models accounting for all the physicochemical aspects operative in a reactor (and which have been preferably tested on industrial data) are a prerequisite for such optimization studies. q 1998 John Wiley & Sons, Inc. J Appl Polym Sci 69: 69-87, 1998
This paper describes the combination of several optimization technologies that can be used to tackle challenging design problems. The approach, that uses a multi-objective genetic algorithm, a neural network, and a gradient-based optimizer, is first outlined with the help of a computationally inexpensive mathematical test function. Then the methodology is applied to the design of a sailing yacht fin keel, coupling the optimization codes to 3D Navier–Stokes simulations. To perform the multi-objective optimization task a parallel computer is employed.
Many real-world search and optimization problems involve inequality and/or equality constraints and are thus posed as constrained optimization problems. In trying to solve constrained optimization problems using genetic algorithms (GAs) or classical optimization methods, penalty function methods have been the most popular approach, because of their simplicity and ease of implementation. However, since the penalty function approach is generic and applicable to any type of constraint (linear or nonlinear), their performance is not always satisfactory. Thus, researchers have developed sophisticated penalty functions specific to the problem at hand and the search algorithm used for optimization. However, the most difficult aspect of the penalty function approach is to find appropriate penalty parameters needed to guide the search towards the constrained optimum. In this paper, GA's population-based approach and ability to make pair-wise comparison in tournament selection operator are exploited to devise a penalty function approach that does not require any penalty parameter. Careful comparisons among feasible and infeasible solutions are made so as to provide a search direction towards the feasible region. Once sufficient feasible solutions are found, a niching method (along with a controlled mutation operator) is used to maintain diversity among feasible solutions. This allows a real-parameter GA's crossover operator to continuously find better feasible solutions, gradually leading the search near the true optimum solution. GAs with this constraint handling approach have been tested on nine problems commonly used in the literature, including an engineering design problem. In all cases, the proposed approach has been able to repeatedly find solutions closer to the true optimum solution than that reported earlier.
This paper describes and analyzes CHC, a nontraditional genetic algorithm which combines a conservative selection strategy that always preserves the best individuals found so far with a radical (highly disruptive) recombination operator that produces offspring that are maximally different from both parents. The traditional reasons for preferring a recombination operator with a low probability of disrupting schemata may not hold when such a conservative selection strategy is used. On the contrary, certain highly disruptive crossover operators provide more effective search. Empirical evidence is provided to support these claims.
The paper describes a rank-based fitness assignment method for Multiple Objective Genetic Algorithms (MOGAs). Conventional niche formation methods are extended to this class of multimodal problems and theory for setting the niche size is presented. The fitness assignment method is then modified to allow direct intervention of an external decision maker (DM). Finally, the MOGA is generalised further: the genetic algorithm is seen as the optimizing element of a multiobjective optimization loop,...
Several methods have been proposed for handling nonlinear constraints by evolutionary algorithms for numerical optimization
problems; a survey paper [7] provides an overview of various techniques and some experimental results, as well as proposes
a set of eleven test problems. Recently a new, decoder-based approach for solving constrained numerical optimization problems
was proposed [2, 3]. The proposed method defines a homomorphous mapping between n-dimensional cube and a feasible search space. In [3] we have demonstrated the power of this new approach on several test
cases. However, it is possible to enhance the performance of the system even further by introducing additional concepts of
(1) nonlinear mappings with an adaptive parameter, and (2) adaptive location of the reference point of the mapping.
This paper finds sufficient conditions for fully or partially deceptive binary functions by calculating schema average fitness values. Deception conditions are first derived for functions of unitation (functions that depend only on the number of 1s in the string) and then extended for any binary function. The analysis is also extended to find a set of sufficient conditions for fully easy binary functions. It is found that the computational effort required to investigate full or partial deception in a problem of sizel using these sufficient conditions isO(2l
) and using all necessary conditions of deception isO(4l
). This calculation suggests that these sufficient conditions can be used to quickly test deception in a function. Furthermore, it is found that these conditions may also be systematically used to design a fully deceptive function by performing onlyO(l
2) comparisons and to design a partially deceptive function to orderk by performing onlyO(kl) comparisons. The analysis shows that in the class of functions of unitation satisfying these conditions of deception, an order-k partially deceptive function is also partially deceptive to any lower order. Finally, these sufficient conditions are used to investigate deception in a number of currently-used deceptive problems.
This paper defines and explores a somewhat different type of genetic algorithm (GA) - a messy genetic algorithm (mGA). Messy GAs process variable-length strings that may be either under- or over-specified with respect to the problem being solved. As nature has formed its genotypes by progressing from simple to more complex life forms, messy GAs solve problems by combining relatively short, well-tested building blocks to form longer, more complex strings that increasingly cover all features of a problem. This approach stands in contrast to the usual fixed-length, fixed-coding genetic algorithm, where the existence of the requisite tight linkage is taken for granted or ignored altogether. To compare the two approaches, a 30-bit, order-three-deceptive problem is searched using a simple GA and a messy GA. Using a random but fixed ordering of the bits, the simple GA makes errors at roughly three-quarters of its positions; under a worst-case ordering, the simple GA errs at all positions. In contrast to the simple GA results, the messy GA repeatedly solves the same problem to optimality. Prior to this time, no GA had ever solved a provably difficult problem to optimality without prior knowledge of good string arrangements. The mGA presented herein repeadedly achieves globally optimal results without such knowledge, and it does so at the very first generation in which strings are long enough to cover the problem. The solution of a difficult nonlinear problem to optimality suggests that messy GAs can solve more difficult problems than has been possible to date with other genetic algorithms. The ramifications of these techniques in search and machine learning are explored, including the possibility of messy floating-point codes, messy permutations, and messy classifiers.
The classic introduction to engineering optimization theory and practice--now expanded and updated. Engineering optimization helps engineers zero in on the most effective, efficient solutions to problems. This text provides a practical, real-world understanding of engineering optimization. Rather than belaboring underlying proofs and mathematical derivations, it emphasizes optimization methodology, focusing on techniques and stratagems relevant to engineering applications in design, operations, and analysis. It surveys diverse optimization methods, ranging from those applicable to the minimization of a single-variable function to those most suitable for large-scale, nonlinear constrained problems. New material covered includes the duality theory, interior point methods for solving LP problems, the generalized Lagrange multiplier method and generalization of convex functions, and goal programming for solving multi-objective optimization problems. A practical, hands-on reference and text, Engineering Optimization, Second Edition covers: Practical issues, such as model formulation, implementation, starting point generation, and more. Current, state-of-the-art optimization software. Three engineering case studies plus numerous examples from chemical, industrial, and mechanical engineering. Both classical methods and new techniques, such as successive quadratic programming, interior point methods, and goal programming. Excellent for self-study and as a reference for engineering professionals, this Second Edition is also ideal for senior and graduate courses on engineering optimization, including television and online instruction, as well as for in-plant training.
David Goldberg's Genetic Algorithms in Search, Optimization and Machine Learning is by far the bestselling introduction to genetic algorithms. Goldberg is one of the preeminent researchers in the field--he has published over 100 research articles on genetic algorithms and is a student of John Holland, the father of genetic algorithms--and his deep understanding of the material shines through. The book contains a complete listing of a simple genetic algorithm in Pascal, which C programmers can easily understand. The book covers all of the important topics in the field, including crossover, mutation, classifier systems, and fitness scaling, giving a novice with a computer science background enough information to implement a genetic algorithm and describe genetic algorithms to a friend.
Introduces the Gene Expression Messy Genetic Algorithm (GEMGA)-a new generation of messy genetic algorithms that directly search for relations among the members of the search space. GEMGA is an O[Λ k(l2+k)] sample complexity algorithm for the class of order-k delineable problems (problems that can be solved by considering no higher than order-k relations). GEMGA is designed based on an alternate perspective of natural evolution, as proposed by the SEARCH (Search Envisioned As Relation and Class Hierarchizing) framework, that emphasizes the role of gene expression. GEMGA uses the transcription operator to search for relations. This paper also presents test results of the GEMGA for large multimodal order-k delineable problems
For part I see ibid., 26-37. The evolutionary approach to multiple
function optimization formulated in the first part of the paper is
applied to the optimization of the low-pressure spool speed governor of
a Pegasus gas turbine engine. This study illustrates how a technique
such as the multiobjective genetic algorithm can be applied, and
exemplifies how design requirements can be refined as the algorithm
runs. Several objective functions and associated goals express design
concerns in direct form, i.e., as the designer would state them. While
such a designer-oriented formulation is very attractive, its practical
usefulness depends heavily on the ability to search and optimize cost
surfaces in a class much broader than usual, as already provided to a
large extent by the genetic algorithm (GA). The two instances of the
problem studied demonstrate the need for preference articulation in
cases where many and highly competing objectives lead to a nondominated
set too large for a finite population to sample effectively. It is shown
that only a very small portion of the nondominated set is of practical
relevance, which further substantiates the need to supply preference
information to the GA. The paper concludes with a discussion of the
results
The concept of Pareto optimality is applied to the study of choice
tradeoffs between reflectivity and thickness in the design of multilayer
microwave absorbers. Absorbers composed of a given number of layers of
absorbing materials selected from a predefined database of available
materials are considered. Three types of Pareto genetic algorithms for
absorber synthesis are introduced and compared to each other, as well as
to methods operating with the weighted Tchebycheff method for Pareto
optimization. The Pareto genetic algorithms are applied to construct
Pareto fronts for microwave absorbers with five layers of materials
selected from a representative database of available materials in the
0.2-2 GHz, 2-8 GHz, and 9-11 GHz bands
Although computational techniques for solving Multiobjective Optimization Problems (MOPs) have been available for many years, the recent application of Evolutionary Algorithms (EAs) to such problems provides a vehicle with which to solve very large scale MOPs. Thus, the intent of this paper is to organize, present, and analyze contemporary Multiobjective Evolutionary Algorithm (MOEA) research and associated MOPs. Under the umbrella of a priori, progressive, and a posteriori algorithms, all known MOEA techniques are discussed. Each MOEA proposed in the literature is classified and cataloged based upon this umbrella and more detailed algorithmic characteristics; among others these include objective aggregation, interactive methods, sampling, search, ranking, and niching. The classification, incorporating a consistent MOEA notation, is presented in tabular form for ease of MOEA identification and selection. A detailed quantitative and qualitative analysis is presented. The tabular data g...
. Since 1985 various evolutionary approaches to multiobjective optimization have been developed, capable of searching for multiple solutions concurrently in a single run. But the few comparative studies of different methods available to date are mostly qualitative and restricted to two approaches. In this paper an extensive, quantitative comparison is presented, applying four multiobjective evolutionary algorithms to an extended 0/1 knapsack problem. 1 Introduction Many real-world problems involve simultaneous optimization of several incommensurable and often competing objectives. Usually, there is no single optimal solution, but rather a set of alternative solutions. These solutions are optimal in the wider sense that no other solutions in the search space are superior to them when all objectives are considered. They are known as Pareto-optimal solutions. Mathematically, the concept of Pareto-optimality can be defined as follows: Let us consider, without loss of generality, a ...
The application of evolutionary algorithms (EAs) in multiobjective optimization is currently receiving growing interest from researchers with various backgrounds. Most research in this area has understandably concentrated on the selection stage of EAs, due to the need to integrate vectorial performance measures with the inherently scalar way in which EAs reward individual performance, that is, number of offspring.
In this review, current multiobjective evolutionary approaches are discussed, ranging from the conventional analytical aggregation of the different objectives into a single function to a number of population-based approaches and the more recent ranking schemes based on the definition of Pareto optimality. The sensitivity of different methods to objective scaling and/or possible concavities in the trade-off surface is considered, and related to the (static) fitness landscapes such methods induce on the search space. From the discussion, directions for future research in multiobjective fitness assignment and search strategies are identified, including the incorporation of decision making in the selection procedure, fitness sharing, and adaptive representations.
A central problem in the theory of genetic algorithms is the characterization of problems that are difficult for GAs to optimize. Many attempts to characterize such problems focus on the notion of Deception, defined in terms of the static average fitness of competing schemas. This article examines the Static Building Block Hypothesis (SBBH), the underlying assumption used to define Deception. Exploiting contradictions between the SBBH and the Schema Theorem, we show that Deception is neither necessary nor sufficient for problems to be difficult for GAs. This article argues that the characterization of hard problems must take into account the basic features of genetic algorithms, especially their dynamic, biased sampling strategy. Keywords: Deception, building block hypothesis 1 INTRODUCTION Since Holland's early work on the analysis of genetic algorithms (GAs), the usual approach has been to focus on the allocation of search effort to subspaces described by schemas representing hyper...
Multiobjective optimization using nondominated sorting annealing genetic algorithms
- K.-S Leung
- Z.-Y Zhu
- Z.-B Xu
- Y Leung
Leung, K.-S, Zhu, Z.-Y, Xu, Z.-B., and Leung, Y. (1998). Multiobjective optimization using nondominated sorting annealing genetic algorithms. (Unpublished document).
Optimization of single screw extrusion: Theoretical and experimental results
- J A Covas
- A G Cunha
- P Oliveira
Covas, J. A., Cunha, A. G., and Oliveira, P. (in press). Optimization of single screw extrusion: Theoretical
and experimental results. International Journal of Forming Processes.