Set Of Test Functions Research Articles

A Comprehensive Survey of Test Functions for Evaluating the Performance of Particle Swarm Optimization Algorithm

Test functions play an important role in validating and comparing the performance of optimization algorithms. The test functions should have some diverse properties, which can be useful in testing of any new algorithm. The efficiency, reliability and validation of optimization algorithms can be done by using a set of standard benchmarks or test functions. For any new optimization, it is necessary to validate its performance and compare it with other existing algorithms using a good set of test functions. Optimization problems are widely used in various fields of science and technology. Sometimes such problems can be very complex. Particle Swarm Optimization is a stochastic algorithm used for solving such optimization problems. This paper transplants some of the test functions which can be used to test the performance of Particle Swarm Optimization (PSO) algorithm, in order to improve its performance and have better results. Different test functions can be used for different types of problems. These test functions have a specific range and values, which can be applied in different situations. These functions, when applied to the PSO algorithm, can give the better comparison of results. The test functions that have been the most commonly adopted to assess performance of PSO-based algorithms and details of each of them are provided, such as the search range, the position of their known optima, and other relevant properties.

Hybridization of floating-point genetic algorithms using Hooke-Jeeves algorithm as an intelligent mutation operator

Hybridization of genetic algorithms increases the search capabilities by means of convergence rate and speed. In this paper, we suggest to use Hooke-Jeeves algorithm as a genetic operator which performs a local search using the best chromosome in a generation as the base point. As Hooke-Jeeves algorithm searches a subspace in all directions of parameters for a given starting point, it can be considered as an intelligent mutation operator, whereas, the classical mutation operator is totally blind. The operator is applied within a predefined probability. Simulation studies performed on optimizing some well-known set of test functions show that using such an intelligent mutation operator has significant effects even for small number of iterations.

Sharp extensions for convoluted solutions of wave equations

In this paper, we give sharp extensions of convoluted solutions of wave equations in abstract Banach spaces. The main technique is to use the algebraic structure, for convolution products $\ast$ and $\ast_c$, of these solutions which are defined by a version of the Duhamel's formula. We define algebra homomorphisms, for the convolution product $\ast_c$, from a certain set of test-functions and apply our results to concrete examples of abstract wave equations.

A filling function method for unconstrained global optimization

We consider the problem of finding a global minimum point of a given continuously differentiable function. The strategy is adopted of a sequential nonmonotone improvement of local optima. In particular, to escape the basin of attraction of a local minimum, a suitable Gaussian-based filling function is constructed using the quadratic model (possibly approximated) of the objective function, and added to the objective to fill the basin. Then, a procedure is defined where some new minima are determined, and that of them with the lowest function value is selected as the subsequent restarting point, even if its basin is higher than the starting one. Moreover, a suitable device employing repeatedly the centroid of all the minima determined, is introduced in order to improve the efficiency of the method in the solution of difficult problems where the number of local minima is very high. The algorithm is applied to a set of test functions from the literature and the numerical results are reported along with those obtained by applying a standard Monotonic Basin Hopping method for comparison.

Shifted robust multi-objective test problems

In 2013 Gaspar-Cunha et al. proposed a set of novel robust multi-objective benchmark functions to increase the difficulty of the current test problems and effectively mimic the characteristics of real search spaces. Despite the merits of the proposed benchmark problems, it is observed that the robust Pareto optimal fronts are located on the boundaries of the search space, which may result in the infeasibility of solutions obtained in case of perturbations along the negative side of the second parameter. This paper modifies the proposed test functions by Gaspar-Cunha et al. to mimic real problems better and allow the parameters to be fluctuated by any degree of perturbations. In fact, the robust fronts are shifted to the centre of the search space, so that any degree of uncertainties can be considered. The paper considers theoretical and experimental analysis of both set of test functions as well.

Parallel and Cooperative Particle Swarm Optimizer for Multimodal Problems

Although the original particle swarm optimizer (PSO) method and its related variant methods show some effectiveness for solving optimization problems, it may easily get trapped into local optimum especially when solving complex multimodal problems. Aiming to solve this issue, this paper puts forward a novel method called parallel and cooperative particle swarm optimizer (PCPSO). In case that the interacting of the elements inD-dimensional function vectorX=[x1,x2,…,xd,…,xD]is independent, cooperative particle swarm optimizer (CPSO) is used. Based on this, the PCPSO is presented to solve real problems. Since the dimension cannot be split into several lower dimensional search spaces in real problems because of the interacting of the elements, PCPSO exploits the cooperation of two parallel CPSO algorithms by orthogonal experimental design (OED) learning. Firstly, the CPSO algorithm is used to generate two locally optimal vectors separately; then the OED is used to learn the merits of these two vectors and creates a better combination of them to generate further search. Experimental studies on a set of test functions show that PCPSO exhibits better robustness and converges much closer to the global optimum than several other peer algorithms.

Revisiting random deposition with surface relaxation: approaches from growth rules to the Edwards-Wilkinson equation

We present several approaches for deriving the coarse-grained continuous Langevin equation (or Edwards-Wilkinson equation) from a random deposition with surface relaxation (RDSR) model. First we introduce a novel procedure to divide the first transition moment into the three fundamental processes involved: deposition, diffusion and volume conservation. We show how the diffusion process is related to the antisymmetric contribution and the volume conservation process is related to the symmetric contribution, which renormalizes to zero in the coarse-grained limit. In another approach, we find the coefficients of the continuous Langevin equation, by regularizing the discrete Langevin equation. Finally, in a third approach, we derive these coefficients from the set of test functions supported by the stationary probability density function (SPDF) of the discrete model. The applicability of the approaches used to other discrete random deposition models with instantaneous relaxation to a neighboring site is discussed.

How important is a transfer function in discrete heuristic algorithms

Transfer functions are considered the simplest and cheapest operators in designing discrete heuristic algorithms. The main advantage of such operators is the maintenance of the structure and other continuous operators of a continuous algorithm. However, a transfer function may show different behaviour in various heuristic algorithms. This paper investigates the behaviour and importance of transfer functions in improving performance of heuristic algorithms. As case studies, two algorithms with different mechanisms of optimisation were chosen: Gravitational Search Algorithm and Particle Swarm Optimisation. Eight transfer functions were integrated in these two algorithms and compared on a set of test functions. The results show that transfer functions may show diverse behaviours and have different impacts on the performance of algorithms, which should be considered when designing a discrete algorithm. The results also demonstrate the significant role of the transfer function in terms of improved exploration and exploitation of a heuristic algorithm.

Algoritmo de optimización basado en enjambres de partículas con comportamiento de vorticidad y búsqueda individual y grupal

En este documento se realiza la propuesta de un algoritmo de optimización basado en un enjam­bre de partículas con características de vorticidad, donde se considera una búsqueda grupal asociada al punto medio del enjambre y una búsqueda in­dividual dada por la mejor posición de cada indi­viduo. Con la búsqueda grupal se espera lograr la convergencia de todo el enjambre, mientras que la búsqueda individual permite una mejor explo­ración del espacio de búsqueda. El algoritmo pre­senta fases de convergencia y exploración dadas por el modelo seleccionado, el cual se encuentra basado en el comportamiento de un enjambre de individuos. Para establecer el desempeño del al­goritmo se emplea un conjunto de funciones de prueba en dos dimensiones.

Optimal Regularization in Distribution of Relaxation Times applied to Electrochemical Impedance Spectroscopy: Ridge and Lasso Regression Methods - A Theoretical and Experimental Study

The Distribution of Relaxation Times (DRT) is an approach of particular interest to the interpretation of Electrochemical Impedance Spectroscopy (EIS) measurements. The DRT allows direct access to the time characteristics of the electrochemical system under study. Obtaining the DRT from noisy EIS data requires solving an ill-posed problem using regularization methods. In this work we investigate the optimal choice of the regularization parameter in the case of ridge regression. For this purpose we propose two novel sets of test functions known as real and imaginary discrepancy test function, and real and imaginary cross-validation test functions. Furthermore, we show the validity of these tests with synthetic and real experiments. By applying the optimal regularization parameters, the DRT results from the synthetic experiments (an RC circuit, a ZARC, and a fractal element) successfully reproduce the known exact DRT. Additionally, in the cases taken from electrochemical practice (a collection of RC circuits, a symmetric cell, and a commercial Li-ion battery) our approach confirms the ability to identify various timescales. Additionally, we study the output of the regularization using Least Absolute Shrinkage and Selection Operator (Lasso) and we highlight the added value given by the synergetic use of Lasso and ridge regression.

Unsupervised nearest neighbor regression for dimensionality reduction

Large numbers of high-dimensional patterns are collected in a variety of disciplines, from astronomy to bioinformatics. In this article, we present an approach to non-linear dimensionality reduction based on fitting nearest neighbor regression to the unsupervised regression framework for learning of low-dimensional manifolds. For each high-dimensional pattern, a low-dimensional latent point is generated. The dimensionality of the induced optimization problem grows with the number of patterns. To cope with the large solution space, an iterative solution construction scheme is proposed. In this paper, we introduce two strategies to embed high-dimensional data. First, the latent sorting approach allows embeddings in a one-dimensional latent space corresponding to a sorting of the high-dimensional patterns. Second, Gaussian embeddings randomly generate candidate positions based on sampling from the Gaussian distribution employing distances on data space as variances. Kernel functions increase the flexibility of the approach by mapping the patterns to feature spaces. We analyze and compare the algorithms experimentally on a set of test functions.

Quasi-Newton methods based on ordinary differential equation approach for unconstrained nonlinear optimization

In this paper, we propose a hybrid ODE-based quasi-Newton (QN) method for unconstrained optimization problems, which combines the idea of low-order implicit Runge–Kutta (RK) techniques for gradient systems with the QN type updates of the Jacobian matrix such as the symmetric rank-one (SR1) update. The main idea of this approach is to associate a QN matrix to approximate numerically the Jacobian matrix in the gradient system. Fundamentally this is a gradient system based on the first order optimality conditions of the optimization problem. To further extend the methods for solving large-scale problems, a feature incorporated to the proposed methods is that a limited memory setting for matrix–vector multiplications is used, thus avoiding the computational and storage issues when computing Jacobian information. Under suitable assumptions, global convergence of the proposed method is proved. Practical insights on the effectiveness of these approaches on a set of test functions are given by a numerical comparison with that of the limited memory BFGS algorithm (L-BFGS) and conjugate gradient algorithm (CG).

A Comparison of Improved Artificial Bee Colony Algorithms Based on Differential Evolution

The Artificial Bee Colony (ABC) algorithm is an active field of optimization based on swarm intelligence in recent years. Inspired by the mutation strategies used in Differential Evolution (DE) algorithm, this paper introduced three types strategies (“rand”,” best”, and “current-to-best”) and one or two numbers of disturbance vectors to ABC algorithm. Although individual mutation strategies in DE have been used in ABC algorithm by some researchers in different occasions, there have not a comprehensive application and comparison of the mutation strategies used in ABC algorithm. In this paper, these improved ABC algorithms can be analyzed by a set of testing functions including the rapidity of the convergence. The results show that those improvements based on DE achieve better performance in the whole than basic ABC algorithm. DOI: http://dx.doi.org/10.11591/telkomnika.v11i10.3343

An Optimization Algorithm Employing Multiple Metamodels and Optimizers

Modern engineering design optimization often relies on computer simulations to evaluate candidate designs, a setup which results in expensive black-box optimization problems. Such problems introduce unique challenges, which has motivated the application of metamodel-assisted computational intelligence algorithms to solve them. Such algorithms combine a computational intelligence optimizer which employs a population of candidate solutions, with a metamodel which is a computationally cheaper approximation of the expensive computer simulation. However, although a variety of metamodels and optimizers have been proposed, the optimal types to employ are problem dependant. Therefore, a priori prescribing the type of metamodel and optimizer to be used may degrade its effectiveness. Leveraging on this issue, this study proposes a new computational intelligence algorithm which autonomously adapts the type of the metamodel and optimizer during the search by selecting the most suitable types out of a family of candidates at each stage. Performance analysis using a set of test functions demonstrates the effectiveness of the proposed algorithm, and highlights the merit of the proposed adaptation approach.

Semantic Search-Based Genetic Programming and the Effect of Intron Deletion

The concept of semantics (in the sense of input-output behavior of solutions on training data) has been the subject of a noteworthy interest in the genetic programming (GP) research community over the past few years. In this paper, we present a new GP system that uses the concept of semantics to improve search effectiveness. It maintains a distribution of different semantic behaviors and biases the search toward solutions that have similar semantics to the best solutions that have been found so far. We present experimental evidence of the fact that the new semantics-based GP system outperforms the standard GP and the well-known bacterial GP on a set of test functions, showing particularly interesting results for noncontinuous (i.e., generally harder to optimize) test functions. We also observe that the solutions generated by the proposed GP system often have a larger size than the ones returned by standard GP and bacterial GP and contain an elevated number of introns, i.e., parts of code that do not have any effect on the semantics. Nevertheless, we show that the deletion of introns during the evolution does not affect the performance of the proposed method.

Block thresholding for wavelet-based estimation of function derivatives from a heteroscedastic multichannel convolution model

We observe $n$ heteroscedastic stochastic processes $\{Y_v(t)\}_{v}$, where for any $v\in\{1,\ldots,n\}$ and $t \in [0,1]$, $Y_v(t)$ is the convolution product of an unknown function $f$ and a known blurring function $g_v$ corrupted by Gaussian noise. Under an ordinary smoothness assumption on $g_1,\ldots,g_n$, our goal is to estimate the $d$-th derivatives (in weak sense) of $f$ from the observations. We propose an adaptive estimator based on wavelet block thresholding, namely the "BlockJS estimator". Taking the mean integrated squared error (MISE), our main theoretical result investigates the minimax rates over Besov smoothness spaces, and shows that our block estimator can achieve the optimal minimax rate, or is at least nearly-minimax in the least favorable situation. We also report a comprehensive suite of numerical simulations to support our theoretical findings. The practical performance of our block estimator compares very favorably to existing methods of the literature on a large set of test functions.

Multi-objective Flower Algorithm for Optimization

Flower pollination algorithm is a new nature-inspired algorithm, based on the characteristics of flowering plants. In this paper, we extend this flower algorithm to solve multi-objective optimization problems in engineering. By using the weighted sum method with random weights, we show that the proposed multi-objective flower algorithm can accurately find the Pareto fronts for a set of test functions. We then solve a bi-objective disc brake design problem, which indeed converges quickly.

Benchmarking of Continuous Black Box Optimization Algorithms

Benchmarking of optimization algorithms is necessary to quantitatively assess the performance of optimizers and to understand their strengths and weaknesses. The Black Box Optimization Benchmarking (BBOB) workshops that took place in 2009, 2010, and 2012 during the Genetic and Evolutionary Computation Conference (GECCO) were set up to benchmark both stochastic and deterministic continuous optimization algorithms. For this purpose, a thorough experimental setting, a set of test functions, and a visualization tool were designed and provided. They are based on the idea that (i) test functions should be representative of typical known difficulties, scalable with dimension, and not too easy to solve, yet comprehensible; and (ii) performance measures should be quantitative. A tool for acquiring and postprocessing data was provided.

Taguchi's method for multi-objective optimization problems

Taguchi's method is a quality design technique whose applications in numerical single-objective optimization have been recently exploited. In this article, a novel multi-objective (MO) algorithm based on Taguchi's technique is illustrated and its performances assessed. Validation is performed through a comparison between the presented algorithm and a MO genetic algorithm (GA) based optimization, first on different sets of test functions and then on a practical antenna array synthesis problem. Results indicate a generally better behavior of the proposed algorithm in terms of convergence and spreading over the Pareto front with respect to the GA benchmark. © 2012 Wiley Periodicals, Inc. Int J RF and Microwave CAE , 2013.

Multiresponse Metamodeling in Simulation-Based Design Applications

The optimal design of complex systems in engineering requires the availability of mathematical models of system’s behavior as a function of a set of design variables; such models allow the designer to search for the best solution to the design problem. However, system models (e.g., computational fluid dynamics (CFD) analysis, physical prototypes) are usually time-consuming and expensive to evaluate, and thus unsuited for systematic use during design. Approximate models of system behavior based on limited data, also known as metamodels, allow significant savings by reducing the resources devoted to modeling during the design process. In this work in engineering design based on multiple performance criteria, we propose the use of multi-response Bayesian surrogate models (MR-BSM) to model several aspects of system behavior jointly, instead of modeling each individually. To this end, we formulated a family of multiresponse correlation functions, suitable for prediction of several response variables that are observed simultaneously from the same computer simulation. Using a set of test functions with varying degrees of correlation, we compared the performance of MR-BSM against metamodels built individually for each response. Our results indicate that MR-BSM outperforms individual metamodels in 53% to 75% of the test cases, though the relative performance depends on the sample size, sampling scheme and the actual correlation among the observed response values. In addition, the relative performance of MR-BSM versus individual metamodels was contingent upon the ability to select an appropriate covariance/correlation function for each application, a task for which a modified version of Akaike’s Information Criterion was observed to be inadequate.