Search

When solving constrained optimization problems (COPs) by evolutionary algorithms, the search algorithm plays a crucial role. In general, we expect that the search algorithm has the capability to balance not only diversity and convergence but also constraints and objective function during the evolution. For this purpose, this paper proposes a composite differential evolution (DE) for constrained optimization, which includes three different trial vector generation strategies with distinct advantages. In order to strike a balance between diversity and convergence, one of these three trial vector generation strategies is able to increase diversity, and the other two exhibit the property of convergence. In addition, to accomplish the tradeoff between constraints and objective function, one of the two trial vector generation strategies for convergence is guided by the individual with the least degree of constraint violation in the population, and the other is guided by the individual with the best objective function value in the population. After producing offspring by the proposed composite DE, the feasibility rule and the $\boldsymbol {\varepsilon }$ constrained method are combined elaborately for selection in this paper. Moreover, a restart scheme is proposed to help the population jump out of a local optimum in the infeasible region for some extremely complicated COPs. By assembling the above techniques together, a constrained composite DE is proposed. The experiments on two sets of benchmark test functions with various features, i.e., 24 test functions from IEEE CEC2006 and 18 test functions with 10 dimensions and 30 dimensions from IEEE CEC2010, have demonstrated that the proposed method shows better or at least competitive performance against other state-of-the-art methods. [ABSTRACT FROM AUTHOR]

With the advent of electric vehicles (EVs), issues connected to the electric vehicle charging scheduling (EVCS) problem, which is $\textsf {NP}$ -hard, have become important. In previous studies, EVCS has been mainly formulated as a constrained shortest-path problem; however, such formulations have not involved variables such as charging rates, traffic congestion, scalability, and waiting time at charging station (CS), that need to be considered in practical settings. Earlier research has also tended to focus on the strengths of particular evolutionary optimization algorithms like differential evolution (DE) or particle swarm optimization (PSO) over others or traditional mathematical programming methods, with only a limited study of hybrid approaches. In this paper, fast and slow charging options at a station are considered in the EVCS problem for practical use. In previous studies, EVs have been considered to have fixed speeds; however, in order to mitigate CS congestion and thus waiting times at CSs, dynamic speed control of EVs has been considered in this work. This work also investigates the scalability of different EVCS solutions. A hybrid approach using PSO and the Firefly algorithm (FFA) with a Lévy flights search strategy is designed and implemented to solve the EVCS. Also, different hybrid methods variants of PSO and FFA have been evaluated in this paper to find the best performing hybrid variant. Experimental results validate the effectiveness of our approach on both synthetic and the real-world transportation networks. [ABSTRACT FROM AUTHOR]

Operational indices optimization is crucial for the global optimization in beneficiation processes. This paper presents a multitasking multiobjective evolutionary method to solve operational indices optimization, which involves a formulated multiobjective multifactorial operational indices optimization (MO-MFO) problem and the proposed multiobjective MFO algorithm for solving the established MO-MFO problem. The MO-MFO problem includes multiple level of accurate models of operational indices optimization, which are generated on the basis of a data set collected from production. Among the formulated models, the most accurate one is considered to be the original functions of the solved problem, while the remained models are the helper tasks to accelerate the optimization of the most accurate model. For the MFO algorithm, the assistant models are alternatively in multitasking environment with the accurate model to transfer their knowledge to the accurate model during optimization in order to enhance the convergence of the accurate model. Meanwhile, the recently proposed two-stage assortative mating strategy for a multiobjective MFO algorithm is applied to transfer knowledge among multitasking tasks. The proposed multitasking framework for operational indices optimization has conducted on 10 different production conditions of beneficiation. Simulation results demonstrate its effectiveness in addressing the operational indices optimization of beneficiation problem. Note to Practitioners—Operational indices optimization is a typical approach to achieve global production optimization by efficiently coordinating all the indices to improve the production indices. In this paper, a multiobjective multitasking framework is developed to address the operational indices optimization, which includes a multitasking multiobjective operational indices optimization problem formulation and a multitasking multiobjective evolutionary optimization to solve the above-formulated optimization problem. The proposed approach can achieve a solution set for the decision-making. The simulation results on a real beneficiation process in China with 10 operational conditions show that the proposed approach is able to obtain a superior solution set, which is associated with a higher grade and yield of the product. [ABSTRACT FROM AUTHOR]

Conventional evolutionary algorithms (EAs) are not well suited for solving expensive optimization problems due to the fact that they often require a large number of fitness evaluations to obtain acceptable solutions. To alleviate the difficulty, this paper presents a multitasking evolutionary optimization framework for solving computationally expensive problems. In the framework, knowledge is transferred from a number of computationally cheap optimization problems to help the solution of the expensive problem on the basis of the recently proposed multifactorial EA (MFEA), leading to a faster convergence of the expensive problem. However, existing MFEAs do not work well in solving multitasking problems whose optimums do not lie in the same location or when the dimensions of the decision space are not the same. To address the above issues, the existing MFEA is generalized by proposing two strategies, one for decision variable translation and the other for decision variable shuffling, to facilitate knowledge transfer between optimization problems having different locations of the optimums and different numbers of decision variables. To assess the effectiveness of the generalized MFEA (G-MFEA), empirical studies have been conducted on eight multitasking instances and eight test problems for expensive optimization. The experimental results demonstrate that the proposed G-MFEA works more efficiently for multitasking optimization and successfully accelerates the convergence of expensive optimization problems compared to single-task optimization. [ABSTRACT FROM AUTHOR]

In offline data-driven optimization, only historical data is available for optimization, making it impossible to validate the obtained solutions during the optimization. To address these difficulties, this paper proposes an evolutionary algorithm assisted by two surrogates, one coarse model and one fine model. The coarse surrogate (CS) aims to guide the algorithm to quickly find a promising subregion in the search space, whereas the fine one focuses on leveraging good solutions according to the knowledge transferred from the CS. Since the obtained Pareto optimal solutions have not been validated using the real fitness function, a technique for generating the final optimal solutions is suggested. All achieved solutions during the whole optimization process are grouped into a number of clusters according to a set of reference vectors. Then, the solutions in each cluster are averaged and outputted as the final solution of that cluster. The proposed algorithm is compared with its three variants and two state-of-the-art offline data-driven multiobjective algorithms on eight benchmark problems to demonstrate its effectiveness. Finally, the proposed algorithm is successfully applied to an operational indices optimization problem in beneficiation processes. [ABSTRACT FROM AUTHOR]

Nonlinear equations system (NES) arises commonly in science and engineering. Repulsion techniques are considered to be the effective methods to locate different roots of NES. In general, the repulsive radius needs to be given by the user before the run. However, its optimal parameter setting is difficult and problem-dependent. To alleviate this drawback, in this paper, we first propose a dynamic repulsion technique, and then a general framework based on the dynamic repulsion technique and evolutionary algorithms (EAs) is presented to effectively solve NES. The major advantages of our framework are: 1) the repulsive radius is controlled dynamically during the evolutionary process; 2) multiple roots of NES can be simultaneously located in a single run; 3) the diversity of the population is preserved due to the population reinitialization; and 4) different repulsion techniques and different EAs can be readily integrated into this framework. To extensively evaluate the performance of our framework, we choose 42 problems with diverse features as the test suite. In addition, some representative differential evolution and particle swarm optimization variants are incorporated into the framework. Our method is also compared with other state-of-the-art methods. Experimental results indicate that the dynamic repulsion technique can improve the performance of the original repulsion technique with static repulsive radius. Moreover, the proposed method is able to yield better results compared with other methods. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a novel method for the discovery of value functions for Markov decision processes (MDPs). This method, which we call value function discovery (VFD), is based on ideas from the evolutionary algorithm field. VFDs key feature is that it discovers descriptions of value functions that are algebraic in nature. This feature is unique, because the descriptions include the model parameters of the MDP. The algebraic expression of the value function discovered by VFD can be used in several scenarios, e.g., conversion to a policy (with one-step policy improvement) or control of systems with time-varying parameters. The work in this paper is a first step toward exploring potential usage scenarios of discovered value functions. We give a detailed description of VFD and illustrate its application on an example MDP. For this MDP, we let VFD discover an algebraic description of a value function that closely resembles the optimal value function. The discovered value function is then used to obtain a policy, which we compare numerically to the optimal policy of the MDP. The resulting policy shows near-optimal performance on a wide range of model parameters. Finally, we identify and discuss future application scenarios of discovered value functions. [ABSTRACT FROM PUBLISHER]

This paper introduces sparse representation into spectral clustering and provides a sparse spectral clustering framework via a multiobjective evolutionary algorithm. In contrast to conventional spectral clustering, the main contribution of this paper is to construct the similarity matrix using a sparse representation approach by modeling spectral clustering as a constrained multiobjective optimization problem. Specific operators are designed to obtain a set of high quality solutions in the optimization process. Furthermore, we design a method to select a tradeoff solution from the Pareto front using a measurement called ratio cut based on an adjacency matrix constructed by all the nondominated solutions. We also extend the framework to the semi-supervised clustering field by using the semi-supervised information brought by the labeled samples to set some constraints or to guide the searching process. Experiments on commonly used datasets show that our approach outperforms four well-known similarity matrix construction methods in spectral clustering, and one multiobjective clustering algorithm. A practical application in image segmentation also demonstrates the efficiency of the proposed algorithm. [ABSTRACT FROM PUBLISHER]

This paper is concerned with the application of an automated hybrid approach in addressing the university timetabling problem. The approach described is based on the nature-inspired artificial bee colony (ABC) algorithm. An ABC algorithm is a biologically-inspired optimization approach, which has been widely implemented in solving a range of optimization problems in recent years such as job shop scheduling and machine timetabling problems. Although the approach has proven to be robust across a range of problems, it is acknowledged within the literature that there currently exist a number of inefficiencies regarding the exploration and exploitation abilities. These inefficiencies can often lead to a slow convergence speed within the search process. Hence, this paper introduces a variant of the algorithm which utilizes a global best model inspired from particle swarm optimization to enhance the global exploration ability while hybridizing with the great deluge (GD) algorithm in order to improve the local exploitation ability. Using this approach, an effective balance between exploration and exploitation is attained. In addition, a traditional local search approach is incorporated within the GD algorithm with the aim of further enhancing the performance of the overall hybrid method. To evaluate the performance of the proposed approach, two diverse university timetabling datasets are investigated, i.e., Carter’s examination timetabling and Socha course timetabling datasets. It should be noted that both problems have differing complexity and different solution landscapes. Experimental results demonstrate that the proposed method is capable of producing high quality solutions across both these benchmark problems, showing a good degree of generality in the approach. Moreover, the proposed method produces best results on some instances as compared with other approaches presented in the literature. [ABSTRACT FROM AUTHOR]

Niching techniques have recently been incorporated into evolutionary algorithms (EAs) for multisolution optimization in multimodal landscape. However, existing niching techniques inevitably increase the time complexity of basic EAs due to the computation of the distance matrix of individuals. In this paper, we propose a fast niching technique. The technique avoids pairwise distance calculations by introducing the locality sensitive hashing, an efficient algorithm for approximately retrieving nearest neighbors. Individuals are projected to a number of buckets by hash functions. The similar individuals possess a higher probability of being hashed into the same bucket than the dissimilar ones. Then, interactions between individuals are limited to the candidates that fall in the same bucket to achieve local evolution. It is proved that the complexity of the proposed fast niching is linear to the population size. In addition, this mechanism induces stable niching behavior and it inherently keeps a balance between the exploration and exploitation of multiple optima. The theoretical analysis conducted in this paper suggests that the proposed technique is able to provide bounds for the exploration and exploitation probabilities. Experimental results show that the fast niching versions of the multimodal algorithms can exhibit similar or even better performance than their original ones. More importantly, the execution time of the algorithms is significantly reduced. [ABSTRACT FROM AUTHOR]

This paper presents a new algorithm, called steady-state and generational evolutionary algorithm, which combines the fast and steadily tracking ability of steady-state algorithms and good diversity preservation of generational algorithms, for handling dynamic multiobjective optimization. Unlike most existing approaches for dynamic multiobjective optimization, the proposed algorithm detects environmental changes and responds to them in a steady-state manner. If a change is detected, it reuses a portion of outdated solutions with good distribution and relocates a number of solutions close to the new Pareto front based on the information collected from previous environments and the new environment. This way, the algorithm can quickly adapt to changing environments and thus is expected to provide a good tracking ability. The proposed algorithm is tested on a number of bi- and three-objective benchmark problems with different dynamic characteristics and difficulties. Experimental results show that the proposed algorithm is very competitive for dynamic multiobjective optimization in comparison with state-of-the-art methods. [ABSTRACT FROM PUBLISHER]

Surrogate-assisted evolutionary algorithms (SAEAs) are promising methods for solving high-dimensional expensive problems. The basic idea of SAEAs is the integration of nature-inspired searching ability of evolutionary algorithms and prediction ability of surrogate models. This paper proposes a novel evolutionary sampling assisted optimization (ESAO) method which combines the two abilities to consider global exploration and local exploitation. Differential evolution is employed to generate offspring using mutation and crossover operators. A global radial basis functions surrogate model is built for prescreening of the offspring’s objective function values and identifying the best one, which will be evaluated with the true function. The best offspring will replace its parent’s position in the population if its function value is smaller than that of its parent. A local surrogate model is then built with selected current best solutions. An optimizer is applied to find the optimum of the local model. The optimal solution is then evaluated with the true function. Besides, a better point found in the local search will be added into the population in the global search. Global and local searches will alternate if one search cannot lead to a better solution. Comprehensive analysis is conducted to study the mechanism of ESAO and insights are gained on different local surrogates. The proposed algorithm is compared with two state-of-the-art SAEAs on a series of high-dimensional problems and results show that ESAO behaves better both in effectiveness and robustness on most of the test problems. Besides, ESAO is applied to an airfoil optimization problem to show its effectiveness. [ABSTRACT FROM AUTHOR]

Convergence and diversity are interdependently handled during the evolutionary process by most existing many-objective evolutionary algorithms (MaOEAs). In such a design, the degraded performance of one would deteriorate the other, and only solutions with both are able to improve the performance of MaOEAs. Unfortunately, it is not easy to constantly maintain a population of solutions with both convergence and diversity. In this paper, an MaOEA based on two independent stages is proposed for effectively solving many-objective optimization problems (MaOPs), where the convergence and diversity are addressed in two independent and sequential stages. To achieve this, we first propose a nondominated dynamic weight aggregation method by using a genetic algorithm, which is capable of finding the Pareto-optimal solutions for MaOPs with concave, convex, linear and even mixed Pareto front shapes, and then these solutions are employed to learn the Pareto-optimal subspace for the convergence. Afterward, the diversity is addressed by solving a set of single-objective optimization problems with reference lines within the learned Pareto-optimal subspace. To evaluate the performance of the proposed algorithm, a series of experiments are conducted against six state-of-the-art MaOEAs on benchmark test problems. The results show the significantly improved performance of the proposed algorithm over the peer competitors. In addition, the proposed algorithm can focus directly on a chosen part of the objective space if the preference area is known beforehand. Furthermore, the proposed algorithm can also be used to effectively find the nadir points. [ABSTRACT FROM AUTHOR]

A comprehensive learning particle swarm optimizer (CLPSO) embedded with local search (LS) is proposed to pursue higher optimization performance by taking the advantages of CLPSO’s strong global search capability and LS’s fast convergence ability. This paper proposes an adaptive LS starting strategy by utilizing our proposed quasi-entropy index to address its key issue, i.e., when to start LS. The changes of the index as the optimization proceeds are analyzed in theory and via numerical tests. The proposed algorithm is tested on multimodal benchmark functions. Parameter sensitivity analysis is performed to demonstrate its robustness. The comparison results reveal overall higher convergence rate and accuracy than those of CLPSO, state-of-the-art particle swarm optimization variants. [ABSTRACT FROM AUTHOR]

The performance of traditional multiobjective evolutionary algorithms (MOEAs) often deteriorates rapidly as the number of decision variables increases. While some efforts were made to design new algorithms by adapting existing techniques to large-scale single-objective optimization to the MOEA context, the specific difficulties that may arise from large-scale multiobjective optimization have rarely been studied. In this paper, the exclusive challenges along with the increase of the number of variables of a multiobjective optimization problem (MOP) are examined empirically, and the popular benchmarks are categorized into three groups accordingly. Problems in the first category only require MOEAs to have stronger convergence, and can thus be mitigated using techniques employed in large-scale single-objective optimization. Problems that require MOEAs to have stronger diversification but ignore a correlation between position and distance functions are grouped as the second. The rest of the problems that pose a great challenge to the balance between diversification and convergence by considering a correlation between position and distance functions are grouped as the third. While existing large-scale MOEAs perform well on the problems in the first two categories, they suffer a significant loss when applied to those in the third category. To solve large-scale MOPs in this category, we have developed a novel indicator-based algorithm with an enhanced diversification mechanism. The proposed algorithm incorporates a new solution generator with an external archive, thus forcing the search toward different subregions of the Pareto front using a dual local search mechanism. The results obtained by applying the proposed algorithm to a wide variety of problems (108 instances in total) with up to 8192 variables demonstrate that it outperforms eight state-of-the-art approaches on the examined problems in the third category and show its advantage in the balance between diversification and convergence. [ABSTRACT FROM AUTHOR]

Decomposition-based many-objective evolutionary algorithms generally decompose the objective space into multiple subregions with the help of a set of reference vectors. The resulting subregions are fixed since the reference vectors are usually predefined. When the optimization problem has a complicated Pareto front (PF), this decomposition may decrease the algorithm performance. To deal with this problem, this paper proposes a dynamical decomposition strategy. Instead of using predefined reference vectors, solution themselves are used as reference vectors. Thus, they are adapted to the shape of PF automatically. Besides, the subregions are produced one by one through successively bipartitioning the objective space. The resulting subregions are not fixed but dynamically determined by the population solutions as well as the subregions produced previously. Based on this strategy, a solution ranking method, named dynamical-decomposition-based ranking method (DDR), is proposed which can be employed in the mating selection and environmental selection in commonly used algorithm frameworks. Compared with those in the other decomposition-based algorithms, DDR has the following properties: 1) no predefined reference vectors are required; 2) less parameters are involved; and 3) the ranking results can not only be utilized directly to select solutions but also serve as a secondary criterion in traditional Pareto-based algorithms. In this paper, DDR is equipped in two algorithm frameworks for handling many-objective optimization problems. Comparisons with five state-of-the-art algorithms on 31 widely used test problems are carried out to test the performance of the proposed approach. The experimental results have shown the effectiveness of the proposed approach in keeping a good tradeoff between convergence and diversity. [ABSTRACT FROM AUTHOR]

In sponsored content and service markets, the content and service providers are able to subsidize their target mobile users through directly paying the mobile network operator to lower the price of the data/service access charged by the network operator to the mobile users. The sponsoring mechanism leads to a surge in mobile data and service demand, which in return compensates for the sponsoring cost and benefits the content/service providers. In this paper, we study the interactions among the three parties in the market, namely, the mobile users, the content/service providers, and the network operator, as a two-level game with multiple Stackelberg (i.e., leader) players. Our study is featured by the consideration of global network effects owning to consumers’ grouping. Since the mobile users may have bounded rationality, we model the service-selection process among them as an evolutionary-population follower sub-game. Meanwhile, we model the pricing-then-sponsoring process between the content/service providers and the network operator as a non-cooperative equilibrium searching problem. By investigating the structure of the proposed game, we reveal a few important properties regarding the equilibrium existence and propose a distributed, projection-based algorithm for iterative equilibrium searching. Simulation results validate the convergence of the proposed algorithm and demonstrate how sponsoring helps improve both the providers’ profits and the users’ experience. [ABSTRACT FROM AUTHOR]

Decomposition-based multiobjective evolutionary algorithms (MOEAs) have been studied a lot and have been widely and successfully used in practice. However, there are no related theoretical studies on this kind of MOEAs. In this paper, we theoretically analyze the MOEAs based on decomposition. First, we analyze the runtime complexity with two basic simple instances. In both cases the Pareto front have one-to-one map to the decomposed subproblems or not. Second, we analyze the runtime complexity on two difficult instances with bad neighborhood relations in fitness space or decision space. Our studies show that: 1) in certain cases, polynomial-sized evenly distributed weight parameters-based decomposition cannot map each point in a polynomial sized Pareto front to a subproblem; 2) an ideal serialized algorithm can be very efficient on some simple instances; 3) the standard MOEA based on decomposition can benefit a runtime cut of a constant fraction from its neighborhood coevolution scheme; and 4) the standard MOEA based on decomposition performs well on difficult instances because both the Pareto domination-based and the scalar subproblem-based search schemes are combined in a proper way. [ABSTRACT FROM AUTHOR]

Over the last 25 years, the multiple sequence alignment (MSA) problem has attracted the attention of biologists because it is one of the major techniques used in several areas of computational biology, such as homology searches, genomic annotation, protein structure prediction, gene regulation networks, or functional genomics. This problem implicates the alignment of more than two biological sequences, and is considered as a nondeterministic polynomial time optimization problem. In this paper, we find a number of different approaches for dealing with this biological sequence alignment problem. Basically, we distinguish six main groups: 1) exact methods; 2) progressive methods; 3) consistency-based methods; 4) iterative methods; 5) evolutionary algorithms; and 6) structure-based methods. In this paper, we propose the use of evolutionary computation and multiobjective optimization for solving this bioinformatics problem. A multiobjective version of a memetic metaheuristic is presented: hybrid multiobjective metaheuristics for MSA. In order to prove the effectiveness of the new proposal, we use three structure-based benchmarks created by using empirical data as input. The results obtained by our method are compared with well-known methods published in this paper, concluding that the new approach presents remarkable accuracy when dealing with sets of sequences with a low sequence similarity, the most frequent ones in real world. [ABSTRACT FROM AUTHOR]

The design of evolutionary algorithms has typically been focused on efficiently solving a single optimization problem at a time. Despite the implicit parallelism of population-based search, no attempt has yet been made to multitask, i.e., to solve multiple optimization problems simultaneously using a single population of evolving individuals. Accordingly, this paper introduces evolutionary multitasking as a new paradigm in the field of optimization and evolutionary computation. We first formalize the concept of evolutionary multitasking and then propose an algorithm to handle such problems. The methodology is inspired by biocultural models of multifactorial inheritance, which explain the transmission of complex developmental traits to offspring through the interactions of genetic and cultural factors. Furthermore, we develop a cross-domain optimization platform that allows one to solve diverse problems concurrently. The numerical experiments reveal several potential advantages of implicit genetic transfer in a multitasking environment. Most notably, we discover that the creation and transfer of refined genetic material can often lead to accelerated convergence for a variety of complex optimization functions. [ABSTRACT FROM PUBLISHER]

Many-objective optimization has posed a great challenge to the classical Pareto dominance-based multiobjective evolutionary algorithms (MOEAs). In this paper, an evolutionary algorithm based on a new dominance relation is proposed for many-objective optimization. The proposed evolutionary algorithm aims to enhance the convergence of the recently suggested nondominated sorting genetic algorithm III by exploiting the fitness evaluation scheme in the MOEA based on decomposition, but still inherit the strength of the former in diversity maintenance. In the proposed algorithm, the nondominated sorting scheme based on the introduced new dominance relation is employed to rank solutions in the environmental selection phase, ensuring both convergence and diversity. The proposed algorithm is evaluated on a number of well-known benchmark problems having 3–15 objectives and compared against eight state-of-the-art algorithms. The extensive experimental results show that the proposed algorithm can work well on almost all the test functions considered in this paper, and it is compared favorably with the other many-objective optimizers. Additionally, a parametric study is provided to investigate the influence of a key parameter in the proposed algorithm. [ABSTRACT FROM PUBLISHER]

Pareto dominance-based multiobjective optimization has been successfully applied to constrained evolutionary optimization during the last two decades. However, as another famous multiobjective optimization framework, decomposition-based multiobjective optimization has not received sufficient attention from constrained evolutionary optimization. In this paper, we make use of decomposition-based multiobjective optimization to solve constrained optimization problems (COPs). In our method, first of all, a COP is transformed into a biobjective optimization problem (BOP). Afterward, the transformed BOP is decomposed into a number of scalar optimization subproblems. After generating an offspring for each subproblem by differential evolution, the weighted sum method is utilized for selection. In addition, to make decomposition-based multiobjective optimization suit the characteristics of constrained evolutionary optimization, weight vectors are elaborately adjusted. Moreover, for some extremely complicated COPs, a restart strategy is introduced to help the population jump out of a local optimum in the infeasible region. Extensive experiments on three sets of benchmark test functions, namely, 24 test functions from IEEE CEC2006, 36 test functions from IEEE CEC2010, and 56 test functions from IEEE CEC2017, have demonstrated that the proposed method shows better or at least competitive performance against other state-of-the-art methods. [ABSTRACT FROM AUTHOR]

Genetic programming (GP) is a powerful evolutionary algorithm that has been widely used for solving many real-world optimization problems. However, traditional GP can only solve a single task in one independent run, which is inefficient in cases where multiple tasks need to be solved at the same time. Recently, multifactorial optimization (MFO) has been proposed as a new evolutionary paradigm toward evolutionary multitasking. It intends to conduct evolutionary search on multiple tasks in one independent run. To enable multitasking GP, in this paper, we propose a novel multifactorial GP (MFGP) algorithm. To the best of our knowledge, this is the first attempt in the literature to conduct multitasking GP using a single population. The proposed MFGP consists of a novel scalable chromosome encoding scheme which is capable of representing multiple solutions simultaneously, and new evolutionary mechanisms for MFO based on self-learning gene expression programming. Further, comprehensive experimental studies are conducted on multitask scenarios consisting of commonly used GP benchmark problems and real world applications. The obtained empirical results confirmed the efficacy of the proposed MFGP. [ABSTRACT FROM AUTHOR]

Real-world optimization problems are usually multimodal which require optimization algorithms to keep a balance between exploration and exploitation. Therefore, multimodal optimization is one of the main opportunities as well as one of the main challenges for evolutionary algorithms. In this paper, a loser-out tournament-based fireworks algorithm (LoTFWA) is proposed for solving multimodal optimization problems. The search manner of the conventional fireworks algorithm (FWA) is based on the cooperation of several fireworks. While in the LoTFWA, we propose competition as a new manner of interaction, in which the fireworks are compared with each other not only according to their current status but also according to their progress rate. If the fitness of a certain firework cannot catch up with the best one with its current progress rate, it is considered a loser in the competition. The losers will be eliminated and reinitialized because it is vain to continue their search processes. Reinitializing these fireworks would greatly reduce the probability of being trapped in local minima for the algorithm. Experimental results show that the proposed algorithm is very powerful in optimizing multimodal functions. It not only outperforms previous versions of the FWA, but also outperforms several famous evolutionary algorithms. [ABSTRACT FROM AUTHOR]

This paper proposes backtracking search optimizer (BSO) for solving the reactive power dispatch (RPD) problem. The RPD problem is highly nonlinear, nonconvex optimization problem and is consisting of both continuous and discrete control variables. It aims to find the optimal settings of the generator voltages, tap positions of tap changing transformers, and the amount of reactive compensation that able to optimize the transmission power losses. BSO has simple structure and single control parameter. It has two new tuned mutation and crossover operators that control the amplitude of the search-direction matrix and search space boundaries. In this paper, five diversified generation strategies of mutation factor of BSO to solve the RPD. The proposed BSO strategies are applied to the IEEE standard 14-, 30-, 57-bus test systems, and a real power system at West Delta region system as a part of the unified Egyptian network. Simulation results show that the BSO strategy, which almost generates mutation factor concentrated on the mean value of the normal distribution (BSO 4) assigns multidirections through the search space and supports the diversity of populations. To show the effectiveness and superiority of proposed BSO, the obtained results are compared with different algorithms reported in the literature. [ABSTRACT FROM PUBLISHER]

Existing studies on dynamic multiobjective optimization (DMO) focus on problems with time-dependent objective functions, while the ones with a changing number of objectives have rarely been considered in the literature. Instead of changing the shape or position of the Pareto-optimal front/set (PF/PS) when having time-dependent objective functions, increasing or decreasing the number of objectives usually leads to the expansion or contraction of the dimension of the PF/PS manifold. Unfortunately, most existing dynamic handling techniques can hardly be adapted to this type of dynamics. In this paper, we report our attempt toward tackling the DMO problems with a changing number of objectives. We implement a dynamic two-archive evolutionary algorithm which maintains two co-evolving populations simultaneously. In particular, these two populations are complementary to each other: one concerns more about the convergence while the other concerns more about the diversity. The compositions of these two populations are adaptively reconstructed once the environment changes. In addition, these two populations interact with each other via a mating selection mechanism. Comprehensive experiments are conducted on various benchmark problems with a time-dependent number of objectives. Empirical results fully demonstrate the effectiveness of our proposed algorithm. [ABSTRACT FROM PUBLISHER]

Pattern mining (PM) refers to the process of discovering patterns of interest to users from data. However, most studies have considered only one pattern, such as frequent pattern or high-utility pattern. With the continuous requirement of businesses in various industries, the single-objective PM methods are difficult to meet the increasingly diverse needs of users. In this paper, a multi-objective problem model for high quality pattern mining (HQPM) is proposed, where the objectives are support, occupancy, and utility. In order to solve the proposed three-objective problem efficiently, an improved multi-objective evolutionary algorithm for HQPM (MOEA-PM) is proposed. Two kinds of population initialization strategies are designed, which is used to ensure the population is effectively distributed in the feasible solution space. By taking the properties of the model into consideration, an auxiliary tool is proposed to accelerate the convergence of the algorithm. Experimental results on real-world datasets show that the proposed three-objective problem model with the MOEA-PM algorithm can discover patterns that are both frequently occurring and has a high utility in the transaction datasets, while at the same time being relatively complete. Compared with the state-of-the-art MOEA-based HQPM algorithms, MOEA-PM has better performance in terms of efficiency, quality, and convergence speed. [ABSTRACT FROM AUTHOR]

A large number of prediction strategies are specific to a dynamic multiobjective optimization problem (DMOP) with only one type of the Pareto set (PS) change. However, a continuous DMOP with more than one type of the unknown PS change has been seldom investigated. We present a multimodel prediction approach (MMP) realized in the framework of evolutionary algorithms (EAs) to tackle the problem. In this paper, we first detect the type of the PS change, followed by the selection of an appropriate prediction model to provide an initial population for the subsequent evolution. To observe the influence of MMP on EAs, optimal solutions obtained by three classical dynamic multiobjective EAs with and without MMP are investigated. Furthermore, to investigate the performance of MMP, three state-of-the-art prediction strategies are compared on a large number of dynamic test instances under the same particle swarm optimizer. The experimental results demonstrate that the proposed approach outperforms its counterparts under comparison on most optimization problems. [ABSTRACT FROM AUTHOR]

Neighborhood information plays an important role in improving the performance of evolutionary computation in various optimization scenarios, particularly in the context of multimodal optimization. Several neighborhood concepts, i.e., index-based neighborhood, nearest neighborhood, and fuzzy neighborhood, have been studied and engaged in the design of niching methods. However, the use of these neighborhood concepts requires the specification of some problem-related parameters, which is difficult to determine without a prior knowledge. In this paper, we introduce a new neighborhood concept based on a geometrical construction called Voronoi diagram. The new concept offers two advantages at the expense of increasing the computational complexity to a higher level. It eliminates the need of additional parameters and it is more informative than the existing ones. The information provided by the Voronoi neighbors of an individual can be exploited to estimate the evolutionary state. Based on the information, we divide the population into three groups and assign each group a different reproduction strategy to support the exploration and exploitation of the search space. We show the use of the concept in the design of an effective evolutionary algorithm for multimodal optimization. The experiments have been conducted to investigate the performance of the algorithm. The results reveal that the proposed algorithm compare favorably with the state-of-the-art algorithms designed based on other types of neighborhood concepts. [ABSTRACT FROM AUTHOR]

Two-echelon vehicle routing problem (2E-VRP) is a challenging problem that involves both the strategic and tactical planning decisions on both echelons. The satellite locations and the customer distribution affect the cost of different components on the second echelon, thus the possibilities of satellite-to-customer assignment complicates the problem. In this paper, we propose a graph-based fuzzy evolutionary algorithm for solving 2E-VRP. The proposed method integrates a graph-based fuzzy assignment scheme into an iteratively evolutionary learning process to minimize the total cost. To resolve the possibilities of the satellite-to-customer assignment, graph-based fuzzy operator is used to take advantage of population evolution and avoid excessive fitness evaluations of unpromising moves in different satellites. Each offspring is produced via graph-based fuzzy assignment procedure out of an assignment graph from parent individuals, and fuzzy local search procedure is used to further improve the offspring. The experimental results on the public test sets demonstrate the competitiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Decomposition of a multiobjective optimization problem (MOP) into several simple multiobjective subproblems, named multiobjective evolutionary algorithm based on decomposition (MOEA/D)-M2M, is a new version of multiobjective optimization-based decomposition. However, it fails to consider different contributions from each subproblem but treats them equally instead. This paper proposes a collaborative resource allocation (CRA) strategy for MOEA/D-M2M, named MOEA/D-CRA. It allocates computational resources dynamically to subproblems based on their contributions. In addition, an external archive is utilized to obtain the collaborative information about contributions during a search process. Experimental results indicate that MOEA/D-CRA outperforms its peers on 61% of the test cases in terms of three metrics, thereby validating the effectiveness of the proposed CRA strategy in solving MOPs. [ABSTRACT FROM AUTHOR]

Optimization problems with variable-length decision space are a class of challenging optimization problems derived from some real-world applications, such as the composite laminate stacking problem and the sensor coverage problem. Unlike other optimization problems, the solutions in these problems might be represented as the vectors with different variable size (i.e., dimensionality). So far, some research efforts have been done on the use of evolutionary algorithms (EAs) for solving single objective variable-length optimization problems. In fact, the variable-length problem difficulty can also exist in multiobjective optimization. However, such challenging problems have not yet gained much attention in the area of evolutionary multiobjective optimization. To facilitate the research on the variable-length Pareto optimization, we first suggest a systematic toolkit for constructing benchmark multiobjective test problems with variable-length feature in this paper. Then, we also propose a variable-length multiobjective EA based on a two-level decomposition strategy, which decomposes a multiobjective optimization problem in terms of the penalty boundary intersection search directions and the dimensionality of variables. The performance of our proposed algorithm and the other three state-of-the-art algorithms on these problems are compared. To further show the effectiveness of our proposed algorithm, some experimental results on a bi-objective laminate stacking optimization problem are also reported and analyzed. [ABSTRACT FROM AUTHOR]

Metaheuristics have yielded very promising results for the frequency assignment problem (FAP). However, the results obtainable using currently published methods are far from ideal in complex, large-scale instances. This paper applies and extends some of the most recent advances in evolutionary algorithms to two common variants of the FAP, and shows how, in traditional techniques, two common issues affect their performance: 1) premature convergence and 2) the way in which neutral networks are handled. A recent replacement-based diversity management strategy is successfully applied to alleviate the premature convergence drawback. Additionally, by properly defining a distance metric, the performance in the presence of neutrality can also be greatly improved. The replacement strategy combines the principle of transforming a single-objective problem into a multiobjective one by considering diversity as an additional objective, with the idea of adapting the balance induced between exploration and exploitation to the requirements of the different optimization stages. Tests with 44 publicly available instances yield very competitive results. New best-known frequency plans were generated for 11 instances, whereas in the remaining ones the best-known solutions were replicated. Comparisons with a large number of strategies designed to delay convergence of the population clearly show the advantages of our novel proposals. [ABSTRACT FROM PUBLISHER]

While Pareto-based multiobjective optimization algorithms continue to show effectiveness for a wide range of practical problems that involve mostly two or three objectives, their limited application for many-objective problems, due to the increasing proportion of nondominated solutions and the lack of sufficient selection pressure, has also been gradually recognized. In this paper, we revive an early developed and computationally expensive strength Pareto-based evolutionary algorithm by introducing an efficient reference direction-based density estimator, a new fitness assignment scheme, and a new environmental selection strategy, for handling both multiobjective and many-objective problems. The performance of the proposed algorithm is validated and compared with some state-of-the-art algorithms on a number of test problems. Experimental studies demonstrate that the proposed method shows very competitive performance on both multiobjective and many-objective problems considered in this paper. Besides, our extensive investigations and discussions reveal an interesting finding, that is, diversity-first-and-convergence-second selection strategies may have great potential to deal with many-objective optimization. [ABSTRACT FROM AUTHOR]

This paper proposes an application of an evolutionary game dynamics called the escort dynamics (ED) for the decentralized load management of plug-in electric vehicles (PEV). Different from earlier contributions, in the present approach, PEVs work together in a fair scheme in order to provide several ancillary services to the grid: Load shifting, active power balancing, and partial supply of reactive power demand on each phase of the distribution transformer. Meanwhile, batteries are guaranteed to be fully charged according to the constraints imposed by the owners. In the proposed formulation, chargers can be either three phase or single phase; however, in this paper, only three-phase chargers are considered. The key concepts behind ED, especially for escort functions, are provided at the beginning of this paper. Based on these concepts, the assumptions and analogies followed for the construction of the proposed approach are explained in detail, especially for the proposed definition of escort functions. A multipopulation scenario is proposed for the interaction of several PEVs using local ED routines. This interaction among populations follows another well-known evolutionary game dynamics called the best reply dynamics. Performance is evaluated using real data measured from a distribution transformer from the SOREA utility grid company in the region of Savoie, France. [ABSTRACT FROM AUTHOR]

Finding the overall Pareto optimal front while addressing the effect of an increasing number of objectives has become an essential and challenging issue for multiobjective optimization in real-world applications. Preference information provided by a decision maker can guide the search for preferred regions of the Pareto front and accelerate the convergence of the population. In this paper, a new variant of the Pareto dominance relation, called preference angle and reference information-based dominance, is proposed to create a stricter partial order among nondominated solutions. In the proposed method, the Euclidean distance and angle information between candidate solutions and reference points are calculated to evaluate the degree of convergence and population diversity, respectively. In addition, an adaptive threshold is designed to adjust the judgment condition of ar-dominance using an iterative process in a prespecified interval. The proposed algorithm increases the convergence speed of the population and reduces the number of solutions in the nonpreferred region. Comparative evaluation experiments are presented with respect to two performance metrics for a variety of benchmark test problems and real-world aluminum electrolytic production cases. The results demonstrate that the proposed approach is effective for highly complex, multiobjective optimization problems when compared with five state-of-the-art evolutionary algorithms. [ABSTRACT FROM AUTHOR]

Real-world applications typically have multiple sparse reconstruction tasks to be optimized. In order to exploit the similar sparsity pattern between different tasks, this paper establishes an evolutionary multitasking framework to simultaneously optimize multiple sparse reconstruction tasks using a single population. In the proposed method, the evolutionary algorithm aims to search the locations of nonzero components or rows instead of searching sparse vector or matrix directly. Then the within-task and between-task genetic transfer operators are employed to reinforce the exchange of genetic material belonging to the same or different tasks. The proposed method can solve multiple measurement vector problems efficiently because the length of decision vector is independent of the number of measurement vectors. Finally, a case study on hyperspectral image unmixing is investigated in an evolutionary multitasking setting. It is natural to consider a sparse unmixing problem in a homogeneous region as a task. Experiments on signal reconstruction and hyperspectral image unmixing demonstrate the effectiveness of the proposed multitasking framework for sparse reconstruction. [ABSTRACT FROM AUTHOR]

Researches have shown difficulties in obtaining proximity while maintaining diversity for many-objective optimization problems. Complexities of the true Pareto front pose challenges for the reference vector-based algorithms for their insufficient adaptability to the diverse characteristics with no priori. This paper proposes a many-objective optimization algorithm with two interacting processes: cascade clustering and reference point incremental learning (CLIA). In the population selection process based on cascade clustering (CC), using the reference vectors provided by the process based on incremental learning, the nondominated and the dominated individuals are clustered and sorted with different manners in a cascade style and are selected by round-robin for better proximity and diversity. In the reference vector adaptation process based on reference point incremental learning, using the feedbacks from the process based on CC, proper distribution of reference points is gradually obtained by incremental learning. Experimental studies on several benchmark problems show that CLIA is competitive compared with the state-of-the-art algorithms and has impressive efficiency and versatility using only the interactions between the two processes without incurring extra evaluations. [ABSTRACT FROM AUTHOR]

This paper suggests a novel clustering-based evolutionary algorithm for many-objective optimization problems. Its main idea is to classify the population into a number of clusters, which is expected to solve the difficulty of balancing convergence and diversity in high-dimensional objective space. The individuals showing high similarities on the vector angles are gathered into the same cluster, such that the population’s distribution can be well portrayed by the clusters. To efficiently find these clusters, partitional clustering is first used to classify the union population into ${m}$ main clusters based on the ${m}$ axis vectors (${m}$ is the number of objectives), and then hierarchical clustering is further run on these ${m}$ main clusters to get ${N}$ final clusters (${N}$ is the population size and ${N>m}$). At last, in environmental selection, one individual from each of ${N}$ clusters closest to the axis vectors is selected to maintain diversity, while one individual from each of the other clusters is preferred by a simple convergence indicator to ensure convergence. When tackling some well-known test problems with 5–15 objectives, extensive experiments validate the superiority of our algorithm over six competitive many-objective EAs, especially on problems with incomplete and irregular Pareto-optimal fronts. [ABSTRACT FROM AUTHOR]

Both convergence and diversity are crucial to evolutionary many-objective optimization, whereas most existing dominance relations show poor performance in balancing them, thus easily leading to a set of solutions concentrating on a small region of the Pareto fronts. In this paper, a novel dominance relation is proposed to better balance convergence and diversity for evolutionary many-objective optimization. In the proposed dominance relation, an adaptive niching technique is developed based on the angles between the candidate solutions, where only the best converged candidate solution is identified to be nondominated in each niche. Experimental results demonstrate that the proposed dominance relation outperforms existing dominance relations in balancing convergence and diversity. A modified NSGA-II is suggested based on the proposed dominance relation, which shows competitiveness against the state-of-the-art algorithms in solving many-objective optimization problems. The effectiveness of the proposed dominance relation is also verified on several other existing multi- and many-objective evolutionary algorithms. [ABSTRACT FROM AUTHOR]

Through introducing the divide-and-conquer strategy, cooperative co-evolution (CC) has been successfully employed by many evolutionary algorithms (EAs) to solve large-scale optimization problems. In practice, it is common that different subcomponents of a large-scale problem have imbalanced contributions to the global fitness. Thus, how to utilize such imbalance and concentrate efforts on optimizing important subcomponents becomes an important issue for improving performance of cooperative co-EA, especially in distributed computing environment. In this paper, we propose a two-layer distributed CC (dCC) architecture with adaptive computing resource allocation for large-scale optimization. The first layer is the dCC model which takes charge of calculating the importance of subcomponents and accordingly allocating resources. An effective allocating algorithm is designed which can adaptively allocate computing resources based on a periodic contribution calculating method. The second layer is the pool model which takes charge of making fully utilization of imbalanced resource allocation. Within this layer, two different conformance policies are designed to help optimizers use the assigned computing resources efficiently. Empirical studies show that the two conformance policies and the computing resource allocation algorithm are effective, and the proposed distributed architecture possesses high scalability and efficiency. [ABSTRACT FROM AUTHOR]

When solving constrained multiobjective optimization problems, an important issue is how to balance convergence, diversity, and feasibility simultaneously. To address this issue, this paper proposes a parameter-free constraint handling technique, a two-archive evolutionary algorithm, for constrained multiobjective optimization. It maintains two collaborative archives simultaneously: one, denoted as the convergence-oriented archive (CA), is the driving force to push the population toward the Pareto front; the other one, denoted as the diversity-oriented archive (DA), mainly tends to maintain the population diversity. In particular, to complement the behavior of the CA and provide as much diversified information as possible, the DA aims at exploring areas under-exploited by the CA including the infeasible regions. To leverage the complementary effects of both archives, we develop a restricted mating selection mechanism that adaptively chooses appropriate mating parents from them according to their evolution status. Comprehensive experiments on a series of benchmark problems and a real-world case study fully demonstrate the competitiveness of our proposed algorithm, in comparison to five state-of-the-art constrained evolutionary multiobjective optimizers. [ABSTRACT FROM AUTHOR]

Metaheuristics, being tailored to each particular domain by experts, have been successfully applied to many computationally hard optimization problems. However, once implemented, their application to a new problem domain or a slight change in the problem description would often require additional expert intervention. There is a growing number of studies on reusable cross-domain search methodologies, such as selection hyper-heuristics, which are applicable to problem instances from various domains, requiring minimal expert intervention or even none. This paper introduces a new learning automata-based selection hyper-heuristic controlling a set of multiobjective metaheuristics. The approach operates above three well-known multiobjective evolutionary algorithms and mixes them, exploiting the strengths of each algorithm. The performance and behavior of two variants of the proposed selection hyper-heuristic, each utilizing a different initialization scheme are investigated across a range of unconstrained multiobjective mathematical benchmark functions from two different sets and the real-world problem of vehicle crashworthiness. The empirical results illustrate the effectiveness of our approach for cross-domain search, regardless of the initialization scheme, on those problems when compared to each individual multiobjective algorithm. Moreover, both variants perform significantly better than some previously proposed selection hyper-heuristics for multiobjective optimization, thus significantly enhancing the opportunities for improved multiobjective optimization. [ABSTRACT FROM AUTHOR]

Multimodal optimization problem (MMOP), which targets at searching for multiple optimal solutions simultaneously, is one of the most challenging problems for optimization. There are two general goals for solving MMOPs. One is to maintain population diversity so as to locate global optima as many as possible, while the other is to increase the accuracy of the solutions found. To achieve these two goals, a novel dual-strategy differential evolution (DSDE) with affinity propagation clustering (APC) is proposed in this paper. The novelties and advantages of DSDE include the following three aspects. First, a dual-strategy mutation scheme is designed to balance exploration and exploitation in generating offspring. Second, an adaptive selection mechanism based on APC is proposed to choose diverse individuals from different optimal regions for locating as many peaks as possible. Third, an archive technique is applied to detect and protect stagnated and converged individuals. These individuals are stored in the archive to preserve the found promising solutions and are reinitialized for exploring more new areas. The experimental results show that the proposed DSDE algorithm is better than or at least comparable to the state-of-the-art multimodal algorithms when evaluated on the benchmark problems from CEC2013, in terms of locating more global optima, obtaining higher accuracy solution, and converging with faster speed. [ABSTRACT FROM AUTHOR]

Trust and trustworthiness are of great importance in social and human systems, especially when considering managerial and economic decision-making. In this paper, we investigate the emergent dynamics of an evolutionary game-theoretic model—the ${N}$ -player evolutionary trust game—consisting of three types of players: 1) an investor; 2) a trustee who is trustworthy; and 3) a trustee who is untrustworthy. Here, we limit the interactions between players to local neighborhoods defined by a specific spatial topology or social network. Players are able to adjust their game-playing strategies using an evolutionary update rule based on the payoffs obtained by their neighbors. Through comprehensive simulation experiments, we find that trust can be promoted when players interact on a social network despite a substantial number of untrustworthy individuals in the initial population. These results differ from findings reported for an unstructured population of the same game, where the existence of a single untrustworthy individual would eliminate trust completely. We compare the dynamics of the model with different social network densities and structures (e.g., from regular lattices to scale-free and random networks). We observe that the levels of trust vary under different network structures, and the level is correlated with how “difficult” the game is. When game conditions are easy (i.e., low temptation to defect and/or almost no initial untrustworthy trustees), homogeneous networks with higher densities can promote higher levels of trust. However, when the game becomes harder, heterogeneous social networks with lower densities are able to promote higher levels of trust and global net wealth. [ABSTRACT FROM AUTHOR]

Multiobjective evolutionary algorithms evolve a population of solutions through successive generations toward the Pareto-optimal front (POF). One of the most critical questions faced by the researchers and practitioners in this domain relates to the number of generations that may be sufficient for an algorithm to offer a good approximation of the POF for a given problem. Ironically, to date, this question largely remains unanswered and the number of generations are arbitrarily fixed a priori, with potentially punitive implications. If the a priori fixed generations are insufficient, then the algorithm reports suboptimal solutions. In contrast, if the a priori fixed generations are far too many, it implies waste of computational resources. This paper proposes a novel entropy-based dissimilarity measure that helps identify on the fly the number of generations beyond which an algorithm stabilizes, implying that either a good approximation has been obtained or that it cannot be obtained due to the stagnation of the algorithm in the search space. Given that in either case no further improvement in the approximation can be obtained, despite additional computational expense, the proposed dissimilarity measure provides a termination criterion and facilitates a termination detection algorithm. The generality, on-the-fly implementation, low-computational complexity, and the demonstrated efficacy of the proposed termination detection algorithm, on a wide range of multiobjective and many-objective test problems, define the novel contribution of this paper. [ABSTRACT FROM AUTHOR]

Stability analysis is an important research direction in evolutionary game theory. Evolutionarily stable states have a close relationship with Nash equilibria of repeated games, which are characterized by the folk theorem. When applying the folk theorem, one needs to compute the minimax profile of the game in order to find Nash equilibria. Computing the minimax profile is an NP-hard problem. In this paper, we investigate a new methodology to compute evolutionary stable states based on the level- k equilibrium, a new refinement of Nash equilibrium in repeated games. A level- k equilibrium is implemented by a group of players who adopt reactive strategies and who have no incentive to deviate from their strategies simultaneously. Computing the level- k equilibria is tractable because the minimax payoffs and strategies are not needed. As an application, this paper develops a tractable algorithm to compute the evolutionarily stable states and the Pareto front of n -player symmetric games. Three games, including the iterated prisoner’s dilemma, are analyzed by means of the proposed methodology. [ABSTRACT FROM PUBLISHER]

In many diagnostic imaging settings, including positron emission tomography (PET), images are typically used for multiple tasks such as detecting disease and quantifying disease. Unlike conventional image reconstruction that optimizes a single objective, this work proposes a multi-objective optimization algorithm for PET image reconstruction to identify a set of images that are optimal for more than one task. This work is reliant on a genetic algorithm to evolve a set of solutions that satisfies two distinct objectives. In this paper, we defined the objectives as the commonly used Poisson log-likelihood function, typically reflective of quantitative accuracy, and a variant of the generalized scan-statistic model, to reflect detection performance. The genetic algorithm uses new mutation and crossover operations at each iteration. After each iteration, the child population is selected with non-dominated sorting to identify the set of solutions along the dominant front or fronts. After multiple iterations, these fronts approach a single non-dominated optimal front, defined as the set of PET images for which none the objective function values can be improved without reducing the opposing objective function. This method was applied to simulated 2D PET data of the heart and liver with hot features. We compared this approach to conventional, single-objective approaches for trading off performance: maximum likelihood estimation with increasing explicit regularization and maximum a posteriori estimation with varying penalty strength. Results demonstrate that the proposed method generates solutions with comparable to improved objective function values compared to the conventional approaches for trading off performance amongst different tasks. In addition, this approach identifies a diverse set of solutions in the multi-objective function space which can be challenging to estimate with single-objective formulations. [ABSTRACT FROM AUTHOR]

Decomposition-based evolutionary algorithms have been quite successful in solving optimization problems involving two and three objectives. Recently, there have been some attempts to exploit the strengths of decomposition-based approaches to deal with many objective optimization problems. Performance of such approaches are largely dependent on three key factors: 1) means of reference point generation; 2) schemes to simultaneously deal with convergence and diversity; and 3) methods to associate solutions to reference directions. In this paper, we introduce a decomposition-based evolutionary algorithm wherein uniformly distributed reference points are generated via systematic sampling, balance between convergence and diversity is maintained using two independent distance measures, and a simple preemptive distance comparison scheme is used for association. In order to deal with constraints, an adaptive epsilon formulation is used. The performance of the algorithm is evaluated using standard benchmark problems, i.e., DTLZ1-DTLZ4 for 3, 5, 8, 10, and 15 objectives, WFG1-WFG9, the car side impact problem, the water resource management problem, and the constrained ten-objective general aviation aircraft design problem. Results of problems involving redundant objectives and disconnected Pareto fronts are also included in this paper to illustrate the capability of the algorithm. The study clearly highlights that the proposed algorithm is better or at par with recent reference direction-based approaches for many objective optimization. [ABSTRACT FROM AUTHOR]