Powell's Method Research Articles

Active-set strategy in Powell's method for optimization without derivatives

In this article we present an algorithm for solving bound constrained optimization problems without derivatives based on Powell's method [38] for derivative-free optimization. First we consider the unconstrained optimization problem. At each iteration a quadratic interpolation model of the objective function is constructed around the current iterate and this model is minimized to obtain a new trial point. The whole process is embedded within a trust-region framework. Our algorithm uses infinity norm instead of the Euclidean norm and we solve a box constrained quadratic subproblem using an active-set strategy to explore faces of the box. Therefore, a bound constrained optimization algorithm is easily extended. We compare our im_ plementation with NEWUOA and BOBYQA, Powell's algorithms for unconstrained and bound constrained derivative free optimization respectively. Numerical experiments show that, in general, our algorithm require less functional evaluations than Powell's algorithms.

Study of Computation for Structural Reliability Index Based on Penalty Function Method

According to the geometric meaning of the structural reliability index, an unconstrained optimization model with structural reliability index and design point is obtained by exterior penalty function method. The Powell method, golden section method and extrapolation method are used to solve the unconstrained optimization problem. The proposed method not has to deal with the any derivative of the limited state function, and can been used to obtain structural reliability index and design point of the strong nonlinear limit state function, which first-order reliability method (FORM) may fail to converge. Three examples are given to compare penalty function method with the difference methods. The results show that the given method is simply, effective and precise enough.

Parameter identification in mixed Brownian–fractional Brownian motions using Powell's optimization algorithm

This paper deals with the problem of estimating the parameters of mixed Brownian–fractional Brownian motions with the combination of maximum likelihood approach and Powell's method. The maximum likelihood estimators are obtained based on the approximation by random walks of the driving noise. By adapting the Powell fast optimization algorithm, these estimators can be efficiently computed by computer software. The performance of our method is tested on simulated mixed Brownian–fractional Brownian motion data sets, and is compared with the approach proposed by Filatova (2008).

A High-Accurate Fourier-Galerkin Solution for Buoyancy-Driven Flow in a Square Cavity

In this work, a high-accurate solution is proposed for the verification of coupled fluid flow and heat transfer codes. The solution, based on the Fourier-Galerkin (FG) method, is developed for free convection in a square cavity with Rayleigh numbers ranging from 103 to 108. The Fourier coefficients are calculated by solving a highly nonlinear system of algebraic equations using Powell's method. The FG results are then used as reference solutions to study the performance of a finite-element model and to illustrate the benefit of the developed solutions in benchmarking buoyancy-driven flows.

Existing Railway Horizontal Alignment Reconstruction Algorithm Based on Line Segments Identification

A Line Segments Identification model is put forward to realize the existing railway horizontal alignment reconstruction. Firstly, Hough transform based on constraint adjustment quantity is introduced to reject the abnormal points that will affect the quality of the identification in latter steps. Then total least square method is used to get the linear segments, after which the circular curve segments and the transition curves segments will be got. At last, an evaluation function which has four arguments: circular curve radius , two transition curves length and length of straight line between curves was built and its function value is the sum of squares of all distances from measuring points to reconstruction alignment, based on the function , optimal alignment will be solved out by Powell method. Example shows that the reconstruction result is considerably superior to the traditional method and is feasible for the engineering practice.

Parameter Estimation for First-Order Random Coefficient Autoregressive (RCA) Models Based on Kalman Filter

In this article, we propose a new estimate algorithm for the parameters of a first-order Random Coefficient Autoregressive (RCA) Model. This algorithm turns out to be very reliable in estimating the true parameter values of a given model. It combines quasi-maximum likelihood method, the Kalman filter algorithm, and the Powell's method. Simulation results demonstrate that the algorithm is viable and promising.

Bidirectional Dynamic Programming Approach for Highway Alignment Optimization

In order to automatically generate the optimal highway alignment and avoid omitting valuable scenarios, a bidirectional dynamic programming approach for alignment optimization was put forward. Firstly we established a three-dimensional search grid in the study area. Then dynamic programming was used to find the best alignments within the vast study area. A double-direction searching strategy was proposed to obtain a group of scenarios ranking by their corresponding comprehensive cost. The alignments were then refined with Powell method so that the resulting alignments could deviate from the grid points to reach better solutions. Examples demonstrated that this approach can obtain global optimal alignment with good diversity.

A novel artificial bee colony algorithm with Powell's method

Artificial bee colony (ABC) algorithm is a relatively new optimization technique which has been shown to be competitive to other population-based algorithms. However, there is still an insufficiency in ABC regarding its solution search equation, which is good at exploration but poor at exploitation. To address this concerning issue, we first propose a modified search equation which is applied to generate a candidate solution in the onlookers phase to improve the search ability of ABC. Further, we use the Powell's method as a local search tool to enhance the exploitation of the algorithm. The new algorithm is tested on 22 unconstrained benchmark functions and 13 constrained benchmark functions, and are compared with some other ABCs and several state-of-the-art algorithms. The comparisons show that the proposed algorithm offers the highest solution quality, fastest global convergence, and strongest robustness among all the contenders on almost all test functions.

An algorithm for online optimization of accelerators

A general algorithm is developed for online optimization of accelerator performance, i.e., online tuning, using the performance measure as the objective function. This method, named robust conjugate direction search (RCDS), combines the conjugate direction set approach of Powell's method with a robust line optimizer which considers the random noise in bracketing the minimum and uses parabolic fit of data points that uniformly sample the bracketed zone. It is much more robust against noise than traditional algorithms and is therefore suitable for online application. Simulation and experimental studies have been carried out to demonstrate the strength of the new algorithm.

Driven eigenproblem computation in periodic structures accelerated with Brent's method

The driven eigenproblem computation opens up the possibility to solve eigenproblems as excitation problems by utilising the analogy to resonators, especially for periodic structures like substrate integrated waveguides or metamaterials which can be modelled as a unit cell. The eigenvalue (wavevector, resonance frequency) can be efficiently searched by monitoring a field observable, for example, based on the electric field or the transmitted power. Therefore, the eigenvalue searching procedure can be converted to a maximisation of the observable over the complex plane of the wavevector at each frequency. The Brent's and Powell's method are employed to accelerate the maximisation. A grounded corrugated leaky-wave antenna is analysed in both infinite- and finite-width cases to verify the method. As a further example, a doubly periodic mushroom struture is analysed.

2D-3D rigid registration to compensate for prostate motion during 3D TRUS-guided biopsy

Three-dimensional (3D) transrectal ultrasound (TRUS)-guided systems have been developed to improve targeting accuracy during prostate biopsy. However, prostate motion during the procedure is a potential source of error that can cause target misalignments. The authors present an image-based registration technique to compensate for prostate motion by registering the live two-dimensional (2D) TRUS images acquired during the biopsy procedure to a preacquired 3D TRUS image. The registration must be performed both accurately and quickly in order to be useful during the clinical procedure. The authors implemented an intensity-based 2D-3D rigid registration algorithm optimizing the normalized cross-correlation (NCC) metric using Powell's method. The 2D TRUS images acquired during the procedure prior to biopsy gun firing were registered to the baseline 3D TRUS image acquired at the beginning of the procedure. The accuracy was measured by calculating the target registration error (TRE) using manually identified fiducials within the prostate; these fiducials were used for validation only and were not provided as inputs to the registration algorithm. They also evaluated the accuracy when the registrations were performed continuously throughout the biopsy by acquiring and registering live 2D TRUS images every second. This measured the improvement in accuracy resulting from performing the registration, continuously compensating for motion during the procedure. To further validate the method using a more challenging data set, registrations were performed using 3D TRUS images acquired by intentionally exerting different levels of ultrasound probe pressures in order to measure the performance of our algorithm when the prostate tissue was intentionally deformed. In this data set, biopsy scenarios were simulated by extracting 2D frames from the 3D TRUS images and registering them to the baseline 3D image. A graphics processing unit (GPU)-based implementation was used to improve the registration speed. They also studied the correlation between NCC and TREs. The root-mean-square (RMS) TRE of registrations performed prior to biopsy gun firing was found to be 1.87 ± 0.81 mm. This was an improvement over 4.75 ± 2.62 mm before registration. When the registrations were performed every second during the biopsy, the RMS TRE was reduced to 1.63 ± 0.51 mm. For 3D data sets acquired under different probe pressures, the RMS TRE was found to be 3.18 ± 1.6 mm. This was an improvement from 6.89 ± 4.1 mm before registration. With the GPU based implementation, the registrations were performed with a mean time of 1.1 s. The TRE showed a weak correlation with the similarity metric. However, the authors measured a generally convex shape of the metric around the ground truth, which may explain the rapid convergence of their algorithm to accurate results. Registration to compensate for prostate motion during 3D TRUS-guided biopsy can be performed with a measured accuracy of less than 2 mm and a speed of 1.1 s, which is an important step toward improving the targeting accuracy of a 3D TRUS-guided biopsy system.

Research on flatness errors evaluation based on artificial fish swarm algorithm and Powell method

In this paper, based on the analysis of existent evaluation methods for form errors, a hybrid evaluation method is provided. The optimum model and the calculation process are introduced in detail. The hybrid optimisation algorithm based on artificial fish swarm algorithm AFSA and Powell optimisation method. As a new heuristic intelligent optimisation algorithm, AFSA has better performances such as good global convergence, strong robustness, insensitive to initial values, simplicity of implementation and faster convergent speed with random initial values compared with genetic algorithm. The Powell method is a classical powerful local descent algorithm, and its advantages are simple and efficient. By integrating Powell optimisation search the precision of AFSA optimisation result is evidently improved. The flatness error is discussed as an example. Finally, a control experiment is carried out, and the simulation result shows that the hybrid evaluation method is feasible and satisfactory in the evaluation of flatness errors.

Study on evaluation of perpendicularity errors with an improved particle swarm optimisation for planar lines

According to characteristics of perpendicularity error evaluation of planar lines, an improved particle swarm optimisation (PSO) is proposed to evaluate the minimum zone error. The evolutional optimum model and the calculation process are introduced in detail. Compared with conventional optimum methods such as simplex search and Powell method, PSO can find the global optimal solution, and the precision of calculating result is very good. Compared to other intelligence computation algorithms such as genetic algorithm (GA), PSO is easier to carry out with fewer parameters to adjust. Then, the objective function calculation approaches for using the particle swarm optimisation algorithm to evaluate minimum zone error are formulated. Finally, the control experiment results evaluated by different method such as the least square, simplex search, Powell optimum methods and genetic algorithm (GA), indicate that the proposed method can provide better accuracy on perpendicularity error evaluation effectively, and is well suited for position error evaluation.

How to Infer the Interactive Large Scale Regulatory Network in ‘Omic’ Studies

Inferring regulatory networks in genetic systems and metabolic pathways is one of the most important problems in systems biology. Inferring network structure from experimentally observed time series data is an inverse problem. To deal with such problems, we have developed an efficient numerical optimization method called the hybrid method, which is a combination of real-coded genetic algorithms and the modified Powell method using the S-system representation. In general, a large regulatory network comprises numerous interactive system components and requires the optimization of a large number of parameters with non-zero interaction coefficients between them. To date, we have succeeded in optimizing 272 real-valued parameters using the hybrid method. Although compared with conventional numerical optimization methods, the hybrid method is powerful but is still insufficient for inferring large-scale networks. Here we discuss the inference of interactive large-scale regulatory networks in ‘omics’ studies based on our hybrid numerical optimization method.

Multiobjective fixed head hydrothermal scheduling using integrated predator-prey optimization and Powell search method

An integrated methodology that embeds global and local search technique is explored for solving multiobjective hydrothermal scheduling. Predator-prey optimization (PPO) is used as a base level search in the global search space and Powell's method as a local search technique. Predator-prey model is based on particle swarm optimization (PSO). In spite of several advantages of PSO as compared to other evolutionary methods, it has some major drawbacks such as local optimal trapping and lack of efficient capability to treat the constraints. PPO model includes predator and prey particles in population. The prey objective is to escape from the predators reaching to its lair, while predator's goal is to capture the prey that increase exploration and exploitation capability of PSO. Powell's method aims to fine tune the solution obtained from PPO and to enhance the search capability of proposed method. In addition, a penalty less constraint handling method is employed for considering equality and inequality constraints. Fuzzy methodology has also been exploited to find the best compromised solution. The proposed integrated technique is applied to three hydrothermal scheduling problems with valve-point effect of thermal generators and transmission losses. The proposed method yields high quality solution while satisfying all constraints.

Optimized Spectral Transformation for Detection and Classification of Buried Radioactive Materials

We investigate the detection and classification of buried radioactive materials of interest using data collected by a sodium iodide (NaI) detector with a short sensor dwell time (i.e., less than or equal to 1 s). The objective of detection is to detect a target from background and nontarget materials, while the objective of classification is to classify targets buried at different depths. Binned energy windows can reduce data dimensionality, help alleviate the negative impact from background, and suppress trivial spectral variations. However, the performance is sensitive to bin partition parameters including the number of bins and their bin widths. We have developed a particle swarm optimization (PSO)-based automatic system to determine these parameters. We also propose to apply a multiobjective PSO to optimize both the detection and classification accuracy simultaneously. The experimental results show that the PSO-based algorithm can outperform the Powell's direction set optimization method. The multiobjective PSO can achieve the balance between the two objectives, and it may provide even better individual accuracy than a single-objective PSO.

Improved Genetic Algorithm for Gas Well Production Dynamic Optimization Model

In this paper, An improved genetic algorithm that Powell genetic annealing exact penalty function method was proposed to deal with the non-linear characteristics in gas well production dynamic optimization model. The direct method with Powell to unconstrained optimization problems as a parallel operator as selection, crossover and mutation operator, was embedded into the basic genetic algorithm. Powell operator was defined in genetic algorithm and using annealing exact penalty function control penalty term, so the hybrid genetic algorithm for the global optimal solution to unconstrained optimization problem was made. The method avoided the difficulties of solving the model gradient, and effectively overcome searching local optimal solution and not high success probability by artificially giving more initial point to calculate and seek the optimal solution with Powell method. And the method significantly improved the convergence probability of the global optimal solution in the genetic algorithm. The method was practical and effective by analysis of examples, and it can give suggestion for gas well reasonable production system.

Nonlinear vibration analysis of double-walled carbon nanotubes based on nonlocal elasticity theory

The nonlinear free vibration of double-walled carbon nanotubes based on the nonlocal elasticity theory is studied in this paper. The nonlinear equations of motion of the double-walled carbon nanotubes are derived by using Euler beam theory and Hamilton principle, with considering the von Kármán type geometric nonlinearity and the nonlinear van der Waals forces. The surrounding elastic medium is formulated as the Winkler model. The harmonic balance method and Davidon–Fletcher–Powell method are utilized for the analysis and simulation of the nonlinear vibration. The simulation results show that the nonlocal parameter, aspect ratio and surrounding elastic medium play more important roles in the nonlinear noncoaxial vibration than those in the coaxial vibration of the double-walled carbon nanotubes. The noncoaxial vibration amplitudes of only considering nonlinear van der Waals forces are larger than those of considering both geometric nonlinearity and nonlinear van der Waals forces.

Sonar Image Registration Based on Improved PSO and Powell Hybrid Algorithm

In order to further improve the accuracy of the sonar image registration, a novel hybrid algorithm was proposed. It proposed the normalized mutual information as similar estimation, drawn on multi-resolution data structure based on wavelet transform, a low precision solution was solved by improved PSO algorithm, which has strong global search capability, firstly and then a high precision solution was acquired by Powell method, which has strong local search capability. The hybrid algorithm is effective to overcome the fall of local maximum mutual information function; it also improves solution’s precision. Since the hybrid algorithm is the initial point of pre-treatment, an effective solution to the Powell method dependence on the initial point. Experiments reveal that the hybrid algorithm is efficiency and effectiveness.

Curvature correction for microiterative optimizations with QM/MM electronic embedding

One of the most common methods to treat the electrostatic effect of the environment in QM/MM calculations is to include the MM atoms as point charges in the QM Hamiltonian. In this case, a microiterative geometry optimization ignoring the QM contributions to the forces in the relaxation of the environment cannot yield exact stationary points. One solution that has been suggested in the literature is based on using a constant additive correction to the MM gradient during the microiterations, determined in the preceding macroiteration. Here, we analyze the convergence properties of the gradient correction method and point out that a smooth relaxation is not ensured if the curvature of the approximate, MM-based description of the potential energy surface of the environment is too small in comparison with the exact one. We suggest a computationally cheap second-order correction that uses an estimated Hessian from the Davidon-Fletcher-Powell method to tackle the problems caused by the too small curvature. Test calculations on four metalloenzymatic systems (∼100 QM atoms, ∼2000 relaxed MM atoms, ∼20,000 atoms in total) show that our approach efficiently restores the convergence where gradient correction alone would lead to oscillations.