Rosenbrock Function Research Articles

Mathematical Modelling and Hierarchical Encourage Particle Swarm Optimization Genetic Algorithm for Jet Pipe Servo Valve.

The jet pipe servo (JPS) valve is one key component, whose dynamic performance directly influences the aircraft's maneuverability. In this paper, a more accurate mathematical model and a novel multiobjective hierarchical encourage particle swarm optimization genetic algorithm (HEPGA) are proposed to improve the dynamic performance of the jet pipe servo valve. By optimizing the main structure parameters of the jet pipe servo valve, the adjustment and overshoot in the dynamic performance are reduced by 24.28% and 51.39%, respectively, compared with the prototype before optimization. To obtain a more accurate mathematical model, the computational fluid dynamics (CFD) is introduced to modify the analytical model considering the turbulent submerged free jet. Different from conventional numerical simulation, the dynamic mesh technique is used to analyze the flow field distribution by considering the force interaction of various parts of the jet pipe servo valve under actual working condition. Then, the HEPGA with better convergence is utilized because of the conflict of adjustment and overshoot. This proposed hybrid algorithm introduces the concept of staff welfare system to divide the population into elite individuals and excellent individuals of particle swarm optimization and general individuals of genetic algorithm. Meanwhile, the convergency performance of the HEPGA is evaluated through the Rosenbrock function by comparing with other particle swarm genetic hybrid methods. Subsequently, the experimental platform is constructed and the dynamic performance tests are conducted on the prototype after optimization. The experimental results verify the accuracy of the established mathematical model and the significant improvement of dynamic performance of the jet pipe servo valve.

Overall Dynamic Optimization of Metal Mine Technical Indicators Considering Spatial Distribution of Ore Grade Using AADE

In optimizing the production of a metal mine, either the overall dynamic relations between technical indicators or the spatial distribution of the ore grade are usually considered, but few studies have considered both factors together. These two factors in combination have a greater effect on the optimization of mine production in terms of economic benefit and resource utilization than they do individually. We proposed an overall dynamic optimization model of technical indicators of metal mine production that considers the spatial distribution of the ore grade to better optimize the technical indicators and improve sustainable development of mineral resources. We incorporated an adaptive mutation strategy and adaptive control parameters into a differential evolution algorithm (AADE) in order to overcome the drawbacks of the differential evolution algorithm in solving this optimization model. The adaptive mutation strategy and adaptive control parameters were used to increase the rate of convergence and improve the search for a global maximum. To assess the performance of AADE, we used a real case and four test functions (the Sphere, Griewank, Rastrigin and Rosenbrock functions) in tests that compared AADE with a standard genetic algorithm, a standard differential evolution algorithm and the recently developed adaptive differential evolution algorithm. The results indicate that the optimization model we created is better aligned with mine production processes than current optimization models. In optimizing the technical indicators of metal mine production to maximize economic benefits, AADE performed significantly better than the other three algorithms tested in terms of convergence rate and global search ability.

Ship Attitude Prediction Model Based on Cross-Parallel Algorithm Optimized Neural Network

In recent years, with the development of modern technology and industrialization, the shipbuilding industry has become an essential field in a country. Offshore operations, such as helicopter take-off and landing, and ship-to-ship cargo replenishment, need an accurate prediction of ship motion attitude, not only to improve efficiency but also to protect crew members’ lives. However, using the single model has limited influence on prediction. Therefore, this paper proposed a Cross-Parallel optimization algorithm framework, which made full use of the development ability of the Harris Hawk Optimization algorithm (HHO) and the searching ability of the Sine-Cosine Algorithm (SCA). In this paper, we use Rosenbrock and Rastrigin functions for testing. The results show that the Cross-Parallel optimization algorithm has faster search speed and stronger optimization capability. The optimization algorithm is used to optimize the hyperparameters of Long Short-Term Memory (LSTM) neural networks. It improves the network’s training ability to predict the ship’s attitude. The proposed method in this paper is compared with 9 models such as BP, LSTM, GRU, BILSTM and TCN. It is tested with three degrees of freedom under different sea states. MAE and RMSE are used as the evaluation indexes of this paper. In the six sets of experimental results, compared with the other nine models, the method in this paper can reduce the MAE by up to 9.2%, 7.2%, 4.7%, 5.4%, 6.4%, and 6.8%, respectively. When using RMSE as the evaluation index, the decrease was at most 10.1%, 8.7%, 5.6%, 7.1%, 7.5%, and 8.2%. The experimental results show that the method in this paper has the highest prediction accuracy.

On Parameter Identification for Reaction-Dominated Pore-Scale Reactive Transport Using Modified Bee Colony Algorithm

Parameter identification is an important research topic with a variety of applications in industrial and environmental problems. Usually, a functional has to be minimized in conjunction with parameter identification; thus, there is a certain similarity between the parameter identification and optimization. A number of rigorous and efficient algorithms for optimization problems were developed in recent decades for the case of a convex functional. In the case of a non-convex functional, the metaheuristic algorithms dominate. This paper discusses an optimization method called modified bee colony algorithm (MBC), which is a modification of the standard bees algorithm (SBA). The SBA is inspired by a particular intelligent behavior of honeybee swarms. The algorithm is adapted for the parameter identification of reaction-dominated pore-scale transport when a non-convex functional has to be minimized. The algorithm is first checked by solving a few benchmark problems, namely finding the minima for Shekel, Rosenbrock, Himmelblau and Rastrigin functions. A statistical analysis was carried out to compare the performance of MBC with the SBA and the artificial bee colony (ABC) algorithm. Next, MBC is applied to identify the three parameters in the Langmuir isotherm, which is used to describe the considered reaction. Here, 2D periodic porous media were considered. The simulation results show that the MBC algorithm can be successfully used for identifying admissible sets for the reaction parameters in reaction-dominated transport characterized by low Pecklet and high Damkholer numbers. Finite element approximation in space and implicit time discretization are exploited to solve the direct problem.

An n‐dimensional Rosenbrock distribution for Markov chain Monte Carlo testing

AbstractThe Rosenbrock function is a ubiquitous benchmark problem in numerical optimization, and variants have been proposed to test the performance of Markov chain Monte Carlo algorithms on distributions with a curved and narrow shape. In this work we discuss the Rosenbrock distribution and the advantages and limitations of its current n‐dimensional extensions. We then propose a new extension to arbitrary dimensions called the Hybrid Rosenbrock distribution, which addresses all the limitations that affect the current extensions. The Hybrid Rosenbrock distribution is composed of conditional normal kernels arranged in such a way that preserves the key features of the original Rosenbrock kernel. Moreover, due to its structure, the Hybrid Rosenbrock distribution is analytically tractable, and possesses several desirable properties which make it an excellent test model for computational algorithms. We conclude with numerical experiments that show how commonly used Markov chain Monte Carlo algorithms may fail to explore densities with curved correlation structure, restating the importance of a reliable benchmark problem for this class of densities.

A global optima search field division method for evolutionary algorithms

This study aims to investigate the deployment of a proposed search field division method (SFDM) within evolutionary algorithms (EAs) to enhance the capability of searching for the global optima in nonlinear problems. The proposed technique is benchmarked against the following eight widely-used single-modal, multi-modal, and unimodal benchmark functions: Sphere, Rosenbrock, Rastringin, Griewank, Ackley, Fletcher, Quartic, and Schwefel functions, and the outcome is compared to their standard EAs counterparts to validate the effectiveness of the deployed approach in EAs. In the proposed method, we apply three low, medium, high field divisions (1, 2, and 5) dimensions on nine different EAs simultaneously with two different scenarios, 10 and 100 variables, to reach the optimal solution. Then for the validity of our proposed SFDM technique, we examined the exploration-exploitation search space rates and diversity behavior. The results of the implementation of SFDM on eight benchmark test functions show that the consideration of dimensions using SFDM for EAs improves the outcomes of all nine tested EAs. In our proposed method, we find better compatibility with the integration of SFDM in the Particle Swarm Optimization Algorithm concerning searching for the optimum solution relative to the other EAs. Highlights A novel Search Field Division Method (SFDM) for evolutionary algorithms. Applying three field divisions (1, 2, and 5) dimensions on nine different Evolutionary Algorithms (EAs) simultaneously to reach the optimal global solution. The implementation of SFDM on eight benchmark functions shows significant improvement in all nine tested EAs. In our proposed method, we find the Particle Swarm Optimization Algorithm has better compatibility with respect to the other EAs.

Optimization of particle swarm algorithm and its usage in calculation of hydrogeological parameter

In order to improve the accuracy, sensitivity, and convergence, the particle swarm optimization (PSO) algorithm is optimized and applied in the calculation of hydrogeological parameters to improve the accuracy of the hydrogeological model. The sensitivity and accuracy of the algorithm are improved through derivation of the appropriate value of the particles. Based on convergency analysis of this algorithm with the three intelligent optimization algorithms (Sphere function, Rosenbrock function, and Griewank function), the convergence of the algorithm is analyzed through different intelligently optimized algorithms. At the same time, the accuracy is validated through comparison between the optimized and unoptimized particle swarm algorithm as well as the actually observed hydrogeological data, and the sensitivity of the water conductivity coefficient and water storage coefficient under the algorithm is analyzed. The results show that the calculated value of the optimized PSO algorithm is very close to the theoretical value 0 of the intelligent optimization algorithm Sphere function and the theoretical value 1 of the Rosenbrock function as well as the theoretical value 1 of the Griewank function. The results reveal that the calculation results are very close to the theoretical value in the intelligent optimization algorithm test of optimized particle swarm algorithm; the maximum absolute error between the calculated value and the observed value of the optimized particle swarm algorithm is 0.011, and the maximum relative error of which is 8.1%; the maximum absolute error between the calculated value of particle swarm algorithm and the observed value is 0.021, and the maximum relative error reaches 29.1%; the number of iterations of the optimized particle swarm algorithm is 87 on average, while it of the unoptimized particle swarm algorithm is 450, reaching the optimal value. In addition, the optimized PSO algorithm has a standard function of 0.00659, which is significantly smaller than that of the PSO algorithm of 0.00684. When the interference coefficient is −20%~20%, the water conductivity coefficient and water storage coefficient are in negative phase with the drop depth of the aquifer. It suggests that the optimized particle swarm optimization algorithm has high accuracy and convergence, and its sensitivity of water conductivity coefficient and sensitivity of water storage coefficient are both good, which can provide a reliable algorithm basis for the construction of hydrogeological model and the establishment of aquifer parameters.

THE METHOD OF ITERATIVE SEARCH FOR THE OPTIMAL FIELD DEVELOPMENT OPTION

3D hydrodynamic modeling is the standard tool for predicting the development of hydrocarbon (HC) fields. The relevance of the work is associated with the need to introduce fast and affordable optimization algorithms into engineering practice, which will reduce the cost of computer time to justify the best and most effective development solutions. The authors have proposed a new express method for finding the optimal option for the development of deposits. The method works with discrete sets of possible variations of the required development parameters (for example, the number of wells, the type of completion, the rate of hydrocarbon reserves withdrawal etc.) and minimizes the number of launches of the reservoir simulation simulator per forecast required for the feasibility of various combinations study of parameters and finding both local and global optimal combinations. Compared with other methods, its advantage is simplicity and realizability in the “manual” mode with a small number of variable parameters, which can be useful for practical problems. The method uses the principle of iterations and is tested on several examples, including the results of hydrodynamic modeling, a comparison is made with known optimization algorithms — in some problems the method allows finding the optimum faster. For example, in the problem of finding the optimal location of horizontal wells, the iterative search turned out to be faster than the “swarm of particles” method. On the other hand, the method does not allow one to reliably determine the optima of complex objective functions that have several local optima. Testing was carried out on the Himmelblau and Rosenbrock functions: in the first case, all five local optima were found, in the second case, seven out of twelve.

A Flow Perspective on Nonlinear Least-Squares Problems

Just as the damped Newton method for the numerical solution of nonlinear algebraic problems can be interpreted as a forward Euler timestepping on the Newton flow equations, the damped Gauß–Newton method for nonlinear least squares problems is equivalent to forward Euler timestepping on the corresponding Gauß–Newton flow equations. We highlight the advantages of the Gauß–Newton flow and the Gauß–Newton method from a statistical and a numerical perspective in comparison with the Newton method, steepest descent, and the Levenberg–Marquardt method, which are respectively equivalent to Newton flow forward Euler, gradient flow forward Euler, and gradient flow backward Euler. We finally show an unconditional descent property for a generalized Gauß–Newton flow, which is linked to Krylov–Gauß–Newton methods for large-scale nonlinear least squares problems. We provide numerical results for large-scale problems: An academic generalized Rosenbrock function and a real-world bundle adjustment problem from 3D reconstruction based on 2D images.

Study of League Championship Algorithm Efficiency for Global Optimization Problem

The article objective is to study a new League Championship Algorithm (LCA) algorithm efficiency by its comparing with the efficiency of the Particle Swarm optimization (PSO) algorithm.The article presents a brief description of the terms used in the League Championship algorithm, describes the basic rules of the algorithm, on the basis of which the iterative process for solving the global optimization problem is built.Gives a detailed description of the League Championship algorithm, which comprises a flowchart of the algorithm, as well as a formalization of all its main steps.Depicts an exhaustive description of the software developed to implement the League Championship algorithm to solve global optimization problems.Briefly describes the modified particle swarm algorithm. Presents the values of all free parameters of the algorithm and the algorithm modifications, which make it different from the classical version, as well.The main part of the article shows the results of a great deal of computational experiments using two abovementioned algorithms. All the performance criteria, used for assessment of the algorithms efficiency, are given.Computational experiments were performed using the spherical function, as well as the Rosenbrock, Rastrigin, and Ackley functions. The results of the experiments are summarized in Tables, and also illustrated in Figures. Experiments were performed for the vector dimension of the variable parameters that is equal to 2, 4, 8, 16, 32, and 64.An analysis of the results of computational experiments involves a full assessment of the efficiency of the League Championship algorithm, and also provides an answer about expediency for further algorithm development.It is shown that the League Championship algorithm presented in the article has a high development potential and needs further work for its study.

A generalized methodology for multidisciplinary design optimization using surrogate modelling and multifidelity analysis

The advantages of multidisciplinary design are well understood, but not yet fully adopted by the industry where methods should be both fast and reliable. For such problems, minimum computational cost while providing global optimality and extensive design information at an early conceptual stage is desired. However, such a complex problem consisting of various objectives and interacting disciplines is associated with a challenging design space. This provides a large pool of possible designs, requiring an efficient exploration scheme with the ability to provide sufficient feedback early in the design process. This paper demonstrates a generalized optimization framework with rapid design space exploration capabilities in which a Multifidelity approach is directly adjusted to the emerging needs of the design. The methodology is developed to be easily applicable and efficient in computationally expensive multidisciplinary problems. To accelerate such a demanding process, Surrogate Based Optimization methods in the form of both Radial Basis Function and Kriging models are employed. In particular, a modification of the standard Kriging approach to account for Multifidelity data inputs is proposed, aiming to increasing its accuracy without increasing its training cost. The surrogate optimization problem is solved by a Particle Swarm Optimization algorithm and two constraint handling methods are implemented. The surrogate model modifications are visually demonstrated in a 1D and 2D test case, while the Rosenbrock and Sellar functions are used to examine the scalability and adaptability behaviour of the method. Our particular Multiobjective formulation is demonstrated in the common RAE2822 airfoil design problem. In this paper, the framework assessment focuses on our infill sampling approach in terms of design and objective space exploration for a given computational cost.

Time-varying continuous-time optimisation with pre-defined finite-time stability

In this paper, we propose a new family of continuous-time optimisation algorithms based on discontinuous second-order gradient optimisation flows, with finite-time convergence guarantees to local optima, for locally strongly convex (time-varying) cost functions. To analyse our flows, we first extend a well-know Lyapunov inequality condition for finite-time stability, to the case of (time-varying) differential inclusions. We then prove the convergence of these second-order flows in finite-time. In some particular cases, we can show that the finite-time convergence can be pre-defined by the user. We propose a robustification of the flows to bounded additive uncertainties and extend some of the results to the case of constrained optimisation. We show the performance of these flows on well-known optimisation benchmarks, namely, the Rosenbrock function, and the Rastringin function.

Robust Time-Varying Continuous-Time Optimization with Pre-Defined Finite-Time Stability

In this paper we propose a new family of continuous-time optimization algorithms based on discontinuous second order gradient optimization flows, with finite-time convergence guarantees to local optima, for locally strongly convex time-varying cost functions. To analyze our flows, we first extend a well-know Lyapunov inequality condition for finite-time stability, to the case of time-varying differential inclusions. We then prove the convergence of these second-order flows in finite-time. We show the performance of these flows on a time-varying quadratic cost and on the nonlinear time-varying Rosenbrock function.

Comparison of methods for start points initializing of a non-parametric optimization algorithm

The problem of global optimization arises in various fields of science and technology, and several different ways of solving it have been proposed. The results of the study of the effectiveness of the non-parametric global optimization algorithm are presented. A comparative analysis of this algorithm is presented. performance analysis of the algorithm based on the Ackley, Rastrigin, Shekel, Griewank and Rosenbrock function. In addition, studies were carried out for the three initial points of the distribution algorithms: the sequence LPτ, the sequence UDC, the uniform random distribution. thus, the best way to initialize the initial points of the non-parametric optimization algorithm on these test functions was identified. According to the research results, the effective parameters of the genetic algorithm were established.

Optimal parameters selection of the genetic algorithm for global optimization

The purpose of this work is to summarize the results of research concerning the application of genetic algorithms, since in solving problems of complex systems optimization situations often make it difficult or impossible to use classical methods. To solve this problem, research is carried out on the functions of Akli, Rastrigin, Shekel, complaints handling functions and Rosenbrock functions. The studies are conducted on three starting point scattering algorithms: LPτ sequence, UDC sequences and universal random variation. As a result of the analysis, the option of initialization, selection, recombination, mutation and coding of this algorithm according to given test functions for the data of the scatter of initial points is chosen. The effective parameters of the genetic algorithm according to the results of research are established.

Comparative Study of Unconstrained Mechanical Optimization Methods Based on Two-variable Rosenbrock Function

The unconstrained optimization method (UOM) plays an important role in the field of physics, mathematics, statistics, etc. Rosenbrock function is a 4-degree function with a curved canyon. It is most suitable for testing UOM. However, the test hasn't done. We have written 14 UOM computer programs, 12 of which contain our new algorithm. From the optimal result, the 2-order approximation direction is the best, followed by the conjugate direction, followed by the negative gradient direction. For quadratic fitting function method, linear fitting gradient method and unbounded polyhedron deformation method, a new algorithm for moving points was proposed. When the new point is better than all of the other points, the other points should close up to it. And when the new point is worse than all of the other points, it should close up to the best point or center point. The improvement of moving points makes these optimization directions be effective for complex objective functions.

Side lobe pattern synthesis using hybrid SSWOA algorithm for conformal antenna array

In this paper, the pattern synthesis of conformal antenna array using Salp Swarm Whale Optimization Algorithm (SSWOA) algorithm is proposed. The proposed hybrid algorithm is the combination of Salp Swarm Optimization (SSA) and Whale Optimization Algorithm (WOA). In the proposed algorithm, the SSA algorithm plays the role of guiding the evolution and the WOA algorithm works as an assistant. Moreover, after updating the salps position, with WOA the best known position are evolved to enhance the optimum search ability. When applied to antenna array pattern synthesis by taking advantages of SSA and WOA, the proposed optimization has high convergence accuracy and faster convergence speed. Several typical test functions and optimization examples of a linear array pattern synthesis are illustrated to show the performance of the hybrid algorithm. Experimental results show that the proposed hybrid algorithm is superior to SSA and WOA when applied to Sphere, Griewank, Rosenbrock, and Rastrigin test functions.

Навчання еволюційних алгоритмів студентів інформатичних спеціальностей

У статті розглядаються основні поняття еволюційних алгоритмів та методичні аспекти їх навчання студентів інформатичних спеціальностей. Наведено приклади програмної реалізації алгоритму роєвого інтелекту та описано модульну структуру генетичного алгоритму. Ефективність алгоритмів перевіряється на класичних тестових функціях таких, як функції Розенброка и Растригіна.

Research on Rosenbrock Function Optimization Problem Based on Improved Differential Evolution Algorithm

The Rosenbrock function optimization belongs to unconstrained optimization problems, and its global minimum value is located at the bottom of a smooth and narrow valley of the parabolic shape. It is very difficult to find the global minimum value of the function because of the little information provided for the optimization algorithm. According to the characteristics of the Rosenbrock function, this paper specifically proposed an improved differential evolution algorithm that adopts the self-adaptive scaling factor F and crossover rate CR with elimination mechanism, which can effectively avoid premature convergence of the algorithm and local optimum. This algorithm can also expand the search range at an early stage to find the global minimum of the Rosenbrock function. Many experimental results show that the algorithm has good performance of function optimization and provides a new idea for optimization problems similar to the Rosenbrock function for some problems of special fields.

Anthropic Principle Algorithm:A new Heuristic Optimization Meth

Heuristic optimization is an appealing method for solving some en- gineering problems, in which gradient information may not be available, or yet, when the problem presents many minima points. Thus, the goal of this paper is to present a new heuristic algorithm based on the Anthropic Prin- ciple, the Anthropic Principle Algorithm (APA). This algorithm is based on the following idea: the universe developed itself in the exact way to allow the existence of all current things, including life. This idea is very similar to the convergence in an optimization process. Arguing about the merit of the An- thropic Principle is not among the goals of this paper. This principle is treated only as an inspiration for heuristic optimization algorithms. In the final of the paper, some applications of the APA are presented. Classical problems such as Rosenbrock function minimization, system identification examples and min- imization of some benchmark functions are also presented. In order to vali- date the APA’s functionality, a comparison between the APA and the classic heuristic algorithms, Genetic Algorithm (GA) and Particle Swarm Optimiza- tion (PSO) is made. In this comparison, the APA presented better results in majority of tested cases, proving that it has a great potential for application in optimization problems.