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This paper is concerned with the lifespan estimate of classical solutions with small initial data to the Cauchy problem of semilinear damped wave equations with the Fujita critical exponent. We establish the following sharp upper bound of the lifespan T ( e ) ≤ exp ⁡ ( C e − 2 n ) in higher dimensions ( n ≥ 4 ) , by using the heat kernel as the test function. Then, together with the previous results, a complete result on the sharp lower and upper bound estimates is obtained in this case.

In this paper, the application of three well-known multi-objective optimization algorithms to water distribution network (WDN) optimum design has been considered. Non-dominated sorting genetic algorithm II (NSGA-II), Multi-objective differential evolution (MODE) and Multi-objective particle swarm optimization (MOPSO) algorithms are applied to benchmark mathematical test function problems for evaluating the performance of these algorithms. The Accuracy and computational runtime are the two indicators used for the comparison of these three algorithms. The optimization results of mathematical test functions show that all three algorithms were able to accurately produce Pareto Front, but the computational time of MODE algorithm to achieve the optimal solutions is lower than the two other algorithms. Then, the discussed algorithms have been used to optimize the WDN design problem. Comparison of the generated solutions on the Pareto Front for WDN design shows that the obtained Pareto Front of MODE is more accurate and faster. Keywords: Multi-objective optimization, Genetic algorithm, Differential evolution, Particle swarm, Water distribution design

This paper implements three dimensional meshless local Petrov–Galerkin method in the linear elastic fracture mechanics analyses of isotropic functionally graded and homogeneous materials under different thermo-mechanical loads. Energy equation based on Lord–Shulman non-Fourier heat conduction law coupled with equation of motion is applied to evaluate the effect of theoretical and realistic relaxation times on transient thermal stress intensity factors along 3D thorough or semielliptical cracks under thermo-mechanical shock. A new coordinate transform method with linear weight function is introduced to evaluate stress intensity factors by generalized interaction integral method based on incompatibility formulation. Simple linear test function is introduced, which is approximated via compatible radial basis functions and leads in eliminating internal boundary integrals of meshless method and reducing computational time. Shape functions are constructed in every Gauss point by determining the closest domain point method in preprocessing. Parametric studies are carried out in order to determine suitable radial basis functions shape parameters and penalty method parameter. Reasonable accuracy and the small number of used points are the main advantages of this method over finite element method.

A multi-objective optimization method for constant stress accelerated degradation test is proposed in order to solve the problem of different or even conflicted test configuration under different optimization objectives. The inverse Gaussian process is used as a degradation model, and the unknown parameters are solved by maximum likelihood estimation. The two optimization criteria of the maximum determinant of the information matrix and the minimum asymptotic variance of P-quantile are considered. The improved multi-objective particle swarm optimization algorithm is proposed to search test optimal configuration, and the Pareto solution set for dual-objectives is obtained. Finally, the effectiveness of the method is illustrated by a group of examples of electrical connectors. Compared with the single-objective optimization design, the proposed method is more reasonable and convenient for test configuration. The performance index of the test function indicates that the optimal algorithm we proposed has some obvious advantages over the NSGA-II in diversity and convergence of the Pareto solutions and it is significant in guiding engineering practice.

We prove boundary higher integrability for the (spatial) gradient of \emph{very weak} solutions of quasilinear parabolic equations of the form $$u_t - \text{div}\,\mathcal{A}(x,t, \nabla u)=0 \quad \text{on} \ \Omega \times \mathbb{R},$$ where the non-linear structure $\text{div}\,\mathcal{A}(x, t,\nabla u)$ is modelled after the $p$-Laplace operator. To this end, we prove that the gradients satisfy a reverse H\"older inequality near the boundary. In order to do this, we construct a suitable test function which is Lipschitz continuous and preserves the boundary values. \emph{These results are new even for linear parabolic equations on domains with smooth boundary and make no assumptions on the smoothness of $\mathcal{A}(x,t,\nabla u)$}. These results are also applicable for systems as well as higher order parabolic equations.

Gene regulatory networks, which are complex high-dimensional stochastic dynamical systems, are often subject to evident intrinsic fluctuations. It is deemed reasonable to model the systems by the chemical Langevin equations. Since the mRNA dynamics are faster than the protein dynamics, we have a two-time scales system. In general, the process of protein degradation involves time delays. In this paper, we take the system memory into consideration in which we consider a model with a complete memory represented by an integral delay from \begin{document}$ 0 $\end{document} to \begin{document}$ t $\end{document} . Based on the averaging principle and perturbed test function method, this work examines the weak convergence of the slow-varying process. By treating the fast-varying process as a random noise, under appropriate conditions, it is shown that the slow-varying process converges weakly to the solution of a stochastic differential delay equation whose coefficients are the average of those of the original slow-varying process with respect to the invariant measure of the fast-varying process.

Aiming at the problem of slow convergence speed and ease of falling into local optimum when solving high dimensional problems, this paper proposes an improved chicken swarm optimization algorithm. The improved chicken swarm optimization includes four aspects, namely, cock position update mode, hen position update mode, chick position update mode, and population update strategy, so it is abbreviated as ICSO-RHC. On the basis of algorithm improvement, the influence of the number of retained elite individuals and control parameters on the convergence speed of the algorithm is discussed. The calculation results of the test function show that when the number of elite individuals in the population is 1, and the control parameters is a random number uniformly distributed between [0, 1], the algorithm has a faster convergence speed. In addition, in order to verify the performance of ICSO-RHC, 30 test functions and CEC 2005 benchmark functions were selected. The calculation results of these test functions show that the success rate of ICSO-RHC is significantly higher than other algorithms, both for low-dimensional and high-dimensional optimization problems. The average iteration number and average running time are significantly lower than other algorithms. Finally, ICSO-RHC and other improved algorithms in the literature are used to optimize the parameters of four practical engineering problems. The optimization results show that the statistical results obtained by ICSO-RHC are significantly better than other algorithms. The calculation results of the test functions and the actual engineering problems show that the performance of ICSO-RHC proposed in this paper is significantly better than other algorithms.

In this paper, we obtain sufficient conditions for the nonexistence of global solutions for some classes of q-difference inequalities. Our approach is based on the weak formulation of the problem, a particular choice of the test function, and some q-integral inequalities.

In the present article, Petrov–Galerkin method has been utilized for the numerical solution of nonlinear time-fractional KdV–Burgers (KdVB) equation. The nonlinear KdV–Burgers equation has been solved numerically through the Petrov–Galerkin approach utilising a quintic B-spline function as the trial function and a linear hat function as the test function . The numerical outcomes are observed in good agreement with exact solutions for classical order. In case of fractional order, the numerical results of KdV–Burgers equations are compared with those obtained by new method proposed in [1] . Numerical experiments exhibit the accuracy and efficiency of the approach in order to solve nonlinear dispersive and dissipative problems like the time-fractional KdV–Burgers equation.

Dynamic multi-objective optimization problems (DMOPs) not only involve multiple conflicting objectives, but these objectives may also vary with time, raising a challenge for researchers to solve them. This paper presents a cooperative co-evolutionary strategy based on environment sensitivities for solving DMOPs. In this strategy, a new method that groups decision variables is first proposed, in which all the decision variables are partitioned into two subcomponents according to their interrelation with environment. Adopting two populations to cooperatively optimize the two subcomponents, two prediction methods, i.e., differential prediction and Cauchy mutation, are then employed respectively to speed up their responses on the change of the environment. Furthermore, two improved dynamic multi-objective optimization algorithms, i.e., DNSGAII-CO and DMOPSO-CO, are proposed by incorporating the above strategy into NSGA-II and multi-objective particle swarm optimization, respectively. The proposed algorithms are compared with three state-of-the-art algorithms by applying to seven benchmark DMOPs. Experimental results reveal that the proposed algorithms significantly outperform the compared algorithms in terms of convergence and distribution on most DMOPs.

The article touches upon the possibility of using the method of gradientless n-parametric minimization of Powell'64 in tasks of monitoring the working process of marine diesel engines. There is given an example of finding a global minimum of the Rosenbrock test function. Using the Powell'64 method, the Least-squares functionals in the synchronization and modelling tasks of compression-expansion curves in the working cylinder are minimized. The calculation results of data synchronization for low-speed two-stroke and medium-speed four-stroke marine diesel engines are shown. The synchronization problem can be solved in terms of equation P' = 0 derived for the sector from compression starting to combustion starting in the cylinder. The selection of the boundary conditions for simulation is shown. The advantage of Powell’64 method is its high efficiency for quadratic functionals. As opposed to gradient methods, the Powell'64 method does not require calculating derivatives and is universal for minimizing complex nonlinear general functionals. The original author's algorithm of data synchronization by analyzing the indicator diagrams using the Powell'64 method has been applied in the latest versions of monitoring systems of D4.0H marine diesel engine.

In this work, a simple but robust 8-node 16-DOF quadrilateral membrane element with exceptional tolerances to mesh distortion is proposed for efficiently analyzing the two-dimensional orthotropic elasticity problems, within the improved framework of the unsymmetric FEM. This unsymmetric element model adopts a self-equilibrium metric stress field, which is formulated at the basis of Airy stress solutions of the plane problem, to be the trial functions. Meanwhile, to ensure the interelement compatibility, the standard isoparametric displacement field is employed to be the element’s test function. Numerical tests reveal that this quadratic element is always capable of producing exact solutions in constant and linear stress/strain problems and delivering quite satisfactory results in other higher-order benchmark problems. In particular, it can still work very well even when the element shape deteriorates severely into a concave quadrangle or degenerated triangle.

According to the characteristics of classical PSO algorithm(s), this paper uses the spatial equal-scale segmentation method (PESS) to effectively reduce the data dimension because of the low efficiency of PSO algorithm caused by big data and large population size. Diversity selection and global optimization, according to the law of evolution selection, the local optimization is focused on the two extremes of minimum value and maximal value, thus reducing the number of calculations of the fitness function of the algorithm. On this basis, we introduce a time-hop and elite Gaussian mixture strategy that can overcome the particle plunging into local optimum, while improving the efficiency of the algorithm, the generalization ability of the algorithm is taken into account. In order to prove the performance of FSPSO, we chose PSO and the self-organizing migration algorithm (SOMA) evolved from PSO that were tested in 30, 60 and 100 comparison tests on 5 different test function set with different degrees of complexity. The experimental results show that we proposed FSPSO algorithm has good effectiveness, robustness, complexity and universality in solving global optimization problems.

This chapter provides sufficient conditions for non existence Global weak solutions for non-local and non-linear equivalent equations on HN × (0,\(\infty\)) × (0,\(\infty\)), where HN is the Heisenberg group. Our method of proof relies on a suitable choice of a test function and the weak formulation approach of the sought for solutions.

Based on various constraints of actual parking problems, this paper constructs an automatic parking optimization model by taking the shortest parking trajectory as the optimization index and combines with the cubic spline theory. Firstly, a strategy of nonlinear decreasing inertia weight with iterative number is designed to enhance the global convergence ability of particle swarm optimization. Then, combined with genetic evolution mechanism, an adaptive mutation strategy is introduced to enhance the particle swarm diversity maintenance ability, so as to effectively improve its global convergence ability and avoid premature convergence in the late iteration. The simulation results of the test function and the actual problem of automatic parking for entering parking space indicate that the improved algorithm has higher searching accuracy and faster convergence speed.

One of the importance of the contamination uncertainty model is to consider in-determinism in the uncertainty. We consider this advanced property and develop two methods. These methods identify if there is imprecision in a given model or data. In the first approach, we build two different—a probability distribution and an interval—models for a test function f via given data/model. Then, we identify the level of imprecision by assessing, so-called model trust, $$\epsilon \in (0,1)$$ ϵ ∈ ( 0 , 1 ) in the contamination model whether the weight is higher for the probabilistic/interval model or not. In the second approach, we calculate the lowest and highest previsions for the test function and identify the imprecision interval out of them. We further discuss and show the idea via two simple production and clutch design problems to illustrate our novel results.

In this paper, we present some properties of integrable distributions which are continuous linear functional on the space of test function D ℝ 2 . Here, it uses two-dimensional Henstock–Kurzweil integral. We discuss integrable distributional solution for Poisson’s equation in the upper half space ℝ + 3 with Dirichlet boundary condition.

With the global development of the third industrial revolution, intelligent manufacturing has received attention from many countries and regions since it was first proposed. In the next ten years, intelligent manufacturing has become an important factor in determining international status, and it is imminent for traditional manufacturing to switch to intelligent manufacturing. Flexible job-shop scheduling is a key research problem in the field of intelligent manufacturing. In this paper, we uses a novel swarm intelligence optimization algorithm-Sparrow Search Algorithm to solve the problem of the longest processing time of workshop scheduling. The experimental results show that compared with other advanced meta-heuristic algorithms, the Sparrow Search Algorithm (SSA) can not only achieve ideal optimization accuracy in the test function, but also can achieve acceleration effects and solving capabilities that other algorithms do not have in actual shop scheduling problems.

Convolutional neural networks (CNNs) are effective models for image classification and recognition. Gradient descent optimization (GD) is the basic algorithm for CNN model optimization. Since GD appeared, a series of improved algorithms have been derived. Among these algorithms, adaptive moment estimation (Adam) has been widely recognized. However, local changes are ignored in Adam to some extent. In this paper, we introduce an adaptive learning rate factor based on current and recent gradients. According to this factor, we can dynamically adjust the learning rate of each independent parameter to adaptively adjust the global convergence process. We use the factor to adjust the learning rate for each parameter. The convergence of the proposed algorithm is proven by using the regret bound approach of the online learning framework. In the experimental section, comparisons are conducted between the proposed algorithm and other existing algorithms, such as AdaGrad, RMSprop, Adam, diffGrad, and AdaHMG, on test functions and the MNIST dataset. The results show that Adam and RMSprop combined with our algorithm can not only find the global minimum faster in the experiment using the test function but also have a better convergence curve and higher test set accuracy in experiments using datasets. Our algorithm is a supplement to the existing gradient descent algorithms, which can be combined with many other existing gradient descent algorithms to improve the efficiency of iteration, speed up the convergence of the cost function, and improve the final recognition rate.

This paper solves the drawbacks of traditional intelligent optimization algorithms relying on 0 and has good results on CEC 2017 and benchmark functions, which effectively improve the problem of algorithms falling into local optimality. The sparrow search algorithm (SSA) has significant optimization performance, but still has the problem of large randomness and is easy to fall into the local optimum. For this reason, this paper proposes a learning sparrow search algorithm, which introduces the lens reverse learning strategy in the discoverer stage. The random reverse learning strategy increases the diversity of the population and makes the search method more flexible. In the follower stage, an improved sine and cosine guidance mechanism is introduced to make the search method of the discoverer more detailed. Finally, a differential-based local search is proposed. The strategy is used to update the optimal solution obtained each time to prevent the omission of high-quality solutions in the search process. LSSA is compared with CSSA, ISSA, SSA, BSO, GWO, and PSO in 12 benchmark functions to verify the feasibility of the algorithm. Furthermore, to further verify the effectiveness and practicability of the algorithm, LSSA is compared with MSSCS, CSsin, and FA-CL in CEC 2017 test function. The simulation results show that LSSA has good universality. Finally, the practicability of LSSA is verified by robot path planning, and LSSA has good stability and safety in path planning.

This paper attempts to improve the learning representation of radial basis function neural network (RBFNN) through metaheuristic algorithm (MHA) and evolutionary algorithm (EA). Next, the ant colony optimization (ACO)-based and genetic algorithm (GA)-based approaches are employed to train RBFNN. The proposed hybridization of ACO-based and GA-based approaches (HAG) algorithm incorporates the complementarity of exploration and exploitation abilities to reach resolution optimization. The property of population diversity has higher chance to search the global optimal instead of being restricted to local optimal extremely in two benchmark problems. The experimental results have shown that ACO-based and GA-based approaches can be integrated intelligently and develop into a hybrid algorithm which aims for receiving the best precise learning expression among relevant algorithms in this paper. Additionally, method assessment results for two benchmark continuous test function experiments and show that the proposed HAG algorithm outperforms relevant algorithms in term of preciseness for learning of function approximation.

Lion swarm optimization (LSO) is a swarm intelligence algorithm that simulates lion king guarding, lioness hunting, and cub following. However, there are problems that lions are easily out of bounds when the range of activity is large and the position update formulas are not universal, which affect the performance of LSO. Aiming at above problems, a swarm intelligence algorithm, lion swarm optimization by reinforcement pattern search (RPSLSO) is proposed. The algorithm is based on the proposed modified lion swarm optimization (MLSO) and reinforcement pattern search (RPS) algorithm. The former solves above two problems, and the latter enhances the local search capability of MLSO, making the search more directional. In order to test the performance of RPSLSO, RPSLSO was compared with MLSO, LSO and the other two algorithms on the CEC2013 test function set. The experimental results show that the performance of RPSLSO is better, and the modifications to LSO and the proposed RPS in this paper are also effective.

The file attached to this record is the author's final peer reviewed version. The Publisher's final version can be found by following the DOI link. Heuristic optimisation algorithms are in high demand due to the overwhelming amount of complex optimisation problems that need to be solved. The complexity of these problems is well beyond the boundaries of applicability of exact optimisation algorithms and therefore require modern heuristics to find feasible solutions quickly. These heuristics and their effects are almost always evaluated and explained by particular problem instances. In previous works, it has been shown that many such algorithms show structural bias, by either being attracted to a certain region of the search space or by consistently avoiding regions of the search space, on. special test function designed to ensure uniform 'exploration' of the domain. In this paper, we analyse the emergence of such structural bias for Differential Evolution (DE) configurations and, specifically, the effect of different mutation, crossover and correction strategies. We also analyse the emergence of the structural bias during the run-time of each algorithm. We conclude with recommendations of which configurations should be avoided in order to run DE unbiased.

In view of the shortcomings of the basic lion swarm optimization, which is prone to local optimality and low convergence accuracy in partial optimization, this paper proposes a lion swarm optimization based on chaotic search and gaussian perturbation. The improved algorithm adds chaos search and gaussian perturbation strategy to the position of lions in the past dynasties, which improves the optimization efficiency of the algorithm in the optimization process. The simulation results of the test function show that the optimization accuracy of the improved algorithm is much higher than that of the basic lion swarm optimization. The improved algorithm effectively prevents the swarm optimization from easily falling into the local optimal value in the extremely difficult optimization function. Finally, an example of job-shop scheduling problem with the goal of minimizing the total time of job processing is tested. The test results verify the effectiveness of the algorithm.

We interpret likelihood-based test functions from a geometric perspective where the Kullback-Leibler (KL) divergence is adopted to quantify the distance from a distribution to another. Such a test function can be seen as a sub-Gaussian random variable, and we propose a principled way to calculate its corresponding sub-Gaussian norm. Then an error bound for binary hypothesis testing can be obtained in terms of the sub-Gaussian norm and the KL divergence, which is more informative than Pinsker's bound when the significance level is prescribed. For $M$-ary hypothesis testing, we also derive an error bound which is complementary to Fano's inequality by being more informative when the number of hypotheses or the sample size is not large.

In view of the drawbacks of chicken swarm algorithm, such as the fall into local optimization and slow optimizing speed, an improved strategy is proposed in this paper. The convergence speed and optimization accuracy of the algorithm are improved by elite reverse learning strategy. Adaptive Cauchy mutation strategy is applied to avoid the algorithm falling into local optimal in the late iteration, that is, elite reverse learning chicken algorithm with adaptive Cauchy mutation. The optimization performance of the improved algorithm is verified by the test function, and the numerical analysis of the simulation experiment shows that the improved algorithm has more powerful optimization capability than other improved algorithms.

We study the limiting distribution of a pair counting statistics of the form [Formula: see text] for the circular [Formula: see text]-ensemble (C[Formula: see text]E) of random matrices for sufficiently smooth test function [Formula: see text] and [Formula: see text] For [Formula: see text] and [Formula: see text] our results are inspired by a classical result of Montgomery on pair correlation of zeros of Riemann zeta function.

This paper proposed a method to determine the optimal location and sizing of Distributed Generation (DG) for power loss minimization using Ant Lion optimizer (ALO). The analysis also covers the effect of DG installation to voltage improvement and maximum system loadability index. ALO is an optimization algorithm that based on the nature interaction between ants and antlions. The nature interaction consists of five steps of hunting prey such as random walk of ants, building traps, entrapment of ants in traps, catching preys, and rebuilding traps. The ALO algorithm is tested on IEEE 69-bus distribution test system. The system will find the optimal location and sizing with corresponding load increase until it reaches the maximum system loadability (MSL) of the network. DG are usually attached to the end terminal at the load side of the system that refers to a technology that generate electricity at or near where it will be used such as solar panels and combined heat and power. The result of test function shows that the proposed algorithm can provide accurate yet competitive result in terms of power loss minimization, maximum system loadability enhancement and consistency.

This book is concerned with tangent cones, duality formulas, a generalized concept of conjugation, and the notion of maxi-minimizing sequence for a saddle-point problem, and deals more with algorithms in optimization. It focuses on the multiple exchange algorithm in convex programming.

We would like to prove a blow-up result for Sobolev solutions to the Cauchy problem for semi-linear structurally damped σ-evolution equations, where σ ≥ 1 and δ ∈ [0, σ) are assumed to be any fractional numbers. To deal with the fractional Laplacian (− Δ)σ and (− Δ)δ as well-known non-local operators, a modified test function method is applied to prove a blow-up result in the subcritical case and in the critical case as well.

In this paper, we discuss the influence of assuming Lm regularity of initial data, instead of L1, on a heat or damped wave equation with nonlinear memory. We find that the interplay between the loss of decay rate due to the presence of the nonlinear memory and to the assumption of initial data in Lm instead of L1, leads to a new critical exponent for the problem, whose shape is quite different from the one of the critical exponent for Lm theory for the corresponding problem with power nonlinearity |u|p. We prove the optimality of the critical exponent using the test function method.

We consider the initial boundary value problem in exterior domain for strongly damped wave equations with power-type nonlinearity $$|u|^p$$ . We will establish blow-up results under some conditions on the initial data and the exponent p, using the method of test function with an appropriate harmonic functions. We also study the existence of mild solution and its relation with the weak formulation.

Aiming at the problem of incomplete evaluation index of relay guidance task assignment, an evaluation function of the situational advantage and performance advantage of the guidance machine and target was designed. A relay assignment model was established. By adding greedy strategy, Tent chaotic initialization strategy and adaptive inertia weight strategy, Multi-Genetic Algorithm Tent- Dragonfly Algorithm (MGAT-DA) is proposed, the solution of the scheme is completed, and simulation is performed. First, the related theory of the single-target intelligent algorithm is introduced; second, the intelligent algorithm, the dragonfly algorithm, is introduced, and some improvements are made; again, the test function is used to test its optimized performance, and the experiment is compared with the traditional algorithm. ; Finally, the improved algorithm is used to solve the relay guidance task assignment scheme.Secondly, based on the linear programming to solve the optimal task allocation scheme, the guidance advantage under this scheme is 1.3667370376.

On the basis of derived bilinear form of \((2+1)\)-dimensional generalized Hirota-Satsuma-Ito equation with general coefficients, we emphasize on obtaining new analytic solutions of the considered equation. A novel test function has been appointed to formally derive various exact solutions containing abundant arbitrary constants. New solutions consist of hyperbolic, trigonometric, and exponential functions. Three-dimensional plots of all exact solutions determined in this research have also been provided in uniform manner.

Particle swarm optimization is one of the most effective optimization algorithms motivated by bird flocking behaviours. Population topology is a key aspect of particle swarm optimization research. However, after more than twenty years of research, the effects of the population topology are still poorly understood. Previous research has established that the information propagation speed determined by the population topology has an important impact on the algorithm performance; however, the impact of information propagation speed on particle swarm optimization and its variants has not yet been investigated. In this paper, information propagation in particle swarms is described and, hence, a method of simulating information propagation in particle swarms is introduced, which is used to obtain the information propagation speed. The correlation between the information propagation speed and algorithm performance is clarified through numerical simulation. The results show that the information propagation speed has a strong negative correlation with the population diversity of particle swarm optimization and its variants in the early iterations, regardless of the adopted test function and population diversity measure. The results also show that when optimizing problems with the same property, the impact of population topology on the optimization results of particle swarm optimization and variant algorithms is similar. Further more, this study provides some guidance on the population topology selection for particle swarm optimization and its variants. These findings contribute to our understanding of the impact of population topology on particle swarm optimization and its variants, and provide a basis for population topology selection for particle swarm optimization and its variants.

With the advancement of computing and inspired by optimal phenomena found in nature, several algorithms capable of solving complex engineering problems have been developed. This work details the development of the Multi-objective Lichtenberg Algorithm, the version capable of dealing with more than one objective of a newly created meta-heuristic inspired by the propagation of radial intra-cloud lightning and Lichtenberg figures. The algorithm considers in its optimization routine a hybrid system based on both the population and the trajectory, demonstrating a great capacity for exploration and exploitation since it distributes points to be evaluated in the objective function through a Lichtenberg figure that is shot in sizes and different rotations at each iteration. The Multi-objective Lichtenberg Algorithm (MOLA) is the first hybrid multi-objective meta-heuristic and was tested against traditional and recent meta-heuristics using famous and complex test function groups and also constrained complex engineering problems. Regarding important metrics for convergence and coverage assessment, the Multi-objective Lichtenberg Algorithm proved to be a promising multi-objective algorithm surpassing others traditional and recent algorithms such as NSGA-II, MOPSO, MOEA/D, MOGOA and MOGWO with expressive values of convergence and maximum spread.

Quantum electron transport in modern semiconductor devices can be described by a Wigner equation which is formally similar to the classical Liouville equation. The stationary Wigner equation has a singularity at zero momentum (k=0). In order to get a non-singular solution it is necessary to impose a constraint for the solution at k=0 which gives the constrained Wigner equation. We introduce a Petrov-Galerkin method for the solution of the corresponding constrained sigma equation. The constraint in the Wigner equation is interpreted as an extra test function and is naturally incorporated in the method.

The discontinuous Galerkin time-domain (DGTD) method used for the computation of electromagnetic fields in 3-D structures obtains the weak solution form of the Maxwell’s equations through spatial semidiscrete approximation and integration of test function. Numerical flux needs to be added between unstructured tetrahedral elements to compensate for boundary conditions. The establishment of corresponding surface mapping between mesh elements is a prerequisite for the numerical flux transfer between meshes. Based on fast sorting algorithm, this paper designs a grid correspondence judgment algorithm called ‘Sorting and Querying’, which reduces computational complexity from O(n^2) to O(nlogn). The numerical results show that the algorithm is faster than the similar algorithms and easier to program even in the case of a large number of grids.

The approximated nonlinear least squares (ALS) method has been used for the estimation of unknown parameters in the complex computer code which is very time-consuming to execute. The ALS calibrates or tunes the computer code by minimizing the squared difference between real observations and computer output using a surrogate such as a Gaussian process model. When the differences (residuals) are correlated or heteroscedastic, the ALS may result in a distorted code tuning with a large variance of estimation. Another potential drawback of the ALS is that it does not take into account the uncertainty in the approximation of the computer model by a surrogate. To address these problems, we propose a generalized ALS (GALS) by constructing the covariance matrix of residuals. The inverse of the covariance matrix is multiplied to the residuals, and it is minimized with respect to the tuning parameters. In addition, we consider an iterative version for the GALS, which is called as the max-minG algorithm. In this algorithm, the parameters are re-estimated and updated by the maximum likelihood estimation and the GALS, by using both computer and experimental data repeatedly until convergence. Moreover, the iteratively re-weighted ALS method (IRWALS) was considered for a comparison purpose. Five test functions in different conditions are examined for a comparative analysis of the four methods. Based on the test function study, we find that both the bias and variance of estimates obtained from the proposed methods (the GALS and the max-minG) are smaller than those from the ALS and the IRWALS methods. Especially, the max-minG works better than others including the GALS for the relatively complex test functions. Lastly, an application to a nuclear fusion simulator is illustrated and it is shown that the abnormal pattern of residuals in the ALS can be resolved by the proposed methods.

We present a correction of the mistake made in the proof concerning the choice of the test function in our published paper in J. Integral Equations Appl. 30:1 (2018), 67–80.

Optimization is a process in which the system is being modified so that the features can work more effectively or by finding any necessity execution. This process is conducted through maximizing and minimizing the parameters that are being involved in the problem. Many successful optimization approaches are available in the recent scenario; however, this research work focuses on cuckoo search technique. A new higher level procedure optimization algorithm called cuckoo search (CS) was developed by Yang and Deb in 2009. Cuckoo search is a swarm intelligence (SI) algorithm. Swarm intelligence techniques are those techniques that are being originated from the behavior of swarm. Cuckoo search algorithm is being inspired by the bird called cuckoo. It is the inspiration of brooding parasitism of cuckoo species and implemented successfully in many areas for optimization. This paper represents a brief review on CS algorithm along with the optimization process implementing some standard test functions. This research work highlights test function optimization for the non constraint function. This work may give an inspiration to apply the CS algorithm to solve the engineering design optimization problems.

Regional integrated energy site layout optimization involves multi-energy coupling, multi-data processing and multi-objective decision making, among other things. It is essentially a kind of non-convex multi-objective nonlinear programming problem, which is very difficult to solve by traditional methods. This paper proposes a decentralized optimization and comprehensive decision-making planning strategy and preprocesses the data information, so as to reduce the difficulty of solving the problem and improve operational efficiency. Three objective functions, namely the number of energy stations to be built, the coverage rate and the transmission load capacity of pipeline network, are constructed, normalized by linear weighting method, and solved by the improved p-median model to obtain the optimal value of comprehensive benefits. The artificial immune algorithm was improved from the three aspects of the initial population screening mechanism, population updating and bidirectional crossover-mutation, and its performance was preliminarily verified by test function. Finally, an improved artificial immune algorithm is used to solve and optimize the regional integrated energy site layout model. The results show that the strategies, models and methods presented in this paper are feasible and can meet the interest needs and planning objectives of different decision-makers.

In geotechnical engineering, investigation of seismic earth pressure coefficient is a fundamental theme of study for retaining wall. During this study, the seismal active earth pressure coefficient on the rear of the wall supporting c − Φ backfill has been formulated by the help of limit equilibrium method. This type of problems is highly complex nonlinear optimization problems. Therefore, it will very be difficult to analyze the problem using classical optimization techniques. A hybrid algorithm called hDEBSA has been discussed in the present study which was proposed by joining the parts of DE with the segments of BSA calculation to investigation the pseudo-static seismic dynamic earth pressure coefficient. In hDEBSA, a modification of parameter of DE and BSA has been performed through self-adaption-based. The proficiency of the hDEBSA has been checked through CEC2014 test functions and applied to analyze the coefficient of seismic active earth pressure on the rear of the retaining wall supportive \(c -\Phi\) backfill. The result obtained by this algorithm is compared with state-of-the-art other algorithms and are found to be in agreement. The achieved results of active earth pressure coefficient are in contrast with different results available found in the literature. Additionally, the impact of seismic parameters, soil and wall parameters on the earth pressure coefficient has been investigated.

When solving design optimization problems using evolutionary algorithms, the optimization process can be computationally expensive. To accelerate the optimization process, ordinary Kriging (OK) surrogate models are often used with the efficient global optimization (EGO) framework. However, in some cases the EGO framework can lead to a globally inaccurate OK surrogate model when many sample points are close to each other. One way to tackle this issue is to use a regression OK model instead of an interpolation OK model. In this paper, we propose an interpolation method which solve the issue by combining a local and a global OK model fitted to different set of the sample points. This paper describes the optimal weighting method used to combine the different Kriging models and compares the performance of the new method to interpolation and regression OK for the modified Branin test function. We find that when many sample points exist close to each other, the combined Kriging method outperform both the interpolation and the regression OK.

Aiming at the defects of chicken swarm optimization algorithm, such as easy to fall into local optimal, premature convergence and slow convergence, a chicken swarm optimization algorithm based on quantum behavior is proposed in this paper. A quantized potential well model is established based on the individual information of chicken swarm. According to the existing individual extremum and global extremum obtained by the original updating formula, Monte Carlo random sampling is adopted to complete the updating of individual extremum, and the search is conducted at a parallel Angle near individual extremum and global extremum, which improves the local search performance of the algorithm. At the same time, the convergence of quantum-behavior chicken swarm optimization algorithm is discussed in this paper, and QCSO is proved to be a globally convergent optimization algorithm. The optimization capability of QCSO is tested by using basic test function, and the results show that the optimization performance of this algorithm is greatly improved compared with the original algorithm.

This paper presents solutions for single objective optimization problems with developed algorithm from simulated annealing based on a simple (μ + λ) -ES, It is divided into two algorithms, separated mutation (SM1) and survival mutation (SM2). After that, compared with randomized local search and simulated annealing. The test function is part of the IEEE WCCI 2020 on the topic of CEC-C2 single objective bound constrained optimization. This research has chosen the basic functions in the test such as Bent cigar function, rastrigin function, high conditioned elliptic function, HGBat function, rosenbrock’s function, griewank’s function, discus function, expanded schaffer’s function, weierstrass function, sphere function, natyas function, levi function N.13, himmelblau’s function, and three-hump camel function. These functions are attract attention and competition. A results of SM1 and SM2 can solve single objective optimization problems better than RLS and SA. In high conditioned elliptic, The fitness value of RLS is equal to 3.96E-11, The fitness value of SA is equal to 8.12E-10, The fitness value of SM1 is equal to 5.39E-14 and The fitness value of SM2 is equal to 9.70E-15, It let us show the efficiency of SM2 that can get better results than SM1.

Finding a large set of optima in a multimodal optimization landscape is a challenging task. Classical population-based evolutionary algorithms (EAs) typically converge only to a single solution. While this can be counteracted by applying niching strategies, the number of optima is nonetheless trivially bounded by the population size. Estimation-of-distribution algorithms (EDAs) are an alternative, maintaining a probabilistic model of the solution space instead of an explicit population. Such a model is able to implicitly represent a solution set that is far larger than any realistic population size. To support the study of how optimization algorithms handle large sets of optima, we propose the test function EqalBlocksOneMax (EBOM). It has an easy to optimize fitness landscape, however, with an exponential number of optima. We show that the bivariate EDA mutual-information-maximizing input clustering (MIMIC), without any problem-specific modification, quickly generates a model that behaves very similarly to a theoretically ideal model for that function, which samples each of the exponentially many optima with the same maximal probability.

According to the characteristics of the localized operating system, the network system division is analyzed and the network protocol test method is proposed. Two methods are used to verify its function and performance, i.e. to check the test data of communication transmission rate through the TCP mode and UDP mode in hub and direct connection environment, and to check the test data of communication transmission rate between different system on the same machine. The dynamic test covers throughput test, packet loss rate test and delay test. The experimental results show that these two methods can ensure the correctness of the network test function and performance of the localized operating system, and meet the relevant network speed test requirements.