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Line search methods with guaranteed asymptotical convergence to an improving local optimum of multimodal functions

EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2014年第1期235卷 38-46页

作者： Gomes Vieira, Douglas Alexandre Lisboa, Adriano Chaves ENACOM Handcrafted Technol Belo Horizonte MG Brazil

This paper considers line search optimization methods using a mathematical framework based on the simple concept of a v-pattern and its properties. This framework provides theoretical guarantees on preserving, in the localizing interval, a local optimum no worse than the starting point. Notably, the framework can be applied to arbitrary unidimensional functions, including multimodal and infinitely valued ones. Enhanced versions of the golden section, bisection and Brent's methods are proposed and analyzed within this framework: they inherit the improving local optimality guarantee. Under mild assumptions the enhanced algorithms are proved to converge to a point in the solution set in a finite number of steps or that all their accumulation points belong to the solution set. (C) 2013 Elsevier B.V. All rights reserved.

关键词： Nonlinear programming Line search Golden section method Brent's algorithm Bisection multimodal functions

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A niche hybrid genetic algorithm for global optimization of continuous multimodal functions

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APPLIED MATHEMATICS AND COMPUTATION 2005年第3期160卷 649-661页

作者： Wei, LY Zhao, M Shanghai Jiao Tong Univ State Key Lab Vibrat Shock & Noise Shanghai 200030 Peoples R China Guangxi Univ Sch Comp & Informat Engn Nanning 530004 Peoples R China

A niche hybrid genetic algorithm (NHGA) is proposed in this paper to solve continuous multimodal optimization problems more efficiently, accurately and reliably. It provides a new architecture of hybrid algorithms, which organically merges the niche techniques and Nelder-Mead's simplex method into GAs. In the new architecture, the simplex search is first performed in the potential niches, which likely contain a global Optimum, to locate the promising zones within search space, quickly and reliably. Then another simplex search is used to quickly discover the global optimum in the located promising zones. The proposed method not only makes the exploration capabilities of GAs stronger through niche techniques, but also has more powerful exploitation capabilities by using simplex search. So it effectively alleviates premature convergence and improves weak exploitation capacities of GAs. A set of benchmark functions is used to demonstrate the validity of NHGA and the role of every component of NHGA. Numerical experiments show that the NHGA may, efficiently and reliably, obtain a more accurate global optimum for the complex and high-dimension multimodal optimization problems. It also demonstrates that the new hybrid architecture is potential and can be used to generate more potential hybrid algorithms. (C) 2003 Elsevier Inc. All rights reserved.

关键词： hybrid genetic algorithms global optimnization niche multimodal functions

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Hybrid simplex search and particle swarm optimization for the global optimization of multimodal functions

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ENGINEERING OPTIMIZATION 2004年第4期36卷 401-418页

作者： Fan, SKS Liang, YC Zahara, E Yuan Ze Univ Dept Ind Engn & Management Chungli 320 Taoyuan Cty Taiwan

This article proposes the hybrid Nelder-Mead (NM)-Particle Swarm Optimization (PSO) algorithm based on the NM simplex search method and PSO for the optimization of multimodal functions. The hybrid NM-PSO algorithm is very easy to implement, in practice, since it does not require gradient computation. This hybrid procedure performed the exploration with PSO and the exploitation with the NM simplex search method. In a suite of 17 multi-optima test functions taken from the literature, the computational results via various experimental studies showed that the hybrid NM-PSO approach is superior to the two original search techniques ( i.e. NM and PSO) in terms of solution quality and convergence rate. In addition, the presented algorithm is also compared with eight other published methods, such as hybrid genetic algorithm (GA), continuous GA, simulated annealing (SA), and tabu search (TS) by means of a smaller set of test functions. On the whole, the new algorithm is demonstrated to be extremely effective and efficient at locating best-practice optimal solutions for multimodal functions.

关键词： simplex search method particle swarm optimization multimodal functions

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Grenade Explosion Method-A novel tool for optimization of multimodal functions

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APPLIED SOFT COMPUTING 2010年第4期10卷 1132-1140页

作者： Ahrari, Ali Atai, Ali A. Univ Tehran Dept Mech Engn Fac Engn Tehran Iran

This work presents a new optimization technique called Grenade Explosion Method (GEM). The fundamental concepts and ideas which underlie the method are fully explained. It is seen that this simple and robust algorithm is quite powerful in finding all global and some local optima of multimodal functions. The method is tested with several multimodal benchmark functions and the results show it usually converges to the global minima faster than other evolutionary methods such as Genetic Algorithm (GA) and Artificial Bee Colony (ABC). Based on the performance on classical benchmark functions, the efficiency of the method in solving engineering applications can be highly appreciated. (C) 2009 Elsevier B. V. All rights reserved.

关键词： Grenade Explosion Method Global optimization multimodal functions Evolutionary algorithm

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A hybrid genetic algorithm and particle swarm optimization for multimodal functions

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APPLIED SOFT COMPUTING 2008年第2期8卷 849-857页

作者： Kao, Yi-Tung Zahara, Erwie St Johns Univ Dept Ind Engn & Management Tamsui 251 Taiwan Tatung Univ Dept Comp Sci & Engn Taipei 104 Taiwan

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. (c) 2007 Published by Elsevier B.V.

关键词： heuristic optimization multimodal functions genetic algorithms particle swarm optimization

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Glowworm swarm based optimization algorithm for multimodal functions with collective robotics applications

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MULTIAGENT AND GRID SYSTEMS 2006年第3期2卷 209-222页

作者： Krishnanand, K. N. Ghose, Debasish Indian Inst Sci Dept Aerosp Engn Guidance Control & Decis Syst Lab Bangalore 560012 Karnataka India

This paper presents multimodal function optimization, using a nature-inspired glowworm swarm optimization (GSO) algorithm, with applications to collective robotics. GSO is similar to ACO and PSO but with important differences. A key feature of the algorithm is the use of an adaptive local-decision domain, which is used effectively to detect the multiple optimum locations of the multimodal function. Agents in the GSO algorithm have a finite sensor range which defines a hard limit on the local-decision domain used to compute their movements. The GSO algorithm is memoryless and the glowworms do not retain any information in their memory. Some theoretical results related to the luciferin update mechanism in order to prove the bounded nature and convergence of luciferin levels of the glowworms are provided. Simulations demonstrate the efficacy of the GSO algorithm in capturing multiple optima of several multimodal test functions. The algorithm can be directly used in a realistic collective robotics task of simultaneously localizing multiple sources of interest such as nuclear spills, aerosol/hazardous chemical leaks, and fire-origins in a fire calamity.

关键词： Glowworm swarm optimization multimodal functions ant colony optimization particle swarm optimization collective robotics

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Optimization of One-Dimensional multimodal functions

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Journal of the Royal Statistical Society: Series C (Applied Statistics) 1978年第3期27卷 367-375页

作者： A. Žilinskas Institute of Mathematics and Cybernetics Academy of Sciences of the Lithuanian SSR Vilnius Lithuania

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Approximating multimodal functions Using Stochastic Search Method

Approximating Multimodal Functions Using Stochastic Search M...

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2010 2nd IEEE International Conference on Information Management and Engineering(2010年IEEE第二届信息管理与工程国际会议 IEEE ICIME 2010)

作者： Jian Guo Hongmin Li Wuhan Polytechnic UniversityWuhan China Wuhan Polytechnic University Wuhan China

The radial basis function (RBF) is well known dynamic recursion neural network. However, RBF weights and thresholds, which are trained by back propagation algorithm, the gradient descent method and genetic algorithm, will be fixed after the training completing. The adaptive ability is bad. To improve RBF identification performance, particle swarm optimization (PSO), which is a stochastic search algorithm, is employed to train and adjust RBF structure parameter online. The simulation experiments show that PSONN has less adjustable parameters, faster convergence speed and higher precision in multimodal functions identification.

关键词： multimodal functions dynamic identification radial basis function particle swarm optimizatio

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Data from multimodal functions based on an array of photovoltaic modules and an approximation with artificial neural networks as a scenario for testing optimization algorithms

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DATA IN BRIEF 2019年 27卷 104669-104669页

作者： Robles-Algarin, Carlos Restrepo-Leal, Diego Ospino Castro, Adalberto Univ Magdalena Fac Ingn Carrera 32 22-08 Santa Marta Colombia Univ Costa Fac Ingn Calle 58 55-66 Barranquilla Colombia

This paper presents the data of multimodal functions that emulate the performance of an array of five photovoltaic modules under partial shading conditions. These functions were obtained through mathematical modeling and represent the P-V curves of a photovoltaic module with several local maximums and a global maximum. In addition, data from a feedforward neural network are shown, which represent an approximation of the multimodal functions that were obtained with mathematical modeling. The modeling of multimodal functions, the architecture of the neural network and the use of the data were discussed in our previous work entitled "Search for Global Maxima in multimodal functions by Applying Numerical Optimization Algorithms: A Comparison Between Golden Section and Simulated Annealing" [1]. Data were obtained through simulations in a C code, which were exported to DAT files and subsequently organized into four Excel tables. Each table shows the voltage and power data for the five modules of the photovoltaic array, for multimodal functions and for the approximation of the multimodal functions implemented by the artificial neural network. In this way, a dataset that can be used to evaluate the performance of optimization algorithms and system identification techniques applied in multimodal functions was obtained. (C) 2019 The Author(s). Published by Elsevier Inc.

关键词： Artificial neural networks multimodal functions Optimization algorithms Partial shading Photovoltaic modules

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Precision, Local Search and Unimodal functions

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ALGORITHMICA 2011年第3期59卷 301-322页

作者： Dietzfelbinger, Martin Rowe, Jonathan E. Wegener, Ingo Woelfel, Philipp Univ Birmingham Sch Comp Sci Birmingham B15 2TT W Midlands England Tech Univ Ilmenau Fak Informat & Automatisierung D-98684 Ilmenau Germany Tech Univ Dortmund FB Informat D-44221 Dortmund Germany Univ Calgary Dept Comp Sci Calgary AB T2N 1N4 Canada

We investigate the effects of precision on the efficiency of various local search algorithms on 1-D unimodal functions. We present a (1+1)-EA with adaptive step size which finds the optimum in O(log n) steps, where n is the number of points used. We then consider binary (base-2) and reflected Gray code representations with single bit mutations. The standard binary method does not guarantee locating the optimum, whereas using the reflected Gray code does so in I similar to((log n)(2)) steps. A(1+1)-EA with a fixed mutation probability distribution is then presented which also runs in O((log n)(2)). Moreover, a recent result shows that this is optimal (up to some constant scaling factor), in that there exist unimodal functions for which a lower bound of Omega((log n)(2)) holds regardless of the choice of mutation distribution. For continuous multimodal functions, the algorithm also locates the global optimum in O((log n)(2)). Finally, we show that it is not possible for a black box algorithm to efficiently optimise unimodal functions for two or more dimensions (in terms of the precision used).

关键词： Local search Precision Computational complexity Unimodal functions multimodal functions

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