An enhanced monarch butterfly optimization with self-adaptive crossover operator for unconstrained and constrained optimization problems

After studying the behavior of monarch butterflies in nature, Wang et al. proposed a new promising swarm intelligence algorithm, called monarch butterfly optimization (MBO), for addressing unconstrained optimization tasks. In the basic MBO algorithm, the fixed butterfly adjusting rate is used to carry out the butterfly adjusting operator. In this paper, the self-adaptive strategy is introduced to adjust the butterfly adjusting rate. In addition, the crossover operator that is generally used in evolutionary algorithms (EAs) is used to further improve the quality of butterfly individuals. The two optimization strategies, self-adaptive and crossover operator, are combined, and then self-adaptive crossover operator is proposed. After incorporating the above strategies into the basic MBO algorithm, a new version of MBO algorithm, called Self-adaptive Monarch Butterfly Optimization (SaMBO), is put forward. Also, few studies of constrained optimization has been done for MBO research. In this paper, in order to verify the performance of our proposed SaMBO algorithm, the proposed SaMBO algorithm is further benchmarked by 21 CEC 2017 constrained optimization problems. The experimental results indicate that the proposed SaMBO algorithm outperforms the basic MBO and other five state-of-the-art metaheuristic algorithms.

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 is inspired by the migration of monarch butterflies in nature: monarch butterflies in eastern North America migrate southward from northern United States and southern Canada to Mexico every summer/autumn. It is a new swarm intelligence metaheuristic algorithm, which is called monarch butterfly optimization (MBO). In MBO, the monarch butterfly population is ideally divided into two populations, one is located in southern Canada and northern United States (land 1), and the other is located in Mexico (land 2). We update the individuals in the population through migration and butterfly shape adjustment. To maintain the diversity of the population and minimize the fitness value, the total number of butterflies generated by the operation should be equal to the original population. In the experiment, MBO is compared with five metaheuristic algorithms through 38 test functions, especially applied to solve a large-scale 0-1 knapsack (KP) problem. Three groups of 15 large-scale 0-1 KP cases from 800 to 2000 dimensions are used to test. It is proved that MBO has strong advantages in solving global numerical optimization and engineering problems, which provides a reference for further discussion of MBO in the future.

As a significant subset of the family of discrete optimization problems, the 0-1 knapsack problem (0-1 KP) has received considerable attention among the relevant researchers. The monarch butterfly optimization (MBO) is a recent metaheuristic algorithm inspired by the migration behavior of monarch butterflies. The original MBO is proposed to solve continuous optimization problems. This paper presents a novel monarch butterfly optimization with a global position updating operator (GMBO), which can address 0-1 KP known as an NP-complete problem. The global position updating operator is incorporated to help all the monarch butterflies rapidly move towards the global best position. Moreover, a dichotomy encoding scheme is adopted to represent monarch butterflies for solving 0-1 KP. In addition, a specific two-stage repair operator is used to repair the infeasible solutions and further optimize the feasible solutions. Finally, Orthogonal Design (OD) is employed in order to find the most suitable parameters. Two sets of low-dimensional 0-1 KP instances and three kinds of 15 high-dimensional 0-1 KP instances are used to verify the ability of the proposed GMBO. An extensive comparative study of GMBO with five classical and two state-of-the-art algorithms is carried out. The experimental results clearly indicate that GMBO can achieve better solutions on almost all the 0-1 KP instances and significantly outperforms the rest.

The Meta heuristic algorithms are the higher level technique which helps to find the best feasible solution out of all possible solution of an optimization problem. There are various different types of meta heuristic algorithms like Ant Colony Optimization (ACO), Genetic Algorithm, Simulated Annealing, Particle Swarm Optimization, etc. Genetic Algorithm is a search-based optimization technique based on the biological principle of Genetics and adaptation. It is a meta-heuristic approach which is used to solve complex combinatorial problem. The integration of Genetic algorithm with machine learning will be helpful in solving unconstrained and constrained optimization problem. The various genetic operator like selection operator, mutation and cross-over are discussed which will be helpful in knowing how these operators significantly improves State Space search. In this paper the various applications of Genetic algorithms which can be used in machine learning has been discussed. In this paper the author discussed how the significance of Genetic algorithm will be improved while solving complex optimization problem in machine learning. In this paper, flow diagram of Genetic Algorithms has been discussed which will ease the understanding of complex optimization problem like 0–1 Knapsack, Traveling Salesman Problem, etc. In this paper a comparative analysis between traditional algorithm and genetic algorithm has been done on the basis of parameters like flow of control, state space search, Complication, Preconditions, CPU utilization etc. The various limitations of Genetic Algorithms in solving problems with optimal solutions has also been discussed.

Generalized Nash equilibrium problems (GNEPs) allow, in contrast to standard Nash equilibrium problems, a dependence of the strategy space of one player from the decisions of the other players. In this paper, we consider jointly convex GNEPs which form an important subclass of the general GNEPs. Based on a regularized Nikaido-Isoda function, we present two (nonsmooth) reformulations of this class of GNEPs, one reformulation being a constrained optimization problem and the other one being an unconstrained optimization problem. While most approaches in the literature compute only a so-called normalized Nash equilibrium, which is a subset of all solutions, our two approaches have the property that their minima characterize the set of all solutions of a GNEP. We also investigate the smoothness properties of our two optimization problems and show that both problems are continuous under a Slater-type condition and, in fact, piecewise continuously differentiable under the constant rank constraint qualification. Finally, we present some numerical results based on our unconstrained optimization reformulation.

Immunity-based hybrid evolutionary algorithm for multi-objective optimization in global container repositioning

Hybrid differential evolution including geometric mean mutation for optimization of biochemical systems

The present work proposes a multi-objective improved teaching-learning based optimization (MO-ITLBO) algorithm for unconstrained and constrained multi-objective function optimization. The MO-ITLBO algorithm is the improved version of basic teaching-learning based optimization (TLBO) algorithm adapted for multi-objective problems. The basic TLBO algorithm is improved to enhance its exploration and exploitation capacities by introducing the concept of number of teachers, adaptive teaching factor, tutorial training and self-motivated learning. The MO-ITLBO algorithm uses a grid-based approach to adaptively assess the non-dominated solutions (i.e. Pareto front) maintained in an external archive. The performance of the MO-ITLBO algorithm is assessed by implementing it on unconstrained and constrained test problems proposed for the Congress on Evolutionary Computation 2009 (CEC 2009) competition. The performance assessment is done by using the inverted generational distance (IGD) measure. The IGD measures obtained by using the MO-ITLBO algorithm are compared with the IGD measures of the other state-of-the-art algorithms available in the literature. Finally, Lexicographic ordering is used to assess the overall performance of competitive algorithms. Results have shown that the proposed MO-ITLBO algorithm has obtained the 1st rank in the optimization of unconstrained test functions and the 3rd rank in the optimization of constrained test functions.

The term, "Waveform inversion" (WFI), refers to a collection of techniques that use the information from seismic data to derive high-fidelity earth models for seismic imaging. The attractiveness of WFI lies mainly in its lack of approximations, at least in a theoretical sense, in contrast to other model determination techniques such as semblance or tomography. However, a whole raft of approximations must be made to make the technique viable with today'scomputing technology and restrictions of seismic acquisition. These are collectively referred to as "waveform inversion strategies" and in this paper we mainly discuss regularization and preconditioning strategies. Because the wavefields need to be accurately modeled to represent the kinematics of all the waves during WFI iterations, the effects of anisotropy often help to improve WFI results. In this paper, forward modeling and its adjoint computation are based on acoustic wave equations in vertical transversely isotropic (VTI)media. We introduce a multi-parameter inversion for P-wave velocity and anisotropy parameters. WFI is a highly nonlinear, ill-posed problem. We introduce additional information and turn the unconstrained optimization problem into a constrained optimization problem in order to reduce the ill-posedness. The geophysics of the problem leads to appropriate constraints, such as restriction of model parameters, or information from well logs. In this paper, we use well logs as constraints and solve the problem using the augmented Lagrangian method (ALM), a mathematical method that replaces a constrained optimization problem by a series of unconstrained problems. The ALM with well constraints aims at preserving velocity characteristics from welllogs and providing us with more reliable velocity updates. This paper presents the acoustic anisotropic WFI implementation using ALM with well constraints. It also discusses practical strategies for regularization and preconditioning and their influences on the models that are obtained from WFI. We illustrate these approaches on a 2D synthetic Marmousi example and another application to 3D VSOOBC data from the Green Canyon area of the Gulf of Mexico. From the results, we show that multi-parameter VTI WFI with ALM provides us with more useful and reliable model updates. To further evaluate our WFI results, we also compare offset gathers and RTM images and illustrate their significant improvement susing updated models generated from WFI.

Symbolic functions, as an important class of nonsmooth functions, play a key role in many fields such as system control and optimization theory. In this study, the special case of nonlinear optimization problems with sign function constraints is discussed in depth. Based on the smooth approximation theory, the study first adopts an approximate substitution method to smooth the nonsmooth constraints and then transforms them into differentiable optimization problems, thus effectively avoiding the numerical computational difficulties caused by the sign function. Second, the constrained optimization problem is transformed into an unconstrained optimization problem by constructing an exact penalty function. Third, two forms of second-order dynamical systems are established to solve the problem, and a rigorous theoretical analysis of the stability of these systems is conducted. Finally, the convergence and computational efficiency of the proposed method are verified by numerical simulation experiments of the system, and the simulation results fully prove the effectiveness and practicality of the algorithm.

An important and challenging data mining application in marketing is to learn models for predicting potential customers who contribute large profit to a company under resource constraints. In this paper, we first formulate this learning problem as a constrained optimization problem and then converse it to an unconstrained Multi-objective Optimization Problem (MOP). A parallel Multi-Objective Evolutionary Algorithm (MOEA) on consumer-level graphics hardware is used to handle the MOP. We perform experiments on a real-life direct marketing problem to compare the proposed method with the parallel Hybrid Genetic Algorithm, the DMAX approach, and a sequential MOEA. It is observed that the proposed method is much more effective and efficient than the other approaches.

Many constrained optimization problems are encountered in engineering field. The problem of multipath repression in high-precision positioning navigation systems can also be regarded as a constrained optimization problem. Multipath repression can be transformed into a parameter estimation problem with a nonlinear state space model. In this paper, the multipath estimation problem in non-Gaussian noises is transferred into a constrained optimization problem. Then, a modified ε-constrained multi-mutant rank-based differential evolution (εCMRDE ) algorithm is explored to solve the optimization problem. For the proposed algorithm, the generalized maximum correntropy criterion (GMCC) is used as the objective function of the formulated optimization problem. The second-order moment of the estimation error and the prior information are taken as constraints to reduce the mean of the estimation error. Simulation results show that the proposed algorithm outperforms εRDE algorithm in terms of global optimization ability and multipath estimation accuracy.

In the dynamic development of technology, the Internet of Things (IoT) stands as a driving force of innovation and opening new frontiers. Data collection and processing in IoT systems provide services to decision making in various fields including home automation, disaster management, healthcare, and so on. In such services, the IoT has certain limitations to perform the required tasks and it can cause extreme consequences in the network. One of the main challenge in IoT is to ensure maximum coverage by optimally placing the sensors. To address the coverage problem, a meta-heuristic algorithm Monarch Butterfly Optimization with greedy strategy and self-adaptive crossover operator (GCMBO) is proposed for Optimal Sensor Placement (OSP) in 3D monitoring region. The performance of proposed algorithm is evaluated by a series of simulation and it provide efficient results when compared to other similar existing optimization algorithms. Additionally, a statistical analysis (ANOVA and LSD post-hoc) is exhibited from the comparative results.

The aim of this paper is to denoise the high noise density image proficiently with minimal computation cost using various techniques. This paper proposes an approach for image denoising based on Discrete Wavelet Transform (DWT) in association with Monarch Butterfly Optimization (MBO) technique. Along with Gaussian noise, salt & pepper and speckle noise are added to the image and DWT is applied on the noisy image. The Haar wavelet is used to segregate the sub bands and threshold operation is carried out in three bands. The wavelet coefficient optimization process is performed for optimizing the coefficient value with the assistance of MBO technique. In this wavelet-optimized parameter, the inverse DWT (IDWT) is applied. The proposed method decreases the noise from images more efficiently. The result exhibits that MBO technique is better than the existing and other traditional techniques as it minimize the mean square error (MSE). This process is validated with PSNR value and implemented in MATLAB.