Multimodal Function Optimization Research Articles

On polling directions for randomized direct-search approaches: application to beam angle optimization in intensity-modulated proton therapy

AbstractDeterministic direct-search methods have been successfully used to address real-world challenging optimization problems, including the beam angle optimization (BAO) problem in radiation therapy treatment planning. BAO is a highly non-convex optimization problem typically treated as the optimization of an expensive multi-modal black-box function which results in a computationally time consuming procedure. For the recently available modalities of radiation therapy with protons (instead of photons) further efficiency in terms of computational time is required despite the success of the different strategies developed to accelerate BAO approaches. Introducing randomization into otherwise deterministic direct-search approaches has been shown to lead to excellent computational performance, particularly when considering a reduced number (as low as two) of random poll directions at each iteration. In this study several randomized direct-search strategies are tested considering different sets of polling directions. Results obtained using a prostate and a head-and-neck cancer cases confirmed the high-quality results obtained by deterministic direct-search methods. Randomized strategies using a reduced number of polling directions showed difficulties for the higher dimensional search space (head-and-neck) and, despite the excellent mean results for the prostate cancer case, outliers were observed, a result that is often ignored in the literature. While, for general global optimization problems, mean results (or obtaining the global optimum once) might be enough for assessing the performance of the randomized method, in real-world problems one should not disregard the worst-case scenario and beware of the possibility of poor results since, many times, it is only possible to run the optimization problem once. This is even more important in healthcare applications where the mean patient does not exist and the best treatment possible must be assured for every patient.

Mother optimization algorithm: a new human-based metaheuristic approach for solving engineering optimization

This article’s innovation and novelty are introducing a new metaheuristic method called mother optimization algorithm (MOA) that mimics the human interaction between a mother and her children. The real inspiration of MOA is to simulate the mother’s care of children in three phases education, advice, and upbringing. The mathematical model of MOA used in the search process and exploration is presented. The performance of MOA is assessed on a set of 52 benchmark functions, including unimodal and high-dimensional multimodal functions, fixed-dimensional multimodal functions, and the CEC 2017 test suite. The findings of optimizing unimodal functions indicate MOA’s high ability in local search and exploitation. The findings of optimization of high-dimensional multimodal functions indicate the high ability of MOA in global search and exploration. The findings of optimization of fixed-dimension multi-model functions and the CEC 2017 test suite show that MOA with a high ability to balance exploration and exploitation effectively supports the search process and can generate appropriate solutions for optimization problems. The outcomes quality obtained from MOA has been compared with the performance of 12 often-used metaheuristic algorithms. Upon analysis and comparison of the simulation results, it was found that the proposed MOA outperforms competing algorithms with superior and significantly more competitive performance. Precisely, the proposed MOA delivers better results in most objective functions. Furthermore, the application of MOA on four engineering design problems demonstrates the efficacy of the proposed approach in solving real-world optimization problems. The findings of the statistical analysis from the Wilcoxon signed-rank test show that MOA has a significant statistical superiority compared to the twelve well-known metaheuristic algorithms in managing the optimization problems studied in this paper.

Nonlinear computer image scene and target information extraction based on big data technology

Abstract To explore the extraction of computer image scene and target information, a nonlinear method based on big data technology is proposed. The method can decompose the computer image into a plurality of components when the SAR computer image is processed such as target extraction and computer image compression, which represent different captured image features, respectively. Selecting the most suitable processing method according to the characteristics of different components can greatly improve the performance. Using nonlinear diffusion method, the computer image is decomposed into structural components representing large-scale structural information and texture components representing small-scale detailed information, and the automatic threshold estimation in the diffusion process is studied. The LAIDA criterion is introduced into the automatic threshold solution of nonlinear diffusion-based computer image decomposition to test and evaluate the diffusion process of various diffusion parameter forms. The results show that the experimental outcome of the diffusion decomposition based on automatic threshold estimation is very close on each index, which shows that using automatic threshold estimation, no matter what diffusion index is used, very close results can be obtained. Specifically, for each algorithm, the parameter estimation threshold l for outliers plays an obvious role. The third is the degree of initiative of the estimation process. The larger the L, the larger the outlier, which will lead to a greater extent of the diffusion process, resulting in a continuous decrease in the structural similarity index and compositional correlation. It is proved that the algorithm has strong global search ability, can effectively avoid premature convergence, has fast convergence speed, and good long stability. It can be widely used for optimization of various multimodal functions.

Memory, evolutionary operator, and local search based improved Grey Wolf Optimizer with linear population size reduction technique

Optimization of multi-modal functions is challenging even for evolutionary and swarm-based algorithms as it requires an efficient exploration for finding the promising region of the search space, and effective exploitation to precisely find the global optimum. Grey Wolf Optimizer (GWO) is a recently developed metaheuristic algorithm that is inspired by nature with a relatively small number of parameters for tuning. However, GWO and most of its variants may suffer from the lack of population diversity, premature convergence, and the inability to preserve a good balance between exploratory and exploitative behaviors. To address these limitations, this work proposes a new variant of GWO incorporating memory, evolutionary operators, and a stochastic local search technique. It further integrates Linear Population Size Reduction (LPSR) technique. The proposed algorithm is comprehensively tested on 23 numerical benchmark functions, high dimensional benchmark functions, 13 engineering case studies, four data classifications, and three function approximation problems. The benchmark functions are mostly taken from the CEC 2005 and CEC 2010 special sessions, and they include rotated, shifted functions. The engineering case studies are from the CEC 2020 real-world non-convex constrained optimization problems. The performance of the proposed GWO is compared with popular metaheuristics, namely, particle swarm optimization (PSO), gravitational search algorithm (GSA), slap swarm algorithm (SSA), differential evolution (DE), self-adaptive differential evolution (SADE), basic GWO and its three recently improved variants. Statistical analysis and Friedman tests have been conducted to thoroughly compare their performance. The obtained results demonstrate that the proposed GWO outperforms the algorithms compared for the benchmark functions and engineering case studies tested.

Explaining Exploration–Exploitation in Humans

Human as well as algorithmic searches are performed to balance exploration and exploitation. The search task in this paper is the global optimization of a 2D multimodal function, unknown to the searcher. Thus, the task presents the following features: (i) uncertainty (i.e., information about the function can be acquired only through function observations), (ii) sequentiality (i.e., the choice of the next point to observe depends on the previous ones), and (iii) limited budget (i.e., a maximum number of sequential choices allowed to the players). The data about human behavior are gathered through a gaming app whose screen represents all the possible locations the player can click on. The associated value of the unknown function is shown to the player. Experimental data are gathered from 39 subjects playing 10 different tasks each. Decisions are analyzed in a Pareto optimality setting—improvement vs. uncertainty. The experimental results show that the most significant deviations from the Pareto rationality are associated with a behavior named “exasperated exploration”, close to random search. This behavior shows a statistically significant association with stressful situations occurring when, according to their current belief, the human feels there are no chances to improve over the best value observed so far, while the remaining budget is running out. To classify between Pareto and Not-Pareto decisions, an explainable/interpretable Machine Learning model based on Decision Tree learning is developed. The resulting model is used to implement a synthetic human searcher/optimizer successively compared against Bayesian Optimization. On half of the test problems, the synthetic human results as more effective and efficient.

A new point-infilling and level-setting algorithm enhanced by heuristic searching and incremental clustering to solve global optimization problems

A new global algorithm of point-infilling and level-setting integrating heuristic searching and incremental clustering is proposed for multi-modal function optimization. This algorithm has an iteration body consisting of point infilling, level-setting, heuristic searching and incremental clustering. Its prime idea is to use continuous multiple rounds of point-infilled sampling plus heuristic search to find the global optimum. In each iteration level-set operations are executed after point-infilled sampling to remove some bad points. Heuristic searching by particle swarm optimization (PSO) or genetic algorithm (GA) are further executed to heighten the precision and meanwhile to find more new possible global optima. Clustering operations, with spectral clustering analysis to determine the cluster number of the initial points set and then performed by adaptive incremental clustering, are further used to classify points close to different optima into different clusters. The prime frame of the proposed algorithm is first presented, its detailed realization is further demonstrated by optimizing Rastrigin’s function, and the theoretical proving is further presented. The experiments of optimizing ten typical multi-modal functions, a combined CEC benchmark set and an engineering problem are done. The statistical comparisons with other eight algorithms are completed and the results indicate that the proposed algorithm has better effect.

OPTIMISASI FUNGSI RASTRIGIN MENGGUNAKAN FLOWER POLLINATION ALGORITHM

The Rastrigin function is a multimodal function. It is difficult to find a global minimum of the function because it has many local minimums. So, we need an effective and efficient algorithm to find a solution to the global minimum of the function without being trapped by the local minimum. The flower pollination algorithm is a metaheuristic algorithm, it is expected to be capable of solving multimodal function optimization problems. In this study flower pollination algorithm is used to find the global minimum of the Rastrigin function of two variables with MATLAB. The Rastrigin function of two variables is used as objective function for the flower pollination algorithm. The parameters are divided into three configurations based on the difference amount of pollen gamets, the probability switch, and the search domain, with two different iterations 300 and 1500. In order, to get the best results each configuration is running for 10 times. The best results from the flower pollination algorithm are obtained from the first configuration and 1500 iterations

A Metaheuristic Algorithm for Coverage Enhancement of Wireless Sensor Networks

When wireless sensors are randomly deployed in natural environments such as ecological monitoring, military monitoring, and disaster monitoring, the initial position of sensors is generally formed through deployment methods such as air-drop, and then, the second deployment is carried out through the existing optimization methods, but these methods will still lead to serious coverage holes. In order to solve this problem, this paper proposes an algorithm to improve the coverage rate for wireless sensor networks based on an improved metaheuristic algorithm. The sensor deployment coverage model was firstly established, and the sensor network coverage problem was transformed into a high-dimensional multimodal function optimization problem. Secondly, the global searching ability and searching range of the algorithm are enhanced by the reverse expansion of the initial populations. Finally, the firefly principle is introduced to reduce the local binding force of sparrows and avoid the local optimization problem of the population in the search process. The experimental results showed that compared with ALO, GWO, BES, RK, and SSA algorithms, the EFSSA algorithm is better than other algorithms in benchmark function tests, especially in the test of high-dimensional multimodal function. In the tests of different monitoring ranges and number of nodes, the coverage of EFSSA algorithm is higher than other algorithms. The result can tell that EFSSA algorithm can effectively enhance the coverage of sensor deployment.

Performance evaluation of the particle swarm optimization for clustering based on different parameter selection

Particle Swamp Optimization (PSO) is an effective method for solving a wide range of problems. However, the most existing PSO algorithms easily trap into local optima when solving complex multimodal function optimization problems. In this paper, we explain the importance of PSO algorithm’s general purposes to optimize strategy which has various parameters that decide its conduct and viability in advancing a given issue. This study gives a rundown of the best selections of parameters for different advancement situations which should enable the specialist to accomplish better outcomes with less exertion. In this paper, we define an important in the PSO algorithm, the parameters and how to apply this algorithm to different type of datasets online and offline. Repeat Consumption Matrices Dataset has been tested with PSO algorithm to define the number of clusters using selection of parameters. The results are then having been compared with the Genetic Algorithm (GA) where the algorithms have been tested based on the number of cluster and DB index for each of the data. Based on the results, it is shown that the PSO algorithm gives good and efficient results compared to GA algorithm. The output of the research is a PSO-based clustering algorithm that can be used in data mining by providing accurate and robust results in clustering, which can be realized in web search engines and automatic doc-ument organization.

A Hybrid Genetic Algorithm and Sperm Swarm Optimization (HGASSO) for Multimodal Functions

In this paper, we propose a hybrid algorithm combining two different metaheuristic methods, “Genetic Algorithms (GA)” and “Sperm Swarm Optimization (SSO)”, for the global optimization of multimodal benchmarks functions. The proposed Hybrid Genetic Algorithm and Sperm Swarm Optimization (HGASSO) operates based on incorporates concepts from GA and SSO in which generates individuals in a new iteration not only by crossover and mutation operations as proposed in GA, but also by techniques of local search of SSO. The main idea behind this hybridization is to reduce the probability of trapping in local optimum of multi modal problem. Our algorithm is compared against GA, and SSO metaheuristic optimization algorithms. The experimental results using a suite of multimodal benchmarks functions taken from the literature have evinced the superiority of the proposed HGASSO approach over the other approaches in terms of quality of results and convergence rates in which obtained good results in solving the multimodal benchmarks functions that include cosine, sine, and exponent in their formulation.

Cooperative particle swarm optimizer with depth first search strategy for global optimization of multimodal functions

Cooperative particle swarm optimizer with depth first search strategy for global optimization of multimodal functions

SSABA: Search Step Adjustment Based Algorithm

Finding the suitable solution to optimization problems is a fundamental challenge in various sciences. Optimization algorithms are one of the effective stochastic methods in solving optimization problems. In this paper, a new stochastic optimization algorithm called Search Step Adjustment Based Algorithm (SSABA) is presented to provide quasi-optimal solutions to various optimization problems. In the initial iterations of the algorithm, the step index is set to the highest value for a comprehensive search of the search space. Then, with increasing repetitions in order to focus the search of the algorithm in achieving the optimal solution closer to the global optimal, the step index is reduced to reach the minimum value at the end of the algorithm implementation. SSABA is mathematically modeled and its performance in optimization is evaluated on twenty-three different standard objective functions of unimodal and multimodal types. The results of optimization of unimodal functions show that the proposed algorithm SSABA has high exploitation power and the results of optimization of multimodal functions show the appropriate exploration power of the proposed algorithm. In addition, the performance of the proposed SSABA is compared with the performance of eight well-known algorithms, including Particle Swarm Optimization (PSO), Genetic Algorithm (GA), Teaching-Learning Based Optimization (TLBO), Gravitational Search Algorithm (GSA), Grey Wolf Optimization (GWO), Whale Optimization Algorithm (WOA), Marine Predators Algorithm (MPA), and Tunicate Swarm Algorithm (TSA). The simulation results show that the proposed SSABA is better and more competitive than the eight compared algorithms with better performance.

Bi-Attempted Based Optimization Algorithm For Numerical Optimization Problems

Numerical optimization is one of the well-known problems in computer science. Day by day, new methods are developed by many researchers. Recently, optimization became an essential task for many disciplines, such as engineering, medicine, management and others. In many cases, optimization problems may require fast and efficient algorithms for real-time implementations. In this paper, a simple, fast and feasible algorithm is presented for the optimization of both uni-modal and multi-modal benchmark functions. A population based Bi-Attempted Based Optimization Algorithm (ABaOA) is a stochastic search method which searches a solution space with two fixed step-size displacement parameters and two mutation operators. The proposed algorithm is derived from Base Optimization Algorithm (BaOA) which uses basic arithmetic operations. The performance of ABaOA is tested on twenty well-known benchmark functions and the results are statistically compared with the seven well-known stochastic optimization algorithms. Three different statistical analyses were done on the results obtained from the ABaOA. Two non-parametric statistical comparisons with the mean values are performed by using Sign and Wilcoxon tests. The non-parametric statistical multiple comparisons of the proposed algorithm is performed by using the Friedman test. The non-parametric Friedman test of differences among repeated measures of these algorithms was conducted and referred a Chi-square value of 67.337, which was significant (p<0.05). Wilcoxon non-parametric pairwise comparison test was applied to analyze the difference of ABaOA statistically among the other algorithms. The test indicates that the introduced algorithm is statistically significant than other algorithms with a level of significance p < 0.05. The experimental results also show that the ABaOA is clearly superior to the compared stochastic optimization algorithms.

Parallel sine cosine algorithm for the dynamic deployment in wireless sensor networks

All along, people have a high enthusiasm for the research of optimization algorithm. A large number of new algorithms and methods have emerged. The sine cosine algorithm (SCA) is an excellent algorithm that has appeared in recent years. It is a stochastic optimization algorithm based on population. Compared with the existing algorithms, SCA is a suitable solution to different optimization problems, especially the optimization of unimodal functions. It is qualified to optimize real-world problems with unknown and limited search space. But sometimes it does not perform satisfactorily when dealing with some specific problems, such as optimization of multimodal functions or composite functions. This paper presents a parallel version of the sine cosine algorithm (PSCA) with three communication strategies. Different strategies can be selected according to the type of optimization function to achieve better results. We have repeatedly tested different types of functions, and the results show that the proposed PSCA can solve the optimization problem more specifically. In the simulation of wireless sensor network (WSN) dynamic deployment optimization, it is found that using this method can get the ideal sensor node distribution, which makes PSCA’s performance in solving other practical problems worth looking forward to.

Adaptive Multi-Level Search for Global Optimization: An Integrated Swarm Intelligence-Metamodelling Technique

Over the last decade, metaheuristic algorithms have emerged as a powerful paradigm for global optimization of multimodal functions formulated by nonlinear problems arising from various engineering subjects. However, numerical analyses of many complex engineering design problems may be performed using finite element method (FEM) or computational fluid dynamics (CFD), by which function evaluations of population-based algorithms are repetitively computed to seek a global optimum. It is noted that these simulations become computationally prohibitive for design optimization of complex structures. To efficiently and effectively address this class of problems, an adaptively integrated swarm intelligence-metamodelling (ASIM) technique enabling multi-level search and model management for the optimal solution is proposed in this paper. The developed technique comprises two steps: in the first step, a global-level exploration for near optimal solution is performed by adaptive swarm-intelligence algorithm, and in the second step, a local-level exploitation for the fine optimal solution is studied on adaptive metamodels, which are constructed by the multipoint approximation method (MAM). To demonstrate the superiority of the proposed technique over other methods, such as conventional MAM, particle swarm optimization, hybrid cuckoo search, and water cycle algorithm in terms of computational expense associated with solving complex optimization problems, one benchmark mathematical example and two real-world complex design problems are examined. In particular, the key factors responsible for the balance between exploration and exploitation are discussed as well.

A Rank based Adaptive Mutation in Genetic Algorithm

Traditionally Genetic Algorithm has been used for optimization of unimodal and multimodal functions. Earlier researchers worked with constant probabilities of GA control operators like crossover, mutation etc. for tuning the optimization in specific domains. Recent advancements in this field witnessed adaptive approach in probability determination. In Adaptive mutation primarily poor individuals are utilized to explore state space, so mutation probability is usually generated proportionally to the difference between fitness of best chromosome and itself (fMAX - f). However, this approach is susceptible to nature of fitness distribution during optimization. This paper presents an alternate approach of mutation probability generation using chromosome rank to avoid any susceptibility to fitness distribution. Experiments are done to compare results of simple genetic algorithm (SGA) with constant mutation probability and adaptive approaches within a limited resource constraint for unimodal, multimodal functions and Travelling Salesman Problem (TSP). Measurements are done for average best fitness, number of generations evolved and percentage of global optimum achievements out of several trials. The results demonstrate that the rank-based adaptive mutation approach is superior to fitness-based adaptive approach as well as SGA in a multimodal problem space.

A Novel Selection Approach for Genetic Algorithms for Global Optimization of Multimodal Continuous Functions.

Genetic algorithms (GAs) are stochastic-based heuristic search techniques that incorporate three primary operators: selection, crossover, and mutation. These operators are supportive in obtaining the optimal solution for constrained optimization problems. Each operator has its own benefits, but selection of chromosomes is one of the most essential operators for optimal performance of the algorithms. In this paper, an improved genetic algorithm-based novel selection scheme, i.e., stairwise selection (SWS) is presented to handle the problems of exploration (population diversity) and exploitation (selection pressure). For its global performance, we compared with several other selection schemes by using ten well-known benchmark functions under various dimensions. For a close comparison, we also examined the significance of SWS based on the statistical results. Chi-square goodness of fit test is also used to evaluate the overall performance of the selection process, i.e., mean difference between observed and expected number of offspring. Hence, the overall empirical results along with graphical representation endorse that the SWS outperformed in terms of robustness, stability, and effectiveness other competitors through authentication of performance index (PI).

A New Variable Step-size and Updating Search Domains Glowworm Swarm Optimization Algorithm

To address slow convergence of Glowworm Swarm Optimization (GSO) algorithm in the late stage of optimizing multimodal functions, a new variable step-size glowworm swarm optimization (VSGSO) algorithm is proposed, which improves convergence performance by adaptively adjusting step-size based on luciferin carried by each glowworm. Based on VSGSO, VSGSO-D algorithm, which is capable of updating search domains, is proposed. It dramatically improves optimization precision of GSO, especially in the case of optimization of complex multimodal functions. Optimization experiments are carried out by ten benchmark functions. The experimental results show that the improved algorithms proposed in this paper improve the global optimization speed, precision, and stability of GSO. At the same time, the enhanced algorithms protect the diversity of glowworm population and improve the multi-local optimization ability of GSO.

Fractional-Order Spatial Steganography and Blind Steganalysis for Printed Matter: Anti-Counterfeiting for Product External Packing in Internet-of-Things

This paper advocates a novel conceptual formulation of the fractional-order spatial steganography (FSS) and blind steganalysis for printed matter, which can be efficiently employed in the anti-counterfeiting for product external packing in Internet-of-Things (IoT). Traditional digital steganography is not printable. Within the limits of our knowledge, until now, there are not a well-established steganography and a corresponding steganalysis for printed matter in IoT, which should receive desired attention. Fractional calculus has potentially received prominence in applications in the domain of image processing mainly because of its strengths like long-term memory, nonlocality, and weak singularity. Therefore, in an attempt to overcome the aforementioned technical limitation of traditional digital steganography, this paper has studied here, as an interesting theoretical problem, would it be possible to apply the capability of preserving the edges and textural details of fractional calculus to the achievement of the steganography and steganalysis for printed matter in IoT. Motivated by this inspiration, in this work, this paper introduces a novel conceptual formulation of an FSS and a fractional-order blind steganalysis (FBS) for printed matter. At first, according to the opponent process theory of color vision, to better achieve the imperceptibility of the hidden secret information, this paper uses both the self-similar complex textures in a neighborhood and the opponent channel of blue versus yellow to implement FSS for printed matter. Second, without requiring a priori knowledge regarding the characteristics of the original carrier image, hidden secret image, and steganography, an FBS, a fractional-order multimodal function optimization algorithm, is proposed. Finally, the efficient capability of hiding secret information of FSS and that of detecting secret information of FBS are analyzed in detail experimentally, respectively. These two important advantages lead to the superiority of the proposed approach for defending against statistics attack, rotation and distortion attack, cropping attack, scaling attack, noise attack, and color copy attack. The main contribution of this paper is the first preliminary attempt of a feasible achievement of a spatial steganography and a blind steganalysis for printed matter.

Research and implementation of parallel artificial bee colony algorithm based on ternary optical computer

The artificial bee colony (ABC) algorithm is a widely used algorithm in the field of function optimization problems. The traditional ABC algorithm has long search time, slow convergence speed and easy to fall into local optimum at the end of the search. In this paper, the design scheme and method of implementing parallel ABC algorithm are studied, which makes use of the characteristics of many data bits and easy expansion of data bits of the ternary optical computer (TOC). First, by analysing the traditional ABC algorithm, we can find the parallel parts and parallel design. Then the detailed algorithm implementation flow is given and the clock cycle of the algorithm is analysed. Finally, the correctness of the parallel scheme is verified by experiments. Compared with the ABC algorithm and parallel ABC algorithms based on computer (PABC), the ABC algorithm based on TOC (TOC-PABC) effectively shortens the search time, improves the optimization performance of complex multimodal function optimization problems and obtains a higher speedup.