In the domain of artificial neural networks, the comprehensive understanding and optimization of neuron dynamics are crucial. The Dendritic Neuron Model (DNM), noted for its distinct architecture and data processing capabilities, exemplifies this. However, the sophistication of the DNM leads to complexities in hyperparameter tuning. Notably, this complexity manifests in the way parameter changes can alter the dimensions of the solution space, a prime example of the Metameric Variable-length Problem. In this study, we have innovatively integrated chaotic maps into the gene expression mechanisms of Evolutionary Algorithms, enabling the incorporation of all DNM hyperparameters into a singular algorithmic framework. This integration allows for iterative adjustments within a variable-dimensional solution space, representing an evolving neuron model that streamlines the tuning process. Our approach, tested against benchmark datasets from the UCI Machine Learning Repository, demonstrates significant improvements in the DNM's performance, highlighting the effectiveness of incorporating biological chaos phenomena into neural network optimization.

An Escher-like tiling is a tiling consisting of one or a few artistic shapes of tile. This article proposes a method for generating Escher-like tilings consisting of two distinct shapes (dihedral Escher-like tilings) that are as similar as possible to the two goal shapes specified by the user. This study is an extension of a previous study that successfully generated Escher-like tilings consisting of a single tile shape for a single goal shape. Building upon the previous study, our method attempts to exhaustively search for which parts of the discretized tile contours are adjacent to each other to form a tiling. For each configuration, two tile shapes are optimized to be similar to the given two goal shapes. By evaluating the similarity based on as-rigid-as possible deformation energy, the optimized tile shapes preserve the local structures of the goal shapes, even if substantial deformations are necessary to form a tiling. However, in the dihedral case, this approach is seemingly unrealistic because it suffers from the complexity of the energy function and the combinatorial explosion of the possible configurations. We developed a method to address these issues and show that the proposed algorithms can generate satisfactory dihedral Escher-like tilings in a realistic computation time, even for somewhat complex goal shapes.

Escher tiling is well known as a tiling that consists of one or a few recognizable figures, such as animals. The Escherization problem involves finding the most similar shape to a given goal figure that can tile the plane. However, it is easy to imagine that there is no similar tile shape for complex goal shapes. This article devises a method for finding a satisfactory tile shape in such a situation. To obtain a satisfactory tile shape, the tile shape is generated by deforming the goal shape to a considerable extent while retaining the characteristics of the original shape. To achieve this, both goal and tile shapes are represented as triangular meshes to consider not only the contours but also the internal similarity of the shapes. To measure the naturalness of the deformation, energy functions based on traditional as-rigid-as-possible shape modeling are incorporated into a recently developed framework of the exhaustive search of the templates for the Escherization problem. The developed algorithms find satisfactory tile shapes with natural deformations for fairly complex goal shapes.

To maintain the population diversity of genetic algorithms (GAs), we are required to employ an appropriate population diversity measure. However, commonly used population diversity measures designed for permutation problems do not consider the dependencies between the variables of the individuals in the population. We propose three types of population diversity measures that address high-order dependencies between the variables to investigate the effectiveness of considering high-order dependencies. The first is formulated as the entropy of the probability distribution of individuals estimated from the population based on an [Formula: see text]-th--order Markov model. The second is an extension of the first. The third is similar to the first, but it is based on a variable order Markov model. The proposed population diversity measures are incorporated into the evaluation function of a GA for the traveling salesman problem to maintain population diversity. Experimental results demonstrate the effectiveness of the three types of high-order entropy-based population diversity measures against the commonly used population diversity measures.

In this paper, we propose a new parallel genetic algorithm (GA) with edge assembly crossover (EAX) for the traveling salesman problem (TSP). GA with EAX (GA-EAX) is one of the promising meta-heuristics for TSP and found best-known tours for several well-known 100,000-city scale TSP instances. However, it takes about ten days to execute this GA just one time using the default configuration on the 120,000-city instance [1]. Therefore, it is crucial to reduce the running time of GA-EAX for 100,000-city scale instances in order to make it possible to improve the algorithm through trial and error. The proposed parallel GA achieves about twenty-times speed up without deteriorating the quality of solutions compared to the original GA-EAX. We also demonstrate that the proposed parallel GA successfully finds new best-known tours for the 120,000-city and 180,000-city instances called vangogh120K and courbet180K, respectively.

This paper presents an extended variant of the distance-weighted exponential natural evolution strategy (DXNES) that works well in uncertain environments. Since we often face objective functions with uncertain parameters in real-world problems, function optimization in uncertain environments is an important problem. The covariance matrix adaptation evolution strategy (CMA-ES) and DX-NES have been proposed as promising methods for function optimization in deterministic environments. The performance of these methods, however, deteriorates in uncertain environments. The uncertain handling CMA-ES (TIH-CMA-ES) has been proposed as an extended variant of CMA-ES for uncertain environments and has shown relatively good performance on problems with uncertain parameters. In this paper, we propose an extended variant of DX-NES named DX-NES for uncertain environments (DX-NES-TIE). DX-NES-TIE approximates the objective function by a quadratic function. DXNES-TIE uses approximation function values for updating the mutation distribution if the noise is strong; otherwise it uses observed objective function values. The strength of the noise is quantified by using the approximation function and the evolution path. Through numerical experiments on 20-dimensional uncertain benchmark problems, we demonstrate that DX-NES-TIE can find ten to 2,000 times as accurate solutions as TIH-CMA-ES can. We also apply DX-NES-TIE to 80-dimensional problems and confirm that DX-NES-TIE is scalable with respect to problem dimensionality.

In this paper, we propose a new real-coded genetic algorithm (RCGA) for implicit constrained black-box function optimization. On implicit constrained problems, there often exist active constraints of which the optima lie on the boundaries, which makes the problem more difficult. Almost all of conventional constraint-handling techniques cannot be applied to implicit constrained black-box function optimization because we cannot get quantities of constraint violations and preference order of infeasible solutions. The resampling technique may be the only available choice to handle the implicit constraint. AREX/JGG is one of the most powerful RCGAs for non-constrained problems. However, AREX/JGG with resampling technique deteriorates on implicit constrained problems because few individuals are generated near the boundaries of active constraints and, thus, a population cannot approach the boundaries quickly. In order to find these optima, we believe that it is necessary to locate the mode of a distribution for generating new individuals nearer the boundaries. Since solutions around the optima on boundaries of active constraints may have better evaluation values, our proposed method employs the weighted mean of the best half individuals in a population as the mode of the distribution. We assess the proposed method through experiments with some benchmark problems and the results show the proposed method succeeds in finding the optimum with about 40-85% of function evaluations compared to AREX/JGG with resampling technique.

This paper presents a genetic algorithm (GA) for solving the traveling salesman problem (TSP). To construct a powerful GA, we use edge assembly crossover (EAX) and make substantial enhancements to it: (i) localization of EAX together with its efficient implementation and (ii) the use of a local search procedure in EAX to determine good combinations of building blocks of parent solutions for generating even better offspring solutions from very high-quality parent solutions. In addition, we develop (iii) an innovative selection model for maintaining population diversity at a negligible computational cost. Experimental results on well-studied TSP benchmarks demonstrate that the proposed GA outperforms state-of-the-art heuristic algorithms in finding very high-quality solutions on instances with up to 200,000 cities. In contrast to the state-of-the-art TSP heuristics, which are all based on the Lin–Kernighan (LK) algorithm, our GA achieves top performance without using an LK-based algorithm.

This paper presents a new evolutionary algorithm for function optimization named the distance-weighted exponential natural evolution strategies (DX-NES). DX-NES remedies two problems of a conventional method, the exponential natural evolution strategies (xNES), that shows good performance when it does not need to move the distribution for sampling individuals down the slope to the optimal point. The first problem of xNES is that the search efficiency deteriorates while the distribution moves down the slope of an ill-scaled function because it degenerates before reaching the optimal point. The second problem is that the settings of learning rates are inappropriate because they do not taking account of some factors affecting the estimate accuracy of the natural gradient. We compared the performance of DX-NES with that of xNES and CMA-ES on typical benchmark functions and confirmed that DX-NES outperformed the xNES on all the benchmark functions and that DX-NES showed better performance than CMA-ES on the almost all functions except the k-tablet function.

In our previous study, we have proposed Adaptive Weighted Aggregation (AWA), a framework of multi-starting optimization methods based on scalarization for solving multi objective function optimization problems. The experiments in the proposal show that AWA outperforms conventional multi starting descent methods at coverage of solutions. However, the suitable termination condition for AWA has not been understood. Coverage of AWA's solutions and computational cost of AWA strongly depends on the termination condition. In this paper, we derive the necessary and sufficient iteration count to achieve high coverage and the number of approximate solutions generated until AWA stops. Numerical experiments show that AWA still achieves better coverage than the conventional methods under the derived termination condition.

Adaptive Weighted Aggregation (AWA) is a frame work of multi-starting optimization methods based on scalarization for solving multiobjective function optimization problems. It progressively generates new solutions to refine the approximation of the Pareto set or the Pareto front by the subdivision, and iteratively estimates the appropriate weight vector for scalarization in each search by the weight adaptation. Our recent study shows that AWA's solution set combinatorially increases for the number of objectives. In this paper, we propose a new subdivision and weight adaptation scheme of AWA to improve its scalability. Numerical experiments show the effectiveness of the proposed method.

In this paper, we propose a new framework taking account of multi-funnel functions for Real-coded Genetic Algorithms (RCGAs). In the continuous function optimization, Evolutionary Algorithms (EAs) are one of the most effective optimization methods. However, most conventional EAs, such as RCGAs and CMA-ES, work efficiently on functions with big-valley landscape and they deteriorate on the multi-funnel functions. Innately Split Model (ISM) has been proposed as a framework of GAs for multi-funnel functions and outperforms conventional GAs on this kind of functions. However, ISM is considered to have two problems in terms of efficiency of the search and difficulty of parameter settings. Our framework repeats a search by RCGAs as ISM does and has two effective mechanisms to remedy the two problems of ISM. We conducted experiments on benchmark functions with multi-funnel and big valley landscapes and our framework outperformed conventional EAs, Multi-start RCGA (MS-RCGA), Multi-start CMA-ES (MS CMA-ES) and ISM, on the multi-funnel functions. Our frame work achieved as good performance as MS-RCGA and MS CMA-ES on the big-valley function where ISM significantly deteriorates.

The multi-starting descent method is a promising approach to unimodal multiobjective function optimization problems because of its precision of obtained solutions. Descent methods can be classified into two categories; the multiobjective descent method directly using the Jacobian matrix of objective functions and the scalarized descent method using the gradient of a scalarized objective function. In the multiobjective descent method and the scalarized descent method, a convergent point depends on an initial solution and a weight vector, respectively. However, it is difficult to choose appropriate initial solutions or weight vectors for obtaining widely and evenly distributed solutions. In order to remedy the problems of the conventional methods, we propose a multi-starting scalarized descent method named AWA that employs the Chebyshev norm method as a scalarization method and an adaptive scheme of weight vectors for the scalarization method. We show the effectiveness of the proposed method through some experiments.

Real-Coded Genetic Algorithms (RCGAs) have been extensively studied for last two decades because RCGAs have advantages over conventional continuous function optimization methods when multimodal functions are optimized. Innately Split Model (ISM) is one of promising approaches to enhance RCGAs where a set of population groups are evolved in parallel and groups are re-initialized if two groups searches a similar region (it is called redundant searches). In this paper, we propose a new strategy for the re-initialization of groups to improve the performance of ISM. In our method, redundant searches are detected by using the information of the search histories of the groups. This information is called traps and is stored as a set of hyper-ellipsoids representing the distributions of the previous groups. We demonstrate that the proposed method is robust and superior to the original ISM.

This paper investigates theoretically the convergence properties of the stochastic algorithms of a class including both CMAESs and EDAs on constrained minimization of continuously differentiable functions. We are interested in algorithms that do not get stuck on a slope of the function, but converge only to local optimal points. Convergence to a point that is neither a stationary point of the function nor a boundary point is evidence that the convergence properties are not well behaved. We investigate what properties are necessary/sufficient for the algorithm to avoid this type of behavior, i.e., what properties are necessary for the algorithm to converge only to local optimal points of the function. We also investigate the analogous conditions on the parameters of two variants of modern EC-based stochastic algorithms, namely, a CMAES employing rank-μ update and an EDA known as EMNAglobal. The comparison between the apparently similar two systems shows that they have significantly different theoretical behaviors. This result presents us with an insight into the way we design well-behaved optimization algorithms.

In this paper, we propose an efficient algorithm for estimating the natural policy gradient with parameter-based exploration; this algorithm samples directly in the parameter space. Unlike previous methods based on natural gradients, our algorithm calculates the natural policy gradient using the inverse of the exact Fisher information matrix. The computational cost of this algorithm is equal to that of conventional policy gradients whereas previous natural policy gradient methods have a prohibitive computational cost. Experimental results show that the proposed method outperforms several policy gradient methods.

Abstract Vehicle routing problems are at the heart of most decision support systems for real‐life distribution problems. In vehicle routing problem a set of routes must be determined at lowest total cost for a number of resources (i.e., fleet of vehicles) located at one or several points (e.g., depots, warehouses) to efficiently service a number of demand or supply points. In this article a new memetic algorithm is suggested for the standard capacitated vehicle routing problem. The proposed algorithm combines the edge assembly (EAX) crossover with well‐known local searches and allows for infeasible solutions with respect to capacity and route duration constraints after invoking the crossover. To address the constraint violation, an efficient modification algorithm is also suggested. Experimental tests on 47 standard benchmarks demonstrate that the suggested method is robust and competitive, finding new best‐known solution to 20 well‐studied instances and repeating the existing best‐known solution for 24 problems in a reasonable computing time. © 2009 Wiley Periodicals, Inc. NETWORKS, 2009

We propose an evolutionary algorithm (EA) for the vehicle routing problem with time windows (VRPTW). In particular, a crossover operator suitable for solving the VRPTW are presented. The proposed crossover is designed by extending edge assembly crossover (EAX) which was originally designed for the traveling salesman problem (TSP). Moreover, we propose an efficient modification procedure to eliminate violations of the capacity constraints and the time window constraints because intermediate solutions generated by EAX violate these constraints. The EA is composed of EAX, the modification procedure and a very simple local search. The EA is applied to the well-known Gering and Homberger benchmarks for the VRPTW. The EA finds new best-known solutions in 119 of the 180 instances (200 - 600 customer benchmarks).

Niching ethods are useful extension of evolutionary algorithm that can permit population to search many peaks in parallel in multimodal domains. Traditional niching methods that can effectively be used to solve function optimization problems (FOPs) are not suitable for solving combinatorial optimization problems (COPs). We propose a new genetic algorithm (GA) for forming niches that can be efficiently applied to COPs. For this, we propose four requirements that are needed for an ideal method of forming niches for COPs. We then discuss a niching method we designed that satisfies these requirements. The proposed GA was applied to job shop scheduling problems (JSP) to demonstrate its effectiveness.

This work proposes criteria for designing crossovers for genetic algorithms (GAs) used to solve traveling salesman problems (TSPs). These criteria are [1] that the ratio at which new edges are introduced into offspring by crossovers should be adjusted appropriately and [2] that the quality of the new edges should be high. The validity of these criteria is demonstrated by comparing several crossovers that were designed for TSP with regard to these criteria. Experimental results indicate that the performances of GAs using some previously proposed crossovers can be explained by considering these criteria. Furthermore, experiments with a new crossover designed by modifying a very simple crossover called edge recombination (EX) in accordance with these criteria demonstrate that the performance of the GA using this modified crossover is improved dramatically.