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Multi-objective Genetic Algorithms: Problem Difficulties and Construction of Test Problems

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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.

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... 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. ...
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... 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. ...
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... 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. ...
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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. ...
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... 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). ...
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... 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. ...
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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. ...
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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 . ...
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... 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 ...
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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). ...
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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. ...
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... 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]. ...
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... 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. ...
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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. ...
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... 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]. ...
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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. ...
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... 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. ...
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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]. ...
Chapter
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: ...
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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.
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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.
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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.
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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.
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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.
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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,...
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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.
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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.
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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.
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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.
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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.
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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
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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
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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
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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...
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. 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 ...
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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.
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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.