Multimodal Benchmark Problems Research Articles

A Multimodal Differential Evolution Algorithm in Initial Orbit Determination for a Space-Based Too Short Arc

Under the too short arc scenario, the evolutionary-based algorithm has more potential than traditional methods in initial orbit determination. However, the underlying multimodal phenomenon in initial orbit determination is ignored by current works. In this paper, we propose a new enhanced differential evolution (DE) algorithm with multimodal property to study the angle-only IOD problem. Specifically, a coarse-to-fine convergence detector is implemented, based on the Boltzmann Entropy, to determine the evolutionary phase of the population, which lays the basis of the balance between the exploration and exploitation ability. A two-layer niching technique clusters the individuals to form promising niches after each convergence detected. The candidate optima from resulting niches are saved as supporting individuals into an external archive for diversifying the population, and a local search within the archive is performed to refine the solutions. In terms of performance validation, the proposed multimodal differential evolution algorithm is evaluated on the CEC2013 multimodal benchmark problems, and it achieved competitive results compared to 11 state-of-the-art algorithms, which present its capability of multimodal optimization. Moreover, several IOD experiments and analyses are carried out on three simulated scenarios of space-based observation. The findings show that, compared to traditional IOD approaches and EA-based IOD algorithms, the proposed algorithm is more successful at finding plausible solutions while improving IOD accuracy.

Whale optimization and sine–cosine optimization algorithms with cellular topology for parameter identification of chaotic systems and Schottky barrier diode models

This research study aims to enhance the optimization accuracy of the two recently emerged metaheuristics of whale and sine–cosine optimizers by means of the balanced improvements in intensification and diversification phases of the algorithms provided by cellular automata (CA). Stagnation at the early phases of the iterations, which leads to entrapment in local optimum points in the search space, is one of the inherent drawbacks of the metaheuristic algorithms. As a favorable solution alternative to this problem, different types of cellular topologies are implemented into these two algorithms with a view to ameliorating their search mechanisms. Exploitation of the fertile areas in the search domain is maintained by the interaction between the topological neighbors, whereas the improved exploration is resulted from the smooth diffusion of the available population information among the structured neighbors. Numerical experiments have been carried out to assess the optimization performance of the proposed cellular-based algorithms. Optimization benchmark problems comprised of unimodal and multimodal test functions have been applied and numerical results have been compared with those found by some of the state-of-the-art literature optimizers including particle swarm optimization, differential evolution, artificial cooperative search and differential search. Cellular variants have been outperformed by the base algorithms for multimodal benchmark problems of Levy and Penalized1 functions. Then, the proposed cellular algorithms have been applied to two different parameter identification cases in order to test their efficiencies on real-world optimization problems. Extensive performance evaluations on different parameter optimization cases reveal that incorporating the CA concepts on these algorithms not only improves the optimization accuracy but also provides considerable robustness to acquired solutions.

Using Covariance Matrix Adaptation Evolutionary Strategy to boost the search accuracy in hierarchic memetic computations

Many global optimization problems arising naturally in science and engineering exhibit some form of intrinsic ill-posedness, such as multimodality and insensitivity. Severe ill-posedness precludes the use of standard regularization techniques and necessitates more specialized approaches, usually comprised of two separate stages – global phase, that determines the problem's modality and provides rough approximations of the solutions, and a local phase, which refines these approximations.In this work, we attempt to improve one of the most efficient currently known approaches - Hierarchic Memetic Strategy (HMS) - by incorporating the Covariance Matrix Adaptation Evolutionary Strategy (CMA-ES) into its local phase. CMA-ES is a stochastic optimization algorithm that in some sense mimics the behavior of population-based evolutionary algorithms without explicitly evolving the population. This way, it avoids, to an extent, the associated cost of multiple evaluations of the objective function.We compare the performance of the HMS on relatively simple multimodal benchmark problems and on an engineering problem. To do so, we consider two configurations: the CMA-ES and the standard SEA (Simple Evolutionary Algorithm). The results demonstrate that the HMS with CMA-ES in the local phase requires less objective function evaluations to provide the same accuracy, making this approach more efficient than the standard SEA.

The Patch-Levy-Based Bees Algorithm Applied to Dynamic Optimization Problems

Many real-world optimization problems are actually of dynamic nature. These problems change over time in terms of the objective function, decision variables, constraints, and so forth. Therefore, it is very important to study the performance of a metaheuristic algorithm in dynamic environments to assess the robustness of the algorithm to deal with real-word problems. In addition, it is important to adapt the existing metaheuristic algorithms to perform well in dynamic environments. This paper investigates a recently proposed version of Bees Algorithm, which is called Patch-Levy-based Bees Algorithm (PLBA), on solving dynamic problems, and adapts it to deal with such problems. The performance of the PLBA is compared with other BA versions and other state-of-the-art algorithms on a set of dynamic multimodal benchmark problems of different degrees of difficulties. The results of the experiments show that PLBA achieves better results than the other BA variants. The obtained results also indicate that PLBA significantly outperforms some of the other state-of-the-art algorithms and is competitive with others.

Improved particle swarm optimization based on particles' explorative capability enhancement

Accelerating the convergence speed and avoiding the local optimal solution are two main goals of particle swarm optimization (PSO). The very basic PSO model and some variants of PSO do not consider the enhancement of the explorative capability of each particle. Thus these methods have a slow convergence speed and may trap into a local optimal solution. To enhance the explorative capability of particles, a scheme called explorative capability enhancement in PSO (ECE-PSO) is proposed by introducing some virtual particles in random directions with random amplitude. The linearly decreasing method related to the maximum iteration and the nonlinearly decreasing method related to the fitness value of the globally best particle are employed to produce virtual particles. The above two methods are thoroughly compared with four representative advanced PSO variants on eight unimodal and multimodal benchmark problems. Experimental results indicate that the convergence speed and solution quality of ECE-PSO outperform the state-of-the-art PSO variants.

A new Reinforcement Learning-based Memetic Particle Swarm Optimizer

Developing an effective memetic algorithm that integrates the Particle Swarm Optimization (PSO) algorithm and a local search method is a difficult task. The challenging issues include when the local search method should be called, the frequency of calling the local search method, as well as which particle should undergo the local search operations. Motivated by this challenge, we introduce a new Reinforcement Learning-based Memetic Particle Swarm Optimization (RLMPSO) model. Each particle is subject to five operations under the control of the Reinforcement Learning (RL) algorithm, i.e. exploration, convergence, high-jump, low-jump, and fine-tuning. These operations are executed by the particle according to the action generated by the RL algorithm. The proposed RLMPSO model is evaluated using four uni-modal and multi-modal benchmark problems, six composite benchmark problems, five shifted and rotated benchmark problems, as well as two benchmark application problems. The experimental results show that RLMPSO is useful, and it outperforms a number of state-of-the-art PSO-based algorithms.

Novel benchmark functions for continuous multimodal optimization with comparative results

Multi-modal optimization is concerned with locating multiple optima in one single run. Finding multiple solutions to a multi-modal optimization problem is especially useful in engineering, as the best solution may not always be the best realizable due to various practical constraints. To compare the performances of multi-modal optimization algorithms, multi-modal benchmark problems are always required. In this paper, 15 novel scalable multi-modal and real parameter benchmark problems are proposed. Among these 15 problems, 8 are extended simple functions while the rest are composition functions. These functions coordinate rotation and shift operations to create linkage among different dimensions and to place the optima at different locations, respectively. Four typical niching algorithms are used to solve the proposed problems. As shown by the experimental results, the proposed problems are challenging to these four recent algorithms.

$$\mu $$ μ JADE: adaptive differential evolution with a small population

This paper proposes a new differential evolution (DE) algorithm for unconstrained continuous optimisation problems, termed $$\mu $$μJADE, that uses a small or `micro' ($$\mu $$μ) population. The main contribution of the proposed DE is a new mutation operator, `current-by-rand-to-pbest.' With a population size less than 10, $$\mu $$μJADE is able to solve some classical multimodal benchmark problems of 30 and 100 dimensions as reliably as some state-of-the-art DE algorithms using conventionally sized populations. The algorithm also compares favourably to other small population DE variants and classical DE.

Ensemble and Arithmetic Recombination-Based Speciation Differential Evolution for Multimodal Optimization.

Multimodal optimization problems consists of multiple equal or comparable spatially distributed solutions. Niching and clustering differential evolution (DE) techniques have been demonstrated to be highly effective for solving such problems. The key challenge in the speciation niching technique is to balance between local solution exploitation and global exploration. Our proposal enhances exploration by applying arithmetic recombination with speciation and improves exploitation of individual peaks by applying neighborhood mutation with ensemble strategies. Our novel algorithm, called ensemble and arithmetic recombination-based speciation DE, is shown to either outperform or perform comparably to the state-of-the-art algorithms on 29 common multimodal benchmark problems. Comparable performance is observed only when some problems are solved perfectly by the algorithms in the literature.

A New Efficient Meta-Heuristic Optimization Algorithm Inspired by Wild Dog Packs

Although meta-heuristic optimization algorithms have been used to solve many optimization problems, they still suffer from two main difficulties: What are the best parameters for a particular problem? How do we escape from the local optima? In this paper, a new, efficient meta-heuristic optimization algorithm inspired by wild dog packs is proposed. The main idea involves using three self-competitive parameters that are similar to the smell strength. The parameters are used to control the movement of the alpha dogs and, consequently, the movement of the whole pack. The rest of the pack is used to explore the neighboring area of the alpha dog, while the hoo procedure is used to escape from the local optima. The suggested method is applied to several unimodal and multimodal benchmark problems and is compared to five modern meta-heuristic algorithms. The experimental results show that the new algorithm outperforms other peer algorithms.

On the efficacy of chaos-enhanced heuristic walks with nature-based controllers for robust and accurate intelligent search, part A: an experimental analysis

In this investigation, the authors intend to find out whether chaos-embedded heuristic walks with nature-based controllers are capable of increasing the performance of the standard random walks of evolutionary algorithms. Indeed, the research motivation emanates in the pursuit of addressing an increasing interest in using chaos sequences for modifying the exploitation capability of evolutionary optimisers. Here, the authors propose a nature-based controller to dynamically balance the exploration/exploitation capability of the evolutionary optimiser. To test the validity of the proposed method, a number of well-known Gaussian distribution-based evolutionary operators are considered, and their random parameters are changed with discrete-time chaotic sequences. Besides, to obtain authentic and reliable results regarding the accuracy and robustness of the rival techniques, the authors went through the existing literature and extracted 23 scalable, multimodal, constraint and unconstraint numerical benchmark problems and one engineering problem, i.e. optimal control of shape memory alloy actuators. The convergence rate and computational time of the resulting self-controlled chaos-based evolutionary approaches are analysed to find out whether they are capable of stabilising the chromosomes within a logical time. The results of the numerical experiments indicate that the self-controlled chaos-enhanced heuristic walks significantly increase the global searching capability of the evolutionary optimisers and have an aptitude to show a robust performance.

An Adaptive Cauchy Differential Evolution Algorithm with Bias Strategy Adaptation Mechanism for Global Numerical Optimization

Appropriately adapting mutation strategies is a challengeable problem of the literature of the Differential Evolution (DE). The Strategy adaptation Mechanism (SaM) can convert a control parameter adaptation algorithm to a strategy adaptation algorithm. To improve the quality of optimization result, the exploration property is important in the early stage of optimization process and the exploitation property is significant in the late stage of optimization process. To ensure these, we modified the SaM for strictly controlling a balance between the exploration and the exploitation properties, which called the bias SaM (bSaM). We extended the Adaptive Cauchy Differential Evolution (ACDE) by attaching the bSaM. We compared the bSaM with SaM and the bSaM extended ACDE with the state-of-the-art DE algorithms in various benchmark problems. The result of the performance evaluation showed that the bSaM and the bSaM extended ACDE performs better than SaM and the state-of-the-art DE algorithms not only unimodal but also multimodal benchmark problems.

Two-layer particle swarm optimization with intelligent division of labor

Early studies in particle swarm optimization (PSO) algorithm reveal that the social and cognitive components of swarm, i.e. memory swarm, tend to distribute around the problem's optima. Motivated by these findings, we propose a two-layer PSO with intelligent division of labor (TLPSO-IDL) that aims to improve the search capabilities of PSO through the evolution memory swarm. The evolution in TLPSO-IDL is performed sequentially on both the current swarm and the memory swarm. A new learning mechanism is proposed in the former to enhance the swarm's exploration capability, whilst an intelligent division of labor (IDL) module is developed in the latter to adaptively divide the swarm into the exploration and exploitation sections. The proposed TLPSO-IDOL algorithm is thoroughly compared with nine well-establish PSO variants on 16 unimodal and multimodal benchmark problems with or without rotation property. Simulation results indicate that the searching capabilities and the convergence speed of TLPSO-IDL are superior to the state-of-art PSO variants.

Dynamic Diversity Enhancement in Particle Swarm Optimization (DDEPSO) Algorithm for Preventing from Premature Convergence

The problem of early convergence in the Particle Swarm Optimization (PSO) algorithm often causes the search process to be trapped in a local optimum. This problem often occurs when the diversity of the swarm decreases and the swarm cannot escape from a local optimal. In this paper, a novel dynamic diversity enhancement particle swarm optimization (DDEPSO) algorithm is introduced. In this variant of PSO, we periodically replace some of the swarm's particles by artificial ones, which are generated based on the history of the search process, in order to enhance the diversity of the swarm and promote the exploration ability of the algorithm. Afterwards, we update the velocity of the artificial particles in corresponding generating period by a new velocity equation with the minimum inertia weight in order to enhance the exploitation potentiality of the swarm. The performance of this approach has been tested on the set of twelve standard unimodal and multimodal (Rotated or unrotated) benchmark problems and the results have been compared with our previous work as well as four other variants of the PSO algorithm. The numerical results demonstrate that the proposed algorithm outperforms others in most of the test cases taken in this study.

Hybrid enhanced continuous tabu search and genetic algorithm for parameter estimation in colored noise environments

Parameter estimation is an important concept in engineering where a mathematical model of a system is identified with the help of input and output signals. Classical parameter estimation algorithms such as Least Squares (LS), Recursive Least Squares (RLS), Least Mean Squares (LMS) and Generalized Least Squares (GLS) give an unbiased estimate of the parameters when the system noise is white. This property is lost when the system noise is colored which is generally the case for many practical situations. In order to overcome the bias problem associated with the colored noise environment, one can use a whitening filter. The cost function of the estimation problem in the case of a colored noise environment becomes multimodal when the signal to noise ratio is high. Hence the motivation to use some intelligent optimization technique for the purpose of finding the global minimum of the parameter estimation problem. A new hybrid algorithm combining intelligent optimization techniques, i.e. enhanced continuous tabu search (ECTS) and elitism based genetic algorithm (GA) is proposed and is applied to the parameter estimation problem. In this work, the ECTS is used to define smaller search spaces, which are investigated in a second stage by a GA to find the respective local minima. Simulation results show that the parameters estimated using the proposed algorithm is unbiased in the presence colored noise. In addition, the hybrid algorithm is also tested with known multimodal benchmark problems.

AN ASYNCHRONOUS PARALLEL EVOLUTION STRATEGY

This work introduces a new asynchronous parallel self-adaptive evolution strategy. An asynchronous scheme is chosen because the load on the processors is usually unbalanced. The proposed algorithm is non-blocking leaving no processor idle at any given time. The motivation and theory for the algorithm is based on the author's previous analysis of continuous evolution strategies9 and the reduction of random fluctuation in mutative self-adaptation.7 The algorithm is implemented in C ++ using Posix threads and tested on a number of unimodal and multimodal benchmark problems. The result is an efficient and effective numerical optimization strategy for multi-processor machines.