Global Optimization Of Multimodal Functions Research Articles

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

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.

Comprehensive learning gravitational search algorithm for global optimization of multimodal functions

In this paper, a new comprehensive learning gravitational search algorithm (CLGSA) is proposed to enhance the performance of basic GSA. The proposed algorithm is a new kind of intelligent optimization algorithm which has better ability to choose good elements. An intensive comprehensive learning methodology is proposed to enrich the optimization ability of the GSA. The efficiency of the proposed algorithm was evaluated by 28 benchmark functions which have been proposed in IEEE-CEC 2013 sessions. The results are compared with eight state-of-the-art algorithms IPOP, BIPOP, NIPOP, NBIPOP, DE/rand, SPSRDEMMS, SPSO-2011 and GSA. A variety of ways are considered to examine the ability of the proposed technique in terms of convergence ability, success rate and statistical behavior of algorithm over dimensions 10, 30 and 50. Apart from experimental studies, theoretical stability of the proposed CLGSA is also proved. It was concluded that the proposed algorithm performed efficiently with good results.

A new swarm algorithm for global optimization of multimodal functions over multi-threading architecture hybridized with simulating annealing

This paper presents a new algorithm, PCLPSO, based on particle swarm optimization, which uses comprehensive learning particle swarm optimizer. Our algorithm executes C parallel CLPSO algorithms. We adopted as a criterion of completion a maximum value of evaluations of the objective function. During the execution of the CLPSO algorithms, when a certain evaluation value of the functions is reached, the best k are selected, and different initialization criteria are applied to continue the execution of the CLPSO algorithms: restarting the worst ones for the best solution or restores the worst ones to a random solution. For this restart, we use the Boltzmann criterion in a similar way as Simulating Annealing (SA) does. In this work, the experimental results obtained for the search of the minimum of 16 multimodal test functions such as Rosenbrock, Griewank, Rastrigin, Brannin, Schwefel, and others. Our algorithm proved to be more efficient than the traditional CLPSO in its experimental results, and the nonparametric Wilcoxon test confirmed this.

Artificial Acari Optimization as a new strategy for global optimization of multimodal functions

In the article, the authors present the application of new optimization methods called Artificial Acari Optimization (AAO) and analyze it with other known algorithms for specific groups of benchmarks. The authors were inspired by the social behavior of Acari (mites) creating a strategy for herd searching for solutions. In the article results obtained for AAO were compared with other algorithms belonging to swarm intelligence: particle swarm optimization (PSO), glowworm swarm based optimization algorithm (GSO) and artificial bee colony (ABC). In a benchmark test four functions often used in testing methods of optimization were applied. These functions include: Ackley – with a relatively uniform surface, having tens of local minimums and one global maximum with a much lower value than most of the local minimums; Sphere with a single plane, having just one minimum; Easom characterized by a constant plane over overwhelming majority of domain with only one steep minimum, and Eggholder function with uneven surface having dozens of local minimums, where local minimums are close to the value of the only one global minimum.

Social group optimization for global optimization of multimodal functions and data clustering problems

Cost and physical constraints in the engineering applied problems obligate finding the best results that global optimization algorithms cannot realize. For accurate and faster optimization, switching between known multiple local/global solutions is necessary. The current work proposed a social group optimization (SGO) for solving multimodal functions as well as data clustering problems. For solving global optimization problems, the SGO inspired by the social behavior of human toward solving a complex problem was applied. The SGO is a population-based optimization algorithm using solution population to reach global solution. The simulation results compared its performance with eight particle swarm optimizer variants. The results demonstrated good performance of the SGO.

Finite life span for improving the selection scheme in evolution strategies

Evolution strategies traditionally employ the comma or the plus scheme for selection. The concept of intermediate selection schemes was introduced in 1995, in which individuals may survive limited but more than one generation; however, research on this topic has remained dormant. During the recent decade, the comma scheme has emerged as the preferred choice at least for continuous optimization. The objective of this article is to explore pros and cons of both extreme as well as intermediate selection schemes in coping with different types of difficulties that may arise in global optimization of multimodal functions. A specific test suite is developed which allows for studying different selection schemes when facing particular challenges. A modified variant of the previously proposed intermediate selection scheme in which fitness of survivors iteratively degrades is proposed. The degrading rate is controlled by a strategy parameter, called the aging factor. The resulting algorithm is tested on the whole suite using different values of the aging factor. The comparison of results reveals the drastic problem dependence of the optimal aging factor; however, when all situations are considered, an intermediate selection scheme significantly outperforms both the comma and the plus schemes.

A composite particle swarm algorithm for global optimization of multimodal functions

During the last decade, many variants of the original particle swarm optimization (PSO) algorithm have been proposed for global numerical optimization, but they usually face many challenges such as low solution quality and slow convergence speed on multimodal function optimization. A composite particle swarm optimization (CPSO) for solving these difficulties is presented, in which a novel learning strategy plus an assisted search mechanism framework is used. Instead of simple learning strategy of the original PSO, the proposed CPSO combines one particle’s historical best information and the global best information into one learning exemplar to guide the particle movement. The proposed learning strategy can reserve the original search information and lead to faster convergence speed. The proposed assisted search mechanism is designed to look for the global optimum. Search direction of particles can be greatly changed by this mechanism so that the algorithm has a large chance to escape from local optima. In order to make the assisted search mechanism more efficient and the algorithm more reliable, the executive probability of the assisted search mechanism is adjusted by the feedback of the improvement degree of optimal value after each iteration. According to the result of numerical experiments on multimodal benchmark functions such as Schwefel, Rastrigin, Ackley and Griewank both with and without coordinate rotation, the proposed CPSO offers faster convergence speed, higher quality solution and stronger robustness than other variants of PSO.

A Hybrid PSO-BFGS Strategy for Global Optimization of Multimodal Functions

Particle swarm optimizer (PSO) is a powerful optimization algorithm that has been applied to a variety of problems. It can, however, suffer from premature convergence and slow convergence rate. Motivated by these two problems, a hybrid global optimization strategy combining PSOs with a modified Broyden-Fletcher-Goldfarb-Shanno (BFGS) method is presented in this paper. The modified BFGS method is integrated into the context of the PSOs to improve the particles' local search ability. In addition, in conjunction with the territory technique, a reposition technique to maintain the diversity of particles is proposed to improve the global search ability of PSOs. One advantage of the hybrid strategy is that it can effectively find multiple local solutions or global solutions to the multimodal functions in a box-constrained space. Based on these local solutions, a reconstruction technique can be adopted to further estimate better solutions. The proposed method is compared with several recently developed optimization algorithms on a set of 20 standard benchmark problems. Experimental results demonstrate that the proposed approach can obtain high-quality solutions on multimodal function optimization problems.

Adaptive fuzzy particle swarm optimization for global optimization of multimodal functions

This paper proposes an adaptive fuzzy PSO (AFPSO) algorithm, based on the standard particle swarm optimization (SPSO) algorithm. The proposed AFPSO utilizes fuzzy set theory to adjust PSO acceleration coefficients adaptively, and is thereby able to improve the accuracy and efficiency of searches. Incorporating this algorithm with quadratic interpolation and crossover operator further enhances the global searching capability to form a new variant, called AFPSO-QI. We compared the proposed AFPSO and its variant AFPSO-QI with SPSO, quadratic interpolation PSO (QIPSO), unified PSO (UPSO), fully informed particle swarm (FIPS), dynamic multi-swarm PSO (DMSPSO), and comprehensive learning PSO (CLPSO) across sixteen benchmark functions. The proposed algorithms performed well when applied to minimization problems for most of the multimodal functions considered.

Global Optimization of Pavement Structural Parameters during Back-Calculation Using Hybrid Shuffled Complex Evolution Algorithm

In this paper, the use of a hybrid evolutionary optimization algorithm is proposed for global optimization of pavement structural parameters through inverse modeling. Shuffled complex evolution (SCE) is a population-based stochastic optimization technique combining the competitive complex evolution with the controlled random search, the implicit clustering, and the complex shuffling. Back-calculation of pavement layer moduli is an ill-posed inverse engineering problem, which involves searching for the optimal combination of pavement layer stiffness solutions in an unsmooth, multimodal, complex search space. SCE is especially considered a robust and efficient approach for global optimization of multimodal functions. A desirable characteristic of the SCE algorithm is that it uses information about the nature of the response surface, extracted using the deterministic Simplex geometric shape, to direct the search into regions with higher posterior probability. The hybrid back-calculation system described in this paper combines the robustness of the SCE in global optimization with the computational efficiency of neural networks and advanced pavement system characterization offered by employing finite-element models. This is the first time the SCE approach is applied to real-time nondestructive evaluation of pavement systems required in the routine maintenance and rehabilitation activities for sustainable transportation infrastructure.

Neural Network–Swarm Intelligence Hybrid Nonlinear Optimization Algorithm for Pavement Moduli Back-Calculation

This paper describes a novel hybrid intelligent system approach to inversion of nondestructive pavement deflection data and back-calculation of nonlinear stress-dependent pavement layer moduli. Particle swarm optimization (PSO), a population-based stochastic optimization technique inspired by social behavior of bird flocking or fish schooling, is fast emerging as an innovative and powerful computational metaphor for solving complex problems in design, optimization, control, management, business, and finance. Back-calculation of pavement layer moduli is an ill-posed inverse engineering problem which involves searching for the optimal combination of pavement layer stiffness solutions in an unsmooth, multimodal, complex search space. PSO is especially considered a robust and efficient approach for global optimization of multimodal functions. The hybrid back-calculation system described in this paper integrates finite element modeling, neural networks, and PSO in an efficient manner to mitigate the limitations and take advantages of the strengths to produce a system that is more effective and powerful than those which could be built with single technique. This is the first time the PSO approach is applied to real-time nondestructive evaluation of pavement systems.

Statistical learning makes the hybridization of particle swarm and differential evolution more efficient—A novel hybrid optimizer

This brief paper reports a hybrid algorithm we developed recently to solve the global optimization problems of multimodal functions, by combining the advantages of two powerful population-based metaheuristics—differential evolution (DE) and particle swarm optimization (PSO). In the hybrid denoted by DEPSO, each individual in one generation chooses its evolution method, DE or PSO, in a statistical learning way. The choice depends on the relative success ratio of the two methods in a previous learning period. The proposed DEPSO is compared with its PSO and DE parents, two advanced DE variants one of which is suggested by the originators of DE, two advanced PSO variants one of which is acknowledged as a recent standard by PSO community, and also a previous DEPSO. Benchmark tests demonstrate that the DEPSO is more competent for the global optimization of multimodal functions due to its high optimization quality.

A hybrid genetic algorithm and particle swarm optimization for multimodal functions

Heuristic optimization provides a robust and efficient approach for solving complex real-world problems. The focus of this research is on a hybrid method combining two heuristic optimization techniques, genetic algorithms (GA) and particle swarm optimization (PSO), for the global optimization of multimodal functions. Denoted as GA-PSO, this hybrid technique incorporates concepts from GA and PSO and creates individuals in a new generation not only by crossover and mutation operations as found in GA but also by mechanisms of PSO. The results of various experimental studies using a suite of 17 multimodal test functions taken from the literature have demonstrated the superiority of the hybrid GA-PSO approach over the other four search techniques in terms of solution quality and convergence rates.

Comprehensive learning particle swarm optimizer for global optimization of multimodal functions

This paper presents a variant of particle swarm optimizers (PSOs) that we call the comprehensive learning particle swarm optimizer (CLPSO), which uses a novel learning strategy whereby all other particles' historical best information is used to update a particle's velocity. This strategy enables the diversity of the swarm to be preserved to discourage premature convergence. Experiments were conducted (using codes available from http://www.ntu.edu.sg/home/epnsugan) on multimodal test functions such as Rosenbrock, Griewank, Rastrigin, Ackley, and Schwefel and composition functions both with and without coordinate rotation. The results demonstrate good performance of the CLPSO in solving multimodal problems when compared with eight other recent variants of the PSO.

A new tabu search method for continuous parameter optimization: application to design problems in electromagnetic

AbstractSpecificities of optimal design problem in electromagnetic are emphasized. To overcome the revolting difficulties met with the classical stochastic methods, a new tabu search algorithm for global optimization of multimodal functions with continuous variables is presented and used for solving some of this kind of electromagnetic optimization problems such as Bruhless DC motor, SMES optimization, and die mold press. It is compared favorably with Hu's tabu search and universal tabu search methods. Copyright © 2005 John Wiley & Sons, Ltd.

Experimental Testing of Advanced Scatter Search Designs for Global Optimization of Multimodal Functions

Scatter search is an evolutionary method that, unlike genetic algorithms, operates on a small set of solutions and makes only limited use of randomization as a proxy for diversification when searching for a globally optimal solution. The scatter search framework is flexible, allowing the development of alternative implementations with varying degrees of sophistication. In this paper, we test the merit of several scatter search designs in the context of global optimization of multimodal functions. We compare these designs among themselves and choose one to compare against a well-known genetic algorithm that has been specifically developed for this class of problems. The testing is performed on a set of benchmark multimodal functions with known global minima.

A New Tabu Search Method for Optimization With Continuous Parameters

A new tabu search algorithm for global optimization of multimodal functions with continuous variables is presented. The taboo list contains all points and a prohibited zone around each point that depends on the value of the objective function and decreases as the number of iteration increases. The numerical results obtained by solving problems 22 and 25 of the TEAM workshop demonstrate the speed effectiveness of the proposed method. It is compared favorably with other tabu search, genetic algorithm, and simulated annealing.

A Continuous Genetic Algorithm Designed for the Global Optimization of Multimodal Functions

Genetic algorithms are stochastic search approaches based on randomized operators, such as selection, crossover and mutation, inspired by the natural reproduction and evolution of the living creatures. However, few published works deal with their application to the global optimization of functions depending on continuous variables. A new algorithm called Continuous Genetic Algorithm (CGA) is proposed for the global optimization of multiminima functions. In order to cover a wide domain of possible solutions, our algorithm first takes care over the choice of the initial population. Then it locates the most promising area of the solution space, and continues the search through an “intensification” inside this area. The selection, the crossover and the mutation are performed by using the decimal code. The efficiency of CGA is tested in detail through a set of benchmark multimodal functions, of which global and local minima are known. CGA is compared to Tabu Search and Simulated Annealing, as alternative algorithms.

A universal tabu search algorithm for global optimization of multimodal functions with continuous variables in electromagnetics

A universal tabu search algorithm for the optimization of functions with continuous variables is presented based on the tabu method so far used, Essentially, the improvements include the transition criterion for different states, the determination of the step vector, the cancellation of tabu list, the stop criteria, and the restart from the optimum, etc. The numerical performances of the present algorithm are investigated using a benchmark problem and the geometry optimization of the multisection arc pole shoe in large salient pole synchronous generators.