Rosenbrock Function Research Articles

Learning and Evolution: Factors Influencing an Effective Combination

(1) Background: The mutual relationship between evolution and learning is a controversial argument among the artificial intelligence and neuro-evolution communities. After more than three decades, there is still no common agreement on the matter. (2) Methods: In this paper, the author investigates whether combining learning and evolution permits finding better solutions than those discovered by evolution alone. In further detail, the author presents a series of empirical studies that highlight some specific conditions determining the success of such combination. Results are obtained in five qualitatively different domains: (i) the 5-bit parity task, (ii) the double-pole balancing problem, (iii) the Rastrigin, Rosenbrock and Sphere optimization functions, (iv) a robot foraging task and (v) a social foraging problem. Moreover, the first three tasks represent benchmark problems in the field of evolutionary computation. (3) Results and discussion: The outcomes indicate that the effect of learning on evolution depends on the nature of the problem. Specifically, when the problem implies limited or absent agent–environment conditions, learning is beneficial for evolution, especially with the introduction of noise during the learning and selection processes. Conversely, when agents are embodied and actively interact with the environment, learning does not provide advantages, and the addition of noise is detrimental. Finally, the absence of stochasticity in the experienced conditions is paramount for the effectiveness of the combination. Furthermore, the length of the learning process must be fine-tuned based on the considered task.

TensorCRO: A TensorFlow‐based implementation of a multi‐method ensemble for optimization

AbstractThis paper presents a novel implementation of the Coral Reef Optimization with Substrate Layers (CRO‐SL) algorithm. Our approach, which we call TensorCRO, takes advantage of the TensorFlow framework to represent CRO‐SL as a series of tensor operations, allowing it to run on GPU and search for solutions in a faster and more efficient way. We evaluate the performance of the proposed implementation across a wide range of benchmark functions commonly used in optimization research (such as the Rastrigin, Rosenbrock, Ackley, and Griewank functions), and we show that GPU execution leads to considerable speedups when compared to its CPU counterpart. Then, when comparing TensorCRO to other state‐of‐the‐art optimization algorithms (such as the Genetic Algorithm, Simulated Annealing, and Particle Swarm Optimization), the results show that TensorCRO can achieve better convergence rates and solutions than other algorithms within a fixed execution time, given that the fitness functions are also implemented on TensorFlow. Furthermore, we also evaluate the proposed approach in a real‐world problem of optimizing power production in wind farms by selecting the locations of turbines; in every evaluated scenario, TensorCRO outperformed the other meta‐heuristics and achieved solutions close to the best known in the literature. Overall, our implementation of the CRO‐SL algorithm in TensorFlow GPU provides a new, fast, and efficient approach to solving optimization problems, and we believe that the proposed implementation has significant potential to be applied in various domains, such as engineering, finance, and machine learning, where optimization is often used to solve complex problems. Furthermore, we propose that this implementation can be used to optimize models that cannot propagate an error gradient, which is an excellent choice for non‐gradient‐based optimizers.

Exploring the Performance of Whale Optimization Algorithm on Rosenbrock's Function

Optimization of complex and nonlinear functions is essential across various domains, from engineering and finance to artificial intelligence and machine learning. Rosenbrock's function stands as a fundamental benchmark for evaluating optimization algorithms due to its highly nonlinear and multimodal nature. Among the multitude of optimization algorithms, the Whale Optimization Algorithm (WOA) has garnered attention for its inspiration from the social behavior of humpback whales. However, its performance on Rosenbrock's function remains relatively unexplored. This paper aims to investigate the effectiveness of the WOA specifically on Rosenbrock's function through rigorous experimentation and analysis. By evaluating convergence speed, solution accuracy, and robustness, this study sheds light on WOA's behavior when confronted with the challenges posed by Rosenbrock's function. Comparative analysis with other optimization algorithms further elucidates WOA's adaptability and scalability. The findings contribute valuable insights for selecting suitable optimization algorithms in real-world applications and advance understanding of optimization algorithms' behavior in challenging landscapes.

COMPARISON OF CLASSICAL AND QUANTUM COMPUTING FOR PARTICLE SWARM OPTIMIZATION

The article explored and delved into the advanced computational strategies of Particle Swarm Optimization (PSO) by contrasting classical and quantum computing paradigms. The advantages of quantum computing lie in its potential to solve computationally complex problems exponentially faster than classical computers. One of the advantages of Particle Swarm Optimization is its ability to find optimal solutions in complex search spaces. The research centers around the performance of PSO algorithms, as a part of the biological swarm optimization algorithms, when applied to a set of single-objective optimization functions, namely the Sphere, Rosenbrock, Booth, and Himmelblau functions. Utilizing a controlled setup of 100 particles, iterating 100 times across various dimensions tailored to each function, our study reveals that quantum Particle Swarm Optimization, implemented via Q# programming language and tested in Azure Quantum Workspace, consistently surpasses classical PSO in precision and convergence to global minima, despite the increased computational demands and error sensitivity inherent to quantum computations. The classical approach facilitated through Python programming language and leveraging deterministic pseudorandom number generators demonstrates robustness and lower computational costs but does not achieve the quantum's level of accuracy. The paper highlights the potential of quantum PSO to achieve superior optimization results in scenarios with smaller datasets and less complex problem spaces, paving the way for future applications where quantum advantages can be fully realized. The analysis goes further to discuss the implications of these findings for the future of optimization in various industries, including logistics, engineering, and finance, where optimization plays a critical role. The potential of quantum Particle Swarm Optimization to achieve superior optimization results in scenarios with smaller datasets and less complex problem spaces is particularly notable. It suggests that quantum computing could soon transform the landscape of computational optimization, providing solutions that are not only quicker but also more accurate.

Controlled gradient descent: A control theoretical perspective for optimization

The Gradient Descent (GD) paradigm is a foundational principle of modern optimization algorithms. The GD algorithm and its variants, including accelerated optimization algorithms, geodesic optimization, natural gradient, and contraction-based optimization, to name a few, are used in machine learning and the system and control domain. Here, we proposed a new algorithm based on the control theoretical perspective, labeled as the Controlled Gradient Descent (CGD). Specifically, this approach overcomes the challenges of the abovementioned algorithms, which rely on the choice of a suitable geometric structure, particularly in machine learning. The proposed CGD approach visualizes the optimization as a Manifold Stabilization Problem (MSP) through the notion of an invariant manifold and its attractivity. The CGD approach leads to an exponential contraction of trajectories under the influence of a pseudo-Riemannian metric generated through the control procedure as an additional outcome. The efficacy of the CGD is demonstrated with various test objective functions like the benchmark Rosenbrock function, objective function with a lack of flatness, and semi-contracting objective functions often encountered in machine learning applications.

A Comparative Study of Optimization Techniques on the Rosenbrock Function

A Comparative Study of Optimization Techniques on the Rosenbrock Function

Wireless Sensor Network Coverage Optimization for Internet of Things

The objective of this work is to improve the existing Wireless Sensor Network coverage optimization method. The pigeon-inspired optimization algorithm was first evaluated, and its shortcomings were noted. The pigeon-inspired optimization method was then enhanced with the good point set, Yin-Yang optimization algorithm, and opposition-based learning. To test the improved algorithm, five representative standard functions were chosen: sphere function (f1), Rosenbrock function (f2), Levy function (f3), Schwefel function (f4), and Levy function N.13 (f5). The algorithm's speed of convergence may be determined by the first two functions, which are unimodal. The final three functions, which are multimodal, can extract several local optimal values from the local optimum. In comparison with other known algorithms, the improved Yin-Yang PIO algorithm showed the highest optimization accuracy and stability. Three sets of experiments were performed to optimize the WSN coverage with different parameters. The first series of experiments suggest that Yin–Yang PIO has the best optimization effect, with a coverage rate of 99.51% (10.22% higher with PIO and 6.41% higher compared with PSO). The second and third series of experiments show that Yin-Yang PIO significantly increased the WSN coverage ratio, up to 99.9%. The algorithm can be applied to optimize WSN coverage in various environments. Future research can extend the research scope to include other optimization problems in IoT. 

Improved decision making with similarity based machine learning: applications in chemistry

Despite the fundamental progress in autonomous molecular and materials discovery, data scarcity throughout chemical compound space still severely hampers the use of modern ready-made machine learning models as they rely heavily on the paradigm, ‘the bigger the data the better’. Presenting similarity based machine learning (SML), we show an approach to select data and train a model on-the-fly for specific queries, enabling decision making in data scarce scenarios in chemistry. By solely relying on query and training data proximity to choose training points, only a fraction of data is necessary to converge to competitive performance. After introducing SML for the harmonic oscillator and the Rosenbrock function, we describe applications to scarce data scenarios in chemistry which include quantum mechanics based molecular design and organic synthesis planning. Finally, we derive a relationship between the intrinsic dimensionality and volume of feature space, governing the overall model accuracy.

A Comparison of Nonlinear Optimization Methods for Rosenbrock, Booth, and Matyas Functions

There is increasing interest in the application of nonlinear optimization techniques across a broad range of real-world problems that is linked to rapid advances in novel technologies and renewed interest in areas like sustainable development. While many algorithms for nonlinear optimization have been developed, they differ widely in their methods of convergence to the solution. Consequently, matching the function to be optimized with the right optimization algorithm is essential. To further explore this relationship, we studied two nonlinear optimization methods – the Nelder-Mead simplex method that only performs function evaluations and the quasi-Newton method which requires estimation of derivatives. These algorithms were used to find the global minima of the Rosenbrock, Booth, and Matyas functions from multiple starting points. The convergence paths for each starting point across both optimization methods for all 3 functions were visualized on contour plots. While convergence on the global minima was observed in all instances, our analysis indicated that the quasi-Newton method was consistently more efficient and needed fewer iterations than the Simplex method. This was especially pronounced for the Booth and Matyas functions where ~10-fold and 20-fold fewer iterations, respectively, were necessary for convergence. Our analysis reinforces the need to carefully match properties of the function being minimized with the performance characteristics of the optimization approach to obtain fast convergence on the global minimum.

Synthesis of Elliptical Antenna Array using Hybrid SSWOA Algorithm

In terms of research, the elliptical antenna arrays (EAA) synthesis is relatively novel. As it does not have to be circular in construction, this novel synthesis can maneuver the primary beam in the right direction, making it easier to realize. The amplitude and angular location of the ellipse, as well as the eccentricity of the ellipse, are all taken into account in the optimization process. The proposed hybrid algorithm is the SSWOA (Salp Swarm Whale Optimization Algorithm), which combines the Salp Swarm Optimization Algorithm (SSA) with the Whale Optimization Algorithm (WOA). The SSA algorithm serves as a guide, while the WOA algorithm serves as a helper in this method. We discover that optimization has a faster convergence time and high convergence accuracy when considering the benefits of SSA and WOA and applying them to the synthesis of antenna array layouts. If Griewank, Rosenbrock, Sphere, and Rastrigin test functions are used, it’s worth noting that the hybrid method outperforms both WOA and SSA.

Study of Jupiter’s Interior with Quadratic Monte Carlo Simulations

We construct models for Jupiter’s interior that match the gravity data obtained by the Juno and Galileo spacecraft. To generate ensembles of models, we introduce a novel quadratic Monte Carlo technique, which is more efficient in confining fitness landscapes than the affine invariant method that relies on linear stretch moves. We compare how long it takes the ensembles of walkers in both methods to travel to the most relevant parameter region. Once there, we compare the autocorrelation time and error bars of the two methods. For a ring potential and the 2d Rosenbrock function, we find that our quadratic Monte Carlo technique is significantly more efficient. Furthermore, we modified the walk moves by adding a scaling factor. We provide the source code and examples so that this method can be applied elsewhere. Here we employ our method to generate five-layer models for Jupiter’s interior that include winds and a prominent dilute core, which allows us to match the planet’s even and odd gravity harmonics. We compare predictions from the different model ensembles and analyze how much an increase in the temperature at 1 bar and ad hoc change to the equation of state affect the inferred amount of heavy elements in the atmosphere and in the planet overall.

APES: approximate posterior ensemble sampler

ABSTRACT This paper proposes a novel approach to generate samples from target distributions that are difficult to sample from using Markov chain Monte Carlo (MCMC) methods. Traditional MCMC algorithms often face slow convergence due to the difficulty in finding proposals that suit the problem at hand. To address this issue, the paper introduces the approximate posterior ensemble sampler (APES) algorithm, which employs kernel density estimation and radial basis interpolation to create an adaptive proposal, leading to fast convergence of the chains. The APES algorithm’s scalability to higher dimensions makes it a practical solution for complex problems. The proposed method generates an approximate posterior probability that closely approximates the desired distribution and is easy to sample from, resulting in smaller autocorrelation times and a higher probability of acceptance by the chain. We compare the performance of the APES algorithm with the affine invariance ensemble sampler with the stretch move in various contexts, demonstrating the efficiency of the proposed method. For instance, on the Rosenbrock function, the APES presented an autocorrelation time 140 times smaller than the affine invariance ensemble sampler. The comparison showcases the effectiveness of the APES algorithm in generating samples from challenging distributions. This paper presents a practical solution to generating samples from complex distributions while addressing the challenge of finding suitable proposals. With new cosmological surveys set to deal with many new systematics, this method offers a practical solution for the upcoming era of cosmological analyses. The algorithms presented in this paper are available at https://github.com/NumCosmo/NumCosmo.

Design space covering for uncertainty

Managing uncertainty in engineering design continues to challenge engineers. By combining the concept of regret from robust optimization with that of design space reduction from set-based design, a new complementary view on uncertainty referred to as Design Space Covering for Uncertainty (DSC-U) is introduced. This method develops a trade-space between reducing the design space and locking in sub-optimal performance while considering interval uncertainty that impacts the problem. A double-loop algorithm and a more efficient set covering algorithm are presented along with the structure of the method. These are demonstrated on three sample problems, a previously-studied cantilever tube optimization, the Rosenbrock function with different levels of coupling between the uncertain parameters, and a ship midship section design problem with thirty-six design variables and four uncertain terms. The DSC-U approach is shown to be practical in providing a complementary insight into the impact of uncertainty for a wide range of design techniques.

A natural Hessian approximation for ensemble based optimization

A key challenge in reservoir management and other fields of engineering involves optimizing a nonlinear function iteratively. Due to the lack of available gradients in commercial reservoir simulators the attention over the last decades has been on gradient free methods or gradient approximations. In particular, the ensemble-based optimization has gained popularity over the last decade due to its simplicity and efficient implementation when considering an ensemble of reservoir models. Typically, a regression type gradient approximation is used in a backtracking or line search setting. This paper introduces an approximation of the Hessian utilizing a Monte Carlo approximation of the natural gradient with respect to the covariance matrix. This Hessian approximation can further be implemented in a trust region approach in order to improve the efficiency of the algorithm. The advantages of using such approximations are demonstrated by testing the proposed algorithm on the Rosenbrock function and on a synthetic reservoir field.

Multi-Time Scale Optimal Scheduling Model of Wind and Hydrogen Integrated Energy System Based on Carbon Trading

In the context of carbon trading, energy conservation and emissions reduction are the development directions of integrated energy systems. In order to meet the development requirements of energy conservation and emissions reduction in the power grid, considering the different responses of the system in different time periods, a wind-hydrogen integrated multi-time scale energy scheduling model was established to optimize the energy-consumption scheduling problem of the system. As the scheduling model is a multiobjective nonlinear problem, the artificial fish swarm algorithm–shuffled frog leaping algorithm (AFS-SFLA) was used to solve the scheduling model to achieve system optimization. In the experimental test process, the Griewank benchmark function and the Rosenbrock function were selected to test the performance of the proposed AFS-SFL algorithm. In the Griewank environment, compared to the SFLA algorithm, the AFS-SFL algorithm was able to find a feasible solution at an early stage, and tended to converge after 110 iterations. The optimal solution was −4.83. In the test of total electric power deviation results at different time scales, the maximum deviation of early dispatching was 14.58 MW, and the minimum deviation was 0.56 MW. The overall deviation of real-time scheduling was the minimum, and the minimum deviation was 0 and the maximum deviation was 1.89 WM. The integrated energy system adopted real-time scale dispatching, with good system stability and low-energy consumption. Power system dispatching optimization belongs to the objective optimization problem. The artificial fish swarm algorithm and frog algorithm were innovatively combined to solve the dispatching model, which improved the accuracy of power grid dispatching. The research content provides an effective reference for the efficient use of clean and renewable energy.

Benchmarking Differential Evolution on a Quantum Simulator

The use of Evolutionary Algorithms (EA) for solving Mathematical/Computational Optimization Problems is inspired by the biological processes of Evolution. Few of the primitives involved in the Evolutionary process/paradigm are selection of ’Fit’ individuals (from a population sample) for retention, cloning, mutation, discarding, breeding, crossover etc. In the Evolutionary Algorithm abstraction, the individuals are deemed to be solution candidates to an Optimization problem and additional solution(/sets) are built by applying analogies to the above primitives (cloning, mutation etc.) by means of evaluating a ’Fitness’ function/criterion. One such algorithm is Differential Evolution(DE) which can be used to compute the minima of functions such as the rastrigin function and rosenbrock function. This work is an attempt to study the result of applying the DE method on these functions with candidate individuals generated on classical Turing modeled computation and comparing the same with those on state of the art Quantum computation.The study benchmarks the convergence of these functions by varying the parameters initialized and reports timing, convergence, and resource utilization results.

Design of two dimensional hyperchaotic system through optimization benchmark function

The hyperchaotic systems are essentially needed for various applications, especially multimedia encryption, watermarking and communications. However, existing chaotic systems have limited chaotic performance in terms of precise chaos measuring tools like bifurcation and attractor diagrams, Lyapunov exponent (LE), 0-1 test, correlation dimension (CD) and Kolmogorov entropy (KE). In this paper, a new hyperchaotic system so-called 2D Rosenbrock map is designed by exploiting the Rosenbrock function, which has perfect swinging characteristics in modular form. In order to manage the map, two control parameters are inserted to the Rosenbrock function. The proposed 2D Rosenbrock map is self-verified and also validated over a comparison with the recently reported results. The 2D Rosenbrock map has excellent ergodicity and diversity properties. Moreover, the 2D Rosenbrock map is implemented to multimedia encryption. The findings manifest that the designed 2D Rosenbrock map owns superior chaotic capability due to its reproduction and oscillation features.

Self-decisive algorithm for unconstrained optimization problems as in biomedical image analysis.

This study describes the construction of a new algorithm where image processing along with the two-step quasi-Newton methods is used in biomedical image analysis. It is a well-known fact that medical informatics is an essential component in the perspective of health care. Image processing and imaging technology are the recent advances in medical informatics, which include image content representation, image interpretation, and image acquisition, and focus on image information in the medical field. For this purpose, an algorithm was developed based on the image processing method that uses principle component analysis to find the image value of a particular test function and then direct the function toward its best method for evaluation. To validate the proposed algorithm, two functions, namely, the modified trigonometric and rosenbrock functions, are tested on variable space.

Development of The Steepest Descent Method for Unconstrained Optimization of Nonlinear Function

The q-gradient method used a Yuan step size for odd steps, and geometric recursion as an even step size (q-GY). This study aimed to accelerate convergence to a minimum point by minimizing the number of iterations, by dilating the parameter q to the independent variable and then comparing the results with three algorithms namely, the classical steepest descent (SD) method, steepest descent method with Yuan Steps (SDY), and q-gradient method with geometric recursion (q-G). The numerical results were presented in tables and graphs. The study used Rosenbrock function f(x)=〖(1-x_1)〗^2+100〖(x_2-〖x_1〗^2)〗^2 and determined μ=1,σ_0=0.5,β=0.999, the starting point (x_0) with a uniform distribution on the interval x_0= (-2.048, 2.048) in R^2, with 49 starting points (x_0) executed using the Python online compiler on a 64bit core i3 laptop. The maximum number of iterations was 58,679. Using tolerance limits as stopping criteria is 10-4 and the inequality 〖f(x〗^*)>f to get numerical results. q-GY method down ward movement towards the minimum point was better than the SD and SDY methods while the numerical results of the Rosenbrock function showed good enough performance to increase convergence to the minimum point

Marketing Archive Management of Drama Intangible Cultural Heritage Based on Particle Swarm Algorithm.

Each Chinese region has its own ancient opera, which is a treasure of folk culture and a living fossil for studying the historical origins of a local culture. It has significant academic value and historical significance at the national and local levels, whether from the perspective of promoting and disseminating national culture or from the perspective of protecting the world's intangible cultural heritage. Based on this practical significance, this paper conducts a study using the particle swarm algorithm to manage the marketing archives of drama intangible cultural heritage. The article employs the PSO algorithm to test the particle swarm optimization algorithm's convergence and conditions. The effectiveness of the algorithm is analysed in the Sphere function, Rosenbrock function, Griewanks function, and Rastrigin noncont function, and then the algorithm is compared, including the calculation speed comparison between the algorithm in this paper and the three optimal fitness functions. The experimental results show that the PSO algorithm has the highest four items in the statistics of the Schwefel function experimental results. About 45.0379 is the best value and 70.5878 is the maximum precision. The optimal average value is 6.1524, while the average value is 56.15245. In comparison to the QPSO and PSO algorithms, the algorithm in this paper has a faster convergence speed and better search accuracy. The topic of the intersection of the disciplines of drama, intangible cultural heritage marketing, and archive management using the particle swarm algorithm is well-developed.