Mead Method Research Articles

Hybrid evolutionary programming using adaptive Lévy mutation and modified Nelder–Mead method

Evolutionary programming has been widely applied to solve global optimization problems. Its performance is related to both mutation operators and fitness landscapes. In order to make evolutionary programming more efficient, its mutation operator should adapt to fitness landscapes. The paper presents novel hybrid evolutionary programming with adaptive Levy mutation, in which the shape parameter of Levy probability distribution adapts to the roughness of local fitness landscapes. Furthermore, a modified Nelder–Mead method is added to evolutionary programming for enhancing its exploitation ability. The proposed algorithm is tested on 39 selected benchmark functions and also benchmark functions in CEC2005 and CEC2017. The experimental results demonstrate that the overall performance of the proposed algorithm is better than other algorithms in terms of the solution accuracy.

An efficient recurrent fuzzy CMAC model based on a dynamic-group–based hybrid evolutionary algorithm for identification and prediction applications

This article presents an efficient TSK-type recurrent fuzzy cerebellar model articulation controller (T-RFCMAC) model based on a dynamic-group--based hybrid evolutionary algorithm (DGHEA) for solving identification and prediction problems. The proposed T-RFCMAC model is based on the traditional CMAC model and the Takagi--Sugeno--Kang (TSK) parametric fuzzy inference system. Otherwise, the recurrent network, which imports feedback links with a receptive field cell, is embedded in the T-RFCMAC model, and the feedback units are used as memory elements. The DGHEA, which is a hybrid of the dynamic-group quantum particle swarm optimization (QPSO) and the Nelder--Mead method, is proposed for adjusting the parameters of the T-RFCMAC model. In DGHEA, an entropy-based grouping technique is adopted to improve the searching capability and the convergent speed of quantum particles swarm optimization. Experimental results show that the proposed DGHEA-based T-RFCMAC model is more effective at identification and prediction than other models.

Scalable Microwave-Assisted Johnson–Claisen Rearrangement with a Continuous Flow Microwave System

We demonstrated the rapid Johnson–Claisen rearrangement of allyl alcohol and triethyl orthoacetate with a continuous flow apparatus combined with a microwave reactor. The reaction could be carried out without solvent, and only a catalytic amount of acetic acid was sufficient to promote the reaction under microwave irradiation conditions. To confirm the optimal reaction conditions found experimentally, we performed Design of Experiments (DoE) by the Nelder–Mead method and a least-squares method regarding the amount of acetic acid and the flow rate. Consequently, the highest yield of the desired γ,δ-unsaturated ester was obtained, and the productivity at the reaction step of the continuous process was 89.5 g/h under the optimal conditions, suggesting that 2.1 kg of the product would be theoretically obtained in 1 day. We also investigated the Johnson–Claisen rearrangement using other allylic alcohols, and the corresponding products were obtained in good to high yields per unit of time.

Parallel two-phase methods for global optimization on GPU

Developing general global optimization algorithms is a difficult task, specially for functions with a huge number of local minima in high dimensions. Stochastic metaheuristic algorithms can provide the only alternative for the solution of such problems since they are aimed at guaranteeing global optimality. However, the main drawback of these algorithms is that they require a large number of function evaluations in order to skip/discard local optima, thus exhibiting a low convergence order and, as a result, a high computational cost. Furthermore, the situation can become even worse with the increase of dimension. Usually the number of local minima highly increases, as well as the computational cost of the function evaluation, thus increasing the difficulty for covering the whole search space. On the other hand, deterministic local optimization methods exhibit faster convergence rates, requiring a lower number of functions evaluations and therefore involving a lower computational cost, although they can get stuck into local minima. A way to obtain faster global optimization algorithms is to mix local and global methods in order to benefit from higher convergence rates of local ones, while retaining the global approximation properties. Another way to speedup global optimization algorithms comes from the use of efficient parallel hardware architectures. Nowadays, a good alternative is to take advantage of graphics processing units (GPUs), which are massively parallel processors and have become quite accessible cheap alternative for high performance computing. In this work a parallel implementation on GPUs of some hybrid two-phase optimization methods, that combine the metaheuristic Simulated Annealing algorithm for finding a global minimum, with different local optimization methods, namely a conjugate gradient algorithm and a version of Nelder–Mead method, is presented. The performance of parallelized versions of the above hybrid methods are analyzed for a set of well known test problems. Results show that GPUs represent an efficient alternative for the parallel implementation of two-phase global optimization methods.

An approach to an optimal T-head jetty: a numerical simulation case in Chamela Bay, Mexico

ABSTRACTA mathematical optimization problem is analyzed to obtain the most suitable T-head jetty design in an artificial tidal inlet system. In this study, the T-head jetty is considered optimal when two constraints are satisfied, specifically (1) the velocity of the currents and (2) the turbulence generated by the flow, both are minimal. These constraints are assessed in a control region around the inlet. To resolve the mathematical optimization problem, the hydrodynamic model Delft3D was combined with a derivative-free optimization method, the Nelder–Mead method. The effectiveness of this methodology was tested in a series of numerical experiments made at Chamela Bay on the Mexican Pacific Coast, where it is planned the construction of an artificial channel to connect the bay with a coastal lagoon. With this approach, future negative effects might be analyzed and reduced before modifying this pristine bay.

Grammatical evolution as a hyper-heuristic to evolve deterministic real-valued optimization algorithms

Hyper-heuristic methodologies have been extensively and successfully used to generate combinatorial optimization heuristics. On the other hand, there have been almost no attempts to build a hyper-heuristic to evolve an algorithm for solving real-valued optimization problems. In our previous research, we succeeded to evolve a Nelder–Mead-like real function minimization heuristic using genetic programming and the primitives extracted from the original Nelder–Mead algorithm. The resulting heuristic was better than the original Nelder–Mead method in the number of solved test problems but it was slower in that it needed considerably more cost function evaluations to solve the problems also solved by the original method. In this paper we exploit grammatical evolution as a hyper-heuristic to evolve heuristics that outperform the original Nelder–Mead method in all aspects. However, the main goal of the paper is not to build yet another real function optimization algorithm but to shed some light on the influence of different factors on the behavior of the evolution process as well as on the quality of the obtained heuristics. In particular, we investigate through extensive evolution runs the influence of the shape and dimensionality of the training function, and the impact of the size limit set to the evolving algorithms. At the end of this research we succeeded to evolve a number of heuristics that solved more test problems and in fewer cost function evaluations than the original Nelder–Mead method. Our solvers are also highly competitive with the improvements made to the original method based on rigorous mathematical convergence proofs found in the literature. Even more importantly, we identified some directions in which to continue the work in order to be able to construct a productive hyper-heuristic capable of evolving real function optimization heuristics that would outperform a human designer in all aspects.

Modified Spider Monkey Optimization based on Nelder–Mead method for global optimization

This paper proposed a modified version of Spider Monkey Optimization (SMO) algorithm for solving global optimization problems. The traditional SMO consists of seven phases where each phase has its characteristics and tasks to be performed. However, the local leader phase (LLP), that is the second phase of the SMO has the most significant effect on the performance of the algorithm. In which if it does not has good exploration and exploitation capability then the SMO might stick at a local point. Therefore, the proposed modified version of SMO (that called SMONM) used the transformations of the Nelder–Mead (NM) method to improve the ability of LLP. The proposed SMONM algorithm contains the same number of phases of the traditional SMO except the LLP that has modified through using the reflection, expansion, and contraction transformations of the NM. These transformations of NM worked if there is no improvement in the fitness function value after the solution is updating using the original LLP. The performance of the proposed algorithm has compared with other four algorithms namely, original SMO, Artificial Bee Colony optimization, Biography Based Optimization and Particle Swarm Optimization. A set of experimental series is performed to evaluate the performance of the proposed algorithm using 23 standard benchmark functions, 15 composite functions, and three classical engineering problems. The preliminary results show that the modified version of SMO has excellent ability to avoid the limitations of the tradition SMO algorithm, as well as, it provides better results than the other comparative algorithms regarding performance measures.

Determination of Meteorite Trajectories: Observations and Modeling

We construct the complete trajectory of a meteorite from observations on the visible (“bright”) section of the trajectory. Mathematically, this is a minimization problem for a functional defined as the deviation of the observed trajectory from the trajectory calculated using the system of meteor physics equations. This differential system has been solved numerically, and the minimization problem has been solved by the Nelder–Mead method. The solution makes it possible to determine the physical parameters of the body entering the Earth’s atmosphere and to complete its trajectory on the unobservable (“dark”) section until its collision with the Earth’s surface.

Updating Finite Element Model Using Stochastic Subspace Identification Method and Bees Optimization Algorithm

Abstract This study investigates the application of operational modal analysis along with bees optimization algorithm for updating the finite element model of structures. Bees algorithm applies instinctive behavior of honeybees as they look for nectar of flowers. The parameters that needed to be updated are uncertain parameters such as geometry and material properties of the structure. To determine these uncertain parameters, local and global sensitivity analyses have been performed. An objective function is defined based on the sum of the squared errors between the natural frequencies obtained from operational modal analysis and finite element method. The natural frequencies of physical structure are determined by stochastic subspace identification method which is considered as a strong and efficient method in operational modal analysis. To verify the accuracy of this method, the proposed algorithm is implemented on a three-story structure to update parameters of its finite element model. Moreover, to study the efficiency of bees algorithm, its results are compared with those of the particle swarm optimization, and Nelder and Mead methods. The comparison indicates that this algorithm leads more accurate results with faster convergence.

Differential evolution algorithm with multiple mutation strategies based on roulette wheel selection

In this paper, we propose a differential evolution (DE) algorithm variant with a combination of multiple mutation strategies based on roulette wheel selection, which we call MMRDE. We first propose a new, reflection-based mutation operation inspired by the reflection operations in the Nelder–Mead method. We design an experiment to compare its performance with seven mutation strategies, and we prove its effectiveness at balancing exploration and exploitation of DE. Although our reflection-based mutation strategy can balance exploration and exploitation of DE, it is still prone to premature convergence or evolutionary stagnation when solving complex multimodal optimization problems. Therefore, we add two basic strategies to help maintain population diversity and increase the robustness. We use roulette wheel selection to arrange mutation strategies based on their success rates for each individual. MMRDE is tested with some improved DE variants based on 28 benchmark functions for real-parameter optimization that have been recommended by the Institute of Electrical and Electronics Engineers CEC2013 special session. Experimental results indicate that the proposed algorithm shows its effectiveness at cooperative work with multiple strategies. It can obtain a good balance between exploration and exploitation. The proposed algorithm can guide the search for a global optimal solution with quick convergence compared with other improved DE variants.

Construction of D-Optimal Experimental Designs for Nonparametric Regression Models

Under study is the problem of a D-optimal experimental design for the problem of nonparametric kernel smoothing. Modification is proposed for the process of calculating the Fisher information matrix. D-optimal designs are constructed for one and several target points for the problems of nonparametric kernel smoothing using a uniform kernel, the Gauss and Epanechnikov kernels. Comparison is performed between Fedorov’s algorithm and direct optimization methods (such as the Nelder–Mead method and the method of differential evolution). The features of the application of the optimality criterion for the experimental design of the problems with several target points were specified for the cases of various kernels and bandwidths.

Application of Hybrid Nelder-Mead Bat Algorithm to Improve the Grasp Quality during the Automated Robotic Grasping

Robots are introduced in the manufacturing industry to automate the manufacturing process to meet the high productivity in the manufacturing industry as well as for manufacturing high quality and low-cost product for the customer. Robots are playing a crucial role in handling the object in pick and place operation, palletizing, welding, handling of materials and other operations in automated manufacturing industries. Robot-mounted with the gripper as the end effector need to identify the best position on the object for positioning the tip of the gripper for a successful grasp. The key to a successful and a stable grasp in automation is to obtain the most appropriate grasp planning and grasp synthesis for the desired task. In this work, a hybrid Meta heuristics method has been developed by combing an efficient local search method known as Nelder Mead method with the Bat algorithm and the new proposed hybrid algorithm is known as the hybrid Nelder-Mead Bat algorithm (HNMBA). The proposed algorithm was introduced for solving the automated grasp synthesis and grasp planning problem in the robotic grasping in an automated manufacturing line. The fitness function for the considered robotic grasp synthesis problem has been formulated using the largest ball grasp quality criteria. The proposed methodology is validated for the tessellated 3D objects, and the obtained results are compared with the previously proposed methods.

Temporal Limitations of the Standard Leaky Integrate and Fire Model

The leaky integrate and fire model of neural spiking has been used extensively to simulate saccadic responses in a variety of tasks from visual search to simple reaction times. Although it has been tested for its neural spiking accuracy and its spatial prediction of fixations in visual salience, it has not been well tested for its temporal accuracy. Saccade generation invariably results in a positively skewed distribution of saccadic reaction times over large numbers of samples, yet we show that the LIF algorithm tends to produce a distribution shifted to shorter fixations (in comparison with human data) in its classic implementation. Further, parameter optimization using a genetic algorithm and Nelder–Mead method does improve the fit of the resulting distribution, but is still unable to match temporal distributions of human responses in a simple visual search task. Further analysis revealed, that the LIF algorithm produces discrete reaction times instead of distributions. Aggregated over many pictures they may be treated as a distribution although the form of this distribution depends on the input images used to create it.

Estimation of Hydrologic Parameters of Kocanaz Watershed by a Hydrologic Model

The main goal of this study is to estimate the hydrologic parameters of Kocanaz watershed located in Western Black Sea Region, using a semi-distributed hydrologic model, Hydrologic Engineering Center – Hydrologic Modelling System (HEC-HMS). In this study, the hydrologic model was set up for two flood events occurred in 2002 and 2013, in which one was used for calibration while the other one was used for validation of the calibrated hydrologic parameters. The watershed was introduced into the model as a single basin. Hydrologic parameters used for transformation from rainfall to runoff and base flow estimations were searched using Nelder and Mead method and calibrated using the peak-weighted root mean square error. The model results were deliberated by statistical indicators such as Nash-Sutcliffe efficiency coefficient (E NS ), coefficient of determination (R 2 ), mass balance error (MBE) and peak flow rate error (PE). The model results were found to be very good for calibration and satisfactory for validation while the peak flow rate was under estimated for both calibration and validation. It can be concluded that small watersheds such as Kocanaz might be modelled as a single basin without sacrificing the estimation capability of the model and increasing the model simulation time. Though, slightly improved peak discharge estimations were obtained in case of using sub basins.

Identification of mathematical model of bacteria population under the antibiotic influence

Abstract This work is devoted to the identification of a mathematical model of bacteria population under the antibiotic influence, based on the solution of the corresponding inverse problems. These problems are solved by the gradient method, genetic algorithm and Nelder–Mead method. Calculations are made using model and real data.

The inverse coefficient problem of heat transfer in layered nanostructures

The rapid development of electronics leads to the creation and use of electronic components of small dimensions, including nanoelements of complex, layered structure. The search for effective methods for cooling electronic systems dictates the need for the development of methods for the numerical analysis of heat transfer in nanostructures. A characteristic feature of energy transfer in such systems is the dominant role of contact thermal resistance at interlayer interfaces. Since the contact resistance depends on a number of factors associated with the technology of heterostructures manufacturing, it is of great importance to determine the corresponding coefficients from the results of temperature measurements.The purpose of this paper is to evaluate the possibility of reconstructing the thermal resistance coefficients at the interfaces between layers by solving the inverse problem of heat transfer.The complex of algorithms includes two major blocks — a block for solving the direct heat transfer problem in a layered nanostructure and an optimization block for solving the inverse problem. The direct problem was formulated in an algebraic (finite difference) form under the assumption of a constant temperature within each layer, which is due to the small thickness of the layers. The inverse problem was solved in the extreme formulation, the optimization was carried out using zero-order methods that do not require the calculation of the derivatives of the optimized function. As a basic optimization algorithm, the Nelder—Mead method was used in combination with random restarts to search for a global minimum.The results of the identification of the contact thermal resistance coefficients obtained in the framework of a quasi-real experiment are presented. The accuracy of the identification problem solution is estimated as a function of the number of layers in the heterostructure and the «measurements» error.The obtained results are planned to be used in the new technique of multiscale modeling of thermal regimes of the electronic component base of the microwave range, when identifying the coefficients of thermal conductivity of heterostructure.

New optimal controller tuning method for an AVR system using a simplified Ant Colony Optimization with a new constrained Nelder–Mead algorithm

In this paper, an optimal gain tuning method for PID controllers is proposed using a novel combination of a simplified Ant Colony Optimization algorithm and Nelder–Mead method (ACO-NM) including a new procedure to constrain NM. To address Proportional-Integral-Derivative (PID) controller tuning for the Automatic Voltage Regulator (AVR) system, this paper presents a meta-analysis of the literature on PID parameter sets solving the AVR problem. The investigation confirms that the proposed ACO-NM obtains better or equivalent PID solutions and exhibits higher computational efficiency than previously published methods. The proposed ACO-NM application is extended to realistic conditions by considering robustness to AVR process parameters, control signal saturation and noisy measurements as well as tuning a two-degree-of-freedom PID controller (2DOF-PID). For this type of PID, a new objective function is also proposed to manage control signal constraints. Finally, real time control experiments confirm the performance of the proposed 2DOF-PIDs in quasi-real conditions. Furthermore, the efficiency of the algorithm is confirmed by comparing its results to other optimization algorithms and NM combinations using benchmark functions.

Метод Нелдера–Мида в обучении искусственной нейронной сети, входящей в состав ШИМ-элемента

Метод Нелдера–Мида в обучении искусственной нейронной сети, входящей в состав ШИМ-элемента

Optimization of thermal modes and cooling systems of the induction traction engines of trams

We developed a procedure for the optimization of thermal modes and parameters of the cooling system of induction traction engines of tram carriages. The procedure includes the following basic steps. The optimization of operating modes of an induction traction drive by the criterion of effectiveness of its work under different modes. The optimization of motion modes of a tram carriage along a track section with the assigned motion schedule and profile based on the curves of the motion of a tram carriage, optimal by the criterion of energy consumption, using the method of Hamilton-Jacobi-Bellman. The optimization of parameters of the cooling fan of traction engines by the criterion of efficiency of a cooling system using the Weyl method by the generalized golden section. It is proposed to conduct determining of operating modes of a traction drive in advance based on the solution to the problem of conditional optimization of its modes. In order to determine the optimal operating modes of a traction drive, we selected a combined method: global search is executed by genetic algorithm with a one-point crossover and by selection on the principle of roulette. At the final stage of an optimization procedure, optimum refining is carried out using the Nelder–Mead method. When a tram carriage moved along a track section, we defined the following. We determined the optimal modes of motion of the tram carriage T-3 VPA with induction traction engines for a track section with the assigned motion schedule. It was found that, compared with the basic design, efficiency of the cooling system increased by 27.6 %, which corresponds to a reduction in the proposed criterion of efficiency. Based on the results of modeling a traction engine with an optimal fan, it was established that the largest overheating is observed in the frontal part of the stator winding. The temperature is 139.6 °C at second 3363 from starting the motion and it does not exceed a permissible value of 140 °C.

Updating finite element model using frequency domain decomposition method and bees algorithm

The following study deals with the updating the finite element model of structures using the operational modal analysis. The updating process uses an evolutionary optimization algorithm, namely bees algorithm which applies instinctive behavior of honeybees for finding food sources. To determine the uncertain updated parameters such as geometry and material properties of the structure, local and global sensitivity analyses have been performed. The sum of the squared errors between the natural frequencies obtained from operational modal analysis and the finite element method is used to define the objective function. The experimental natural frequencies are determined by frequency domain decomposition technique which is considered as an efficient operational modal analysis method. To verify the accuracy of the proposed algorithm, it is implemented on a three-story structure to update its finite element model. Moreover, to study the efficiency of bees algorithm, its results are compared with those particle swarm optimization and Nelder and Mead methods. The results show that this algorithm leads more accurate results with faster convergence. In addition, modal assurance criterion is calculated for updated finite element model and frequency domain decomposition technique. Moreover, finding the best locations of acceleration and shaker mounting in order to accurate experiments are explained.