There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Tables of integrals

Differential Evolution (DE) is arguably one of the most powerful and versatile evolutionary optimizers for the continuous parameter spaces in recent times. Almost 5 years have passed since the first comprehensive survey article was published on DE by Das and Suganthan in 2011. Several developments have been reported on various aspects of the algorithm in these 5 years and the research on and with DE have now reached an impressive state. Considering the huge progress of research with DE and its applications in diverse domains of science and technology, we find that it is a high time to provide a critical review of the latest literatures published and also to point out some important future avenues of research. The purpose of this paper is to summarize and organize the information on these current developments on DE. Beginning with a comprehensive foundation of the basic DE family of algorithms, we proceed through the recent proposals on parameter adaptation of DE, DE-based single-objective global optimizers, DE adopted for various optimization scenarios including constrained, large-scale, multi-objective, multi-modal and dynamic optimization, hybridization of DE with other optimizers, and also the multi-faceted literature on applications of DE. The paper also presents a dozen of interesting open problems and future research issues on DE.

Recent advances in differential evolution – An updated survey

Particle swarm optimization (PSO) is a heuristic global optimization method, proposed originally by Kennedy and Eberhart in 1995. It is now one of the most commonly used optimization techniques. This survey presented a comprehensive investigation of PSO. On one hand, we provided advances with PSO, including its modifications (including quantum-behaved PSO, bare-bones PSO, chaotic PSO, and fuzzy PSO), population topology (as fully connected, von Neumann, ring, star, random, etc.), hybridization (with genetic algorithm, simulated annealing, Tabu search, artificial immune system, ant colony algorithm, artificial bee colony, differential evolution, harmonic search, and biogeography-based optimization), extensions (to multiobjective, constrained, discrete, and binary optimization), theoretical analysis (parameter selection and tuning, and convergence analysis), and parallel implementation (in multicore, multiprocessor, GPU, and cloud computing forms). On the other hand, we offered a survey on applications of PSO to the following eight fields: electrical and electronic engineering, automation control systems, communication theory, operations research, mechanical engineering, fuel and energy, medicine, chemistry, and biology. It is hoped that this survey would be beneficial for the researchers studying PSO algorithms.

/pdf/a-comprehensive-survey-on-particle-swarm-optimization-1d380v5xx1.pdf

A Comprehensive Survey on Particle Swarm Optimization Algorithm and Its Applications

In nature, the eastern North American monarch population is known for its southward migration during the late summer/autumn from the northern USA and southern Canada to Mexico, covering thousands of miles. By simplifying and idealizing the migration of monarch butterflies, a new kind of nature-inspired metaheuristic algorithm, called monarch butterfly optimization (MBO), a first of its kind, is proposed in this paper. In MBO, all the monarch butterfly individuals are located in two distinct lands, viz. southern Canada and the northern USA (Land 1) and Mexico (Land 2). Accordingly, the positions of the monarch butterflies are updated in two ways. Firstly, the offsprings are generated (position updating) by migration operator, which can be adjusted by the migration ratio. It is followed by tuning the positions for other butterflies by means of butterfly adjusting operator. In order to keep the population unchanged and minimize fitness evaluations, the sum of the newly generated butterflies in these two ways remains equal to the original population. In order to demonstrate the superior performance of the MBO algorithm, a comparative study with five other metaheuristic algorithms through thirty-eight benchmark problems is carried out. The results clearly exhibit the capability of the MBO method toward finding the enhanced function values on most of the benchmark problems with respect to the other five algorithms. Note that the source codes of the proposed MBO algorithm are publicly available at GitHub (
 https://github.com/ggw0122/Monarch-Butterfly-Optimization
 , C++/MATLAB) and MATLAB Central (
 http://www.mathworks.com/matlabcentral/fileexchange/50828-monarch-butterfly-optimization
 , MATLAB).

Monarch butterfly optimization

In this paper the maximum sidelobe level (SLL) reductions of three-ring concentric circular antenna arrays (CCAA) are examined using two different classes of evolutionary optimization techniques to finally determine the global optimal three-ring CCAA design. Apart from physical construction of a CCAA, one may broadly classify its design into two major categories: uniformly excited arrays and non-uniformly excited arrays. The present paper assumes non-uniform excitations and uniform spacing of excitation elements in each three-ring CCAA design and a design goal of maximizing SLL reduction associated with optimal beam patterns and beam widths. The design problem is modeled as an optimization problem for each CCAA design. Binary coded Genetic Algorithm (BGA) and Bacterial Foraging Optimization (BFO) are used to determine an optimum set of normalized excitation weights for CCAA elements, which, when incorporated, results in a radiation pattern with optimal (maximum) SLL reduction. Among the various CCAA designs the three-ring CCAA containing (N 1 =4, N 2 =6, N 3 =8) elements proves to be global optimal design. BFO yields global minimum SLL (−14.98 dB) and global minimum BWFN (75.8°) for the optimal design.

Application of Bio-inspired optimization technique for finding the optimal set of concentric circular antenna array

In this paper, a recently proposed global heuristic search algorithm, namely, particle swarm optimization with improved inertia weight (PSOIIW) approach is considered for the design of the 8th order infinite impulse response (IIR) low pass (LP) digital filter. Unlike conventional PSO, inertia weight component in the proposed technique is calculated separately for each particle and modification is based on the iteration dependent velocity and position components. With this approach better global and local exploration is achieved which resulted in faster convergence to near global optimal solution. Performance of the proposed PSOIIW based approach is compared with some well accepted evolutionary algorithms such as particle swarm optimization (PSO) and real coded genetic algorithm (RGA). From the simulation study it is established that the proposed optimization technique PSOIIW outperforms RGA and PSO, not only in the accuracy of the designed filter but also in the convergence speed and solution quality, i.e., the stop band attenuation, transition width, pass band and stop band ripples. Further, the pole zero analysis justifies the stability of the designed optimal IIR filter.

Digital stable IIR low pass filter optimization using particle swarm optimization with improved inertia weight

In this paper, a fractional-order bandpass Butterworth filter (FBPBF) exhibiting symmetric roll-off characteristic is approximated as an integer-order transfer function using a metaheuristic optimization approach. A novel cost function is proposed by considering the exact FBPBF magnitude characteristic obtained through cascading the ( n + α ) , where n is an integer and 0 α 1 , order lowpass and highpass Butterworth filter responses. Design examples considering various values of n and α are presented to illustrate the performance efficacy of the proposed approach. PSPICE simulations for the models realized using the current feedback operational amplifiers confirm proximity to the theoretical characteristics.

Optimal rational approximation of bandpass Butterworth filter with symmetric fractional-order roll-off

Block oriented models are recurrently employed to describe the dynamic characteristics of nonlinear systems. This paper adopts an efficacious block-oriented model called cascaded Wiener-Hammerstein (W-H) structure to model a nonlinear system. The parameters of the W-H model are identified using a nature-inspired evolution algorithm-supported Kalman filter (KF). In the proposed technique, the utility of the employed optimisation algorithm named Harris Hawks optimiser (HHO) is to acquire the optimal global solution of initial state KF parameters for the W-H system identification problem. Then, the conventional KF algorithm utilises these optimised KF parameters to beget the near-global estimated W-H system parameters. The effectiveness of the proposed HHO-based KF algorithm is verified through a comparison study with other variants of particle swarm optimisation (PSO) and harmony search (HS) algorithms-supported KF methods on two numerical W-H problems with different nonlinearity levels and two practical benchmark plants, namely, real electronic diode switching network and flutter clearance of F-18 systems research aircraft. The simulation results ensure that the proposed HHO-KF approach offers significantly improved results over other competing methods in terms of various well known standard metrics. The hypothesis test verifies the stability and consistency of the proposed identification method. Moreover, the proposed approach provides a faster convergence rate and the lowest estimation precision compared with other reported techniques. 

Optimal design of cascaded Wiener-Hammerstein system using a heuristically supervised discrete Kalman filter with application on benchmark problems

This paper presents a novel approach for designing a linear phase digital high pass FIR filter using Improved Particle Swarm Optimization (IPSO) algorithm. Design of FIR filter is a multi-modal optimization problem. The conservative gradient based optimization techniques are not efficient for digital filter design. Given the specifications for the filters to be realized, IPSO algorithm generates a set of optimal filter coefficients and tries to meet the ideal frequency response characteristics. This paper presents the realization of the optimal FIR high pass filter of filter order 20 as per given problem statements. The simulation results have been compared to those obtained from well accepted classical algorithms like Park and McClellan algorithm (PM), and evolutionary algorithms like genetic algorithm (GA) and particle swarm optimization (PSO). The results rationalize that the proposed optimal filter design approach using IPSO outperforms PM, RGA, PSO in the accuracy of the designed filter, as well as in the convergence speed and solution quality.

Durbadal Mandal

Papers

Application of Bio-inspired optimization technique for finding the optimal set of concentric circular antenna array

Digital stable IIR low pass filter optimization using particle swarm optimization with improved inertia weight

Optimal rational approximation of bandpass Butterworth filter with symmetric fractional-order roll-off

Optimal design of cascaded Wiener-Hammerstein system using a heuristically supervised discrete Kalman filter with application on benchmark problems

Design of Optimal Linear Phase FIR High Pass Filter using Improved Particle Swarm Optimization