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

A problem of interference suppression of linear antenna arrays is dealt with in this work. Problem of reduction of sidelobe level relative to main beam is considered. For simplicity antenna arrays are assumed to have ideal elements and hence, mutual coupling effect is neglected. A combination of recently proposed evolutionary algorithm Backtracking Search Algorithm (BSA) with another widely accepted evolutionary algorithm, namely, Differential Evolution (DE) is considered for finding near optimal solutions. It is seen that, combined search outperforms in this interference suppression problem.

Interference suppression of linear antenna arrays with combined Backtracking Search Algorithm and Differential Evolution

In this paper, a steerable isotropic circular array antenna is designed to reduce the side lobe level (SLL), using an evolutionary optimization technique. Particle swarm optimization and the firefly algorithm are used to reduce the SLL, as well as to steer the main beam in a specific direction. In this steerable circular array design, the amplitude excitations are optimized. The results show that the maximum SLL peak of the resultant patterns is as per requirements. Good performance is achieved in the array factor response and suppression of SLL for different numbers of array elements with different main beam steering angles.

Array pattern optimization for a steerable circular isotropic antenna array using the firefly algorithm

A design problem of uniform linear antenna arrays for imposing deeper nulls in the desired direction with the constraint of a reduced Side Lobe Level (SLL) and fixed First Null Beam Width (BWFN) is dealt with. This problem is modelled as a simple optimization problem. The method of Real-coded Genetic Algorithm (RGA) is used to determine an optimum set of current excitation weights of antenna elements and optimum inter-element spacing that satisfy the optimization goal. Three design examples are presented that illustrate the use of the RGA, and the optimization goal in each example is easily achieved. Numerical results show the effectiveness of the proposed method.

Introducing deeper nulls and reduction of Side Lobe Levels in a symmetric linear antenna array using genetic algorithm

This paper proposes an efficient design technique for two commonly used VLSI circuits, namely, CMOS current mirror load-based differential amplifier circuit and CMOS two-stage operational amplifier. The hybrid evolutionary method utilized for these optimal designs is random particle swarm optimization with differential evolution (RPSODE). Random PSO utilizes the weighted particles for monitoring the search directions. DE is a robust evolutionary technique. It has demonstrated an exclusive performance for the optimization problems which are continuous and global but suffers from the uncertainty issues. PSO is a robust optimization method but suffers from sub-optimality problem. This paper effectively hybridizes the random PSO and DE to remove the limitations related to both the techniques individually. In this paper, RPSODE is employed to optimize the sizes of the MOS transistors to reduce the overall area taken by the circuit while satisfying the design constraints. The results obtained from RPSODE technique are validated in SPICE environment. SPICE-based simulation results justify that RPSODE is a much better technique than other formerly reported methods for the designs of the above mentioned circuits in terms of MOS area, gain, power dissipation, etc.

Design of Optimal CMOS Analog Amplifier Circuits Using a Hybrid Evolutionary Optimization Technique

In this paper a design problem of non-uniform circular antenna arrays for maximum side lobe level reduction is dealt with. Circular antenna array has achieved great popularity in the field of Electromagnetics and Communications. Performance comparison of radiation patterns between two cases of circular antenna array as: (i) optimal non-uniform current amplitude excitation weights with optimal uniform spacing, and (ii) non-uniform current amplitude excitation weights with optimal non-uniform spacing. The optimized radiation pattern of the array is obtained using Improved Particle swarm Optimization (IPSO). Circular array antennas lying on x-y plane are assumed. The array is assumed to have the maximum radiation along positive z- axis. Single and multiple nulls are imposed by optimizing the current amplitude excitation weights and optimizing inter-element spacing through IPSO. Various simulated results are presented and hence performances of the radiation patterns are analyzed.

Durbadal Mandal

Papers

Interference suppression of linear antenna arrays with combined Backtracking Search Algorithm and Differential Evolution

Array pattern optimization for a steerable circular isotropic antenna array using the firefly algorithm

Introducing deeper nulls and reduction of Side Lobe Levels in a symmetric linear antenna array using genetic algorithm

Design of Optimal CMOS Analog Amplifier Circuits Using a Hybrid Evolutionary Optimization Technique

IPSO based performance comparison of optimum uniform and non-uniform spacing of circular antenna arrays