Multi-start Method Research Articles

Testing Weyl geometric gravity with the SPARC galactic rotation curves database

We present a detailed investigation of the properties of the galactic rotation curves in the Weyl geometric gravity model, in which the gravitational action is constructed from the square of the Weyl curvature scalar, and of the strength of the Weyl vector. The theory admits a scalar–vector–tensor representation, obtained by introducing an auxiliary scalar field. By assuming that the Weyl vector has only a radial component, an exact solution of the field equations can be obtained, which depends on three integration constants, and, as compared to the Schwarzschild solution, contains two new terms, linear and quadratic in the radial coordinate. In the framework of this solution we obtain the exact general relativistic expression of the tangential velocity of the massive test particles moving in stable circular orbits in the galactic halo. We test the theoretical predictions of the model by using 175 galaxies from the Spitzer Photometry & Accurate Rotation Curves (SPARC) database. We fit the theoretical predictions of the rotation curves in conformal gravity with the SPARC data by using the Multi Start and Global Search methods. In the total expression of the tangential velocity we also include the effects of the baryonic matter, and the mass to luminosity ratio. Our results indicate that the simple solution of the Weyl geometric gravity can successfully account for the large variety of the rotation curves of the SPARC sample, and provide a satisfactory description of the particle dynamics in the galactic halos, without the need of introducing the elusive dark matter particle.

Minimization of the weight of the blade of the aerial installation by an adaptive hybrid optimization method

In various fields of engineering practice, such as turbine building, power engineering, elements of thin-walled structures are widely used, which operate under increased loads when interacting with air or water. These include blades of impellers of radial-axial and rotary-blade hydraulic turbines, blades of wind power plants. Designing highly efficient machines and structures with the required level of reliability requires determining their optimal characteristics. The article proposes an adaptive method for finding the minimum of an arbitrary smooth multivariable function. The method was used to solve the benchmark optimization problem of a function in the form of a valley. The essence of the proposed algorithm lies in the sequential approach to the bottom of the valley and the subsequent movement in the direction of decreasing the objective function. Comparison of the results of calculating the minimum point of the function is performed using both non-gradient and gradient methods, namely: Powell, Hook-Jeeves, the steepest descent method and the developed method. It was found that the effectiveness of the proposed method is greater than the usual search algorithms, but it is not without its drawbacks. The method is proposed that represents a number of hybrid methods, which form a hybrid coalition Proposed hybrid algorithm does not provide a satisfactory result in the "single" search. The search algorithm reaches a point where all the values of the function at the surrounding points are greater than the values at the obtained point, and the algorithm cannot overcome the barrier. To solve the problem, it is necessary to take the obtained point as a new starting point and repeat the algorithm for finding the minimum of the function, that is, use the multistart method. The proposed method was used to solve the problem of optimizing the blade of an air installation, which was reduced to the problem of unconditional optimization using the method of penalty functions, but the goal function had a significantly valley structure. The optimal values of section thicknesses were obtained, which made it possible to build a blade with improved characteristics.

Parallel Algorithm of Balanced Sparse Packing of Rectangular Parallelepipeds

Introduction. Varieties of the problem of packing of rectangular parallelepipeds have a wide practical application in various fields of activity, for example, in the optimal filling of containers, in the design and layout of a wide variety of technological objects and systems, in the creation of backup copies on removable media, in the optimization of storage, protection and transportation of goods, in additive manufacturing, etc. This work continues research on this topic and is devoted to the problem of balanced sparse packing of a given set of identically oriented rectangular parallelepipeds of different sizes into a rectangular parallelepiped of minimum volume. It presents a mathematical model for this packing problem and a parallel algorithm for solving it. This algorithm is based on the reduction of the original problem to an unconditional optimization problem using penalty functions, which is solved by the multistart method, in which r-algorithm is used to find local minima from the set of generated starting points. The purpose. Construction of a mathematical model and development of an algorithm for solving the problem of balanced sparse packing of a given set of identically oriented rectangular parallelepipeds into a rectangular parallelepiped of minimum volume. Results. A mathematical model and a parallel algorithm for balanced sparse packing of identically oriented rectangular parallelepipeds into a rectangular parallelepiped of minimum volume are presented. The algorithm is based on reducing the problem with the help of penalty functions to an unconditional nondifferentiable optimization problem, for finding the solution of which multistart method is used in combination with r-algorithm for finding local minima. The results of numerical experiments are given. Conclusions. The application of the algorithm described in the work based on non-smooth optimization methods allows to improve the results of balanced sparse packing of rectangular parallelepipeds in an acceptable time. Numerical experiments showed effectiveness of the algorithm in practice. Keywords: balanced sparse packing, multistart method, r-algorithm, penalty function, "Master-Slave" procedure, numerical experiments.

Complex error minimization algorithm with adaptive change rate

In this paper, we develop a new algorithm for estimating control parameter values of ordinary differential equation (ODE) models. We call it the complex error minimization algorithm with an adaptive rate of change. It efficiently determines the parameters of ODE models even over extremely short time series. The algorithm is based on the gradient method and successfully solves its major drawback, namely getting stuck in local minima. To detect the getting stuck, we first introduce the calculation of four different errors at once. Our studies show that they achieved the global minimum only when we observe simultaneous minimization of all these errors. When getting stuck in local minima takes place, only one (or several) errors are minimized, while the others remain unchanged or grow. Thus, the algorithm accurately detects getting stuck in local minima. After this, it applies special methods based on simulated annealing to escape from local minima. The algorithm selects the most optimal path for convergence of parameter values to the global minimum by applying random restarts and multi-start methods to avoid local minima. We have tested our method on models with regular and chaotic dynamics and have found that it can work with high accuracy in these cases. Our method is suitable for actual systems with complex dynamics and an extensive set of control parameters but short time series. Thus, the proposed algorithm is an effective method for solving the inverse problem of ODE models.

Bounds on the Optimal Radius When Covering a Set with Minimum Radius Identical Disks

The problem of covering a two-dimensional bounded set with a fixed number of minimum-radius identical disks is studied in the present work. Bounds on the optimal radius are obtained for a certain class of nonsmooth domains, and an asymptotic expansion of the bounds as the number of disks goes to infinity is provided. The proof is based on the approximation of the set to be covered by hexagonal honeycombs and on the thinnest covering property of the regular hexagonal lattice arrangement in the whole plane. The dependence of the optimal radius on the number of disks is also investigated numerically using a shape-optimization approach, and theoretical and numerical convergence rates are compared. An initial point construction strategy is introduced, which, in the context of a multistart method, finds good-quality solutions to the problem under consideration. Extensive numerical experiments with a variety of polygonal regions and regular polygons illustrate the introduced approach. Funding: This work was supported by Fundação de Amparo à Pesquisa do Estado de São Paulo [Grants 2013/07375-0, 2016/01860-1, 2018/24293-0, and 2019/25258-7] and Conselho Nacional de Desenvolvimento Científico e Tecnológico [Grants 302682/2019-8, 303243/2021-0, 304258/2018-0, and 408175/2018-4].

Development of A Multi-Spectral Pyrometry Sensor for High-Speed Transient Surface-Temperature Measurements in Combustion-Relevant Harsh Environments.

Accurate and high-speed transient surface-temperature measurements of combustion devices including internal combustion (IC) engines, gas turbines, etc., provide validation targets and boundary conditions for computational fluid dynamics models, and are broadly relevant to technology advancements such as performance improvement and emissions reduction. Development and demonstration of a multi-infrared-channel pyrometry-based optical instrument for high-speed surface-temperature measurement is described. The measurement principle is based on multi-spectral radiation thermometry (MRT) and uses surface thermal radiation at four discrete spectral regions and a corresponding emissivity model to obtain surface temperature via non-linear least squares (NLLS) optimization. Rules of thumb for specifying the spectral regions and considerations to avoid interference with common combustion products are developed; the impact of these along with linear and non-linear MRT analysis are assessed as a function of temperature and signal-to-noise ratio. A multi-start method to determine the MRT-solution global optimum is described and demonstrated. The resulting multi-channel transient pyrometry instrument is described along with practical considerations including optical-alignment drift, matching intra-channel transient response, and solution-confidence indicators. The instrument demonstrated excellent >97% accuracy and >99% 2-sigma precision over the 400−800 °C range, with ~20 µs (50 kHz, equivalent to 0.2 cad at 2000 RPM IC-engine operation) transient response in the bench validation.

Evaluating the Window Size’s Role in Automatic EEG Epilepsy Detection

Electroencephalography is one of the most commonly used methods for extracting information about the brain's condition and can be used for diagnosing epilepsy. The EEG signal's wave shape contains vital information about the brain's state, which can be challenging to analyse and interpret by a human observer. Moreover, the characteristic waveforms of epilepsy (sharp waves, spikes) can occur randomly through time. Considering all the above reasons, automatic EEG signal extraction and analysis using computers can significantly impact the successful diagnosis of epilepsy. This research explores the impact of different window sizes on EEG signals' classification accuracy using four machine learning classifiers. The machine learning methods included a neural network with ten hidden nodes trained using three different training algorithms and the k-nearest neighbours classifier. The neural network training methods included the Broyden-Fletcher-Goldfarb-Shanno algorithm, the multistart method for global optimization problems, and a genetic algorithm. The current research utilized the University of Bonn dataset containing EEG data, divided into epochs having 50% overlap and window lengths ranging from 1 to 24 s. Then, statistical and spectral features were extracted and used to train the above four classifiers. The outcome from the above experiments showed that large window sizes with a length of about 21 s could positively impact the classification accuracy between the compared methods.

Adaptive hybrid optimization method for valley functions in weight minimization problems for wind turbine blades

The article proposes an adaptive method for finding the minimum of an arbitrary smooth multivariable function. The method has been used to solve the benchmark optimization problem of a valley function. The essence of the proposed algorithm lies in the sequential approach to the bottom of the valley and the subsequent movement in the direction of decreasing the objective function. The comparison of the results of calculating the minimum point of the function is performed by using both non-gradient and gradient methods, namely: Powell, Hook-Jeeves, the steepest descent method and the method developed. It has been found that the effectiveness of the proposed method is greater than the usual search algorithms, but it is not without its drawbacks. The method that represents a number of hybrid methods, which form a hybrid coalition is proposed. The proposed hybrid algorithm does not provide a satisfactory result in the "single" search. The search algorithm reaches a point where all the values of the function at the surrounding points are greater than the values at the obtained point, and the algorithm cannot overcome the barrier. To solve the problem, it is necessary to take the obtained point as a new starting point and repeat the algorithm for finding the minimum of the function, that is, use the multistart method. The proposed method has been used to solve the problem of optimizing the blade of a wind turbine, which was reduced to the problem of unconditional optimization by using the method of penalty functions, but the goal function had a significantly valley structure. The optimal values of section thicknesses have been obtained, which makes it possible to build a blade with improved characteristics.

Математичні моделі та алгоритми оптимальної упаковки куль та кубів у сферичний та кубічний контейнери

Розглянуто математичні моделі та алгоритми оптимальної збалансованоїрозрідженої упаковки куль та кубів у сферичний та кубічний контейнери.Збалансованою розрідженою (задаються допустимі відстані між обʼєктами)упаковкою об’єктів у зовнішній контейнер є така їх упаковка, коли центрваги сімейства об’єктів збігається з центром зовнішнього контейнера, авідстані між об’єктами та відстані від них до зовнішнього контейнера булине менші за наперед задані величини. Наведено математичні моделі, послідовні та паралельні алгоритми розв’язання задач знаходження збалансованої розрідженої упаковки куль різних радіусів у сферичний та кубічнийконтейнери. Наведено математичну модель задачі знаходження збалансованої розрідженої упаковки кубів у куб мінімального обʼєму за умови, щосторони всіх кубів паралельні осям координат, та опис негладкої штрафноїфункції для пошуку локальних мінімумів задачі. Досліджувані задачівідносяться до класу NP-важких задач. Математичні моделі представленібагатоекстремальними задачами нелінійного програмування. Для пошукунайкращого допустимого розвʼязку застосовується метод мультистарту усполученні з r-алгоритмом Шора. Для цього задача зводиться до задачібезумовної оптимізації за допомогою штрафних функцій у вигляді функціймаксимуму, а для пошуку локальних мінімумів із набору стартових точокзастосовуються методи мінімізації негладких функцій, що базуються навикористанні програмних реалізацій r-алгоритму. Математичні моделі тапослідовні і паралельні алгоритми, що розглядаються, можна використатидля розробки програмних засобів розв’язування задач знаходження збалансованої розрідженої упаковки сферичних та кубічних обʼєктів у сферичніта кубічні контейнери. Матеріал представлено в трьох розділах. У розд. 1 наведено математичну модель та алгоритми розв’язання задачі знаходження збалансованої розрідженої упаковки куль різних радіусів у сферичнийконтейнер. Описано послідовний та паралельний алгоритми знаходженнянайкращого допустимого розв’язку задач. У розд. 2 наведено математичну модель та алгоритми розв’язання задачі знаходження збалансованої розрідженоїупаковки куль різних радіусів у кубічний контейнер. Описано послідовний тапаралельний алгоритми знаходження найкращого допустимого розв’язкузадач. У розд. 3 наведено математичну модель задачі знаходження збалансованої розрідженої упаковки кубів у кубічний контейнер. Наведено описнегладкої штрафної функції для пошуку локальних мінімумів задачі.

Design of solvent mixtures for removal of phenol from wastewater using a non-linear programming model with a multi-start method

The optimization-based design of solvent mixtures used for phenolic wastewater treatment was investigated in this work. A nonlinear programming （NLP） model was formulated based on the concepts of computer-aid molecule design (Computer-Aided Molecular Design, CAMD) to select solvent mixtures with the best extraction performance considering the constraints of extraction process and the environmental impact. Due to the complexity of the NLP model, multi-start method was adopted to solve this problem in order to get near global optimal solution. The results of the calculations suggested that the optimal mixture consisted of 70.1% n-octanol and 29.9% 2-octanone (molar fraction).The 119 sets of experimental results showed that the extraction ability of the optimal solvent mixture identified by CAMD technique was among the top 6 sets compared to the experiment results. The results suggested that the developed NLP model could be able to screen the optimal solvent mixture in phenolic wastewater treatment.

An improved multi-start-based method for global optimisation problems

There is a great need to discover the global optimum of multimodal functions that arises from many scientifical and practical problems. A common method that has been used for this kind of problems is the multi-start method. The proposed method extends the traditional method by adding two major features: the first one is a rejection mechanism to prevent the method from spending time in unnecessary function evaluations and the second is a stopping rule aimed to terminate the method when the global minimum has been located with some certainty. The proposed method has been evaluated on some global optimisation functions from the relevant literature and the induced results are compared against some other optimisation methods.

Hybrid Optimization Based Mathematical Procedure for Dimensional Synthesis of Slider-Crank Linkage

In this paper, an optimization procedure for path generation synthesis of the slider-crank mechanism will be presented. The proposed approach is based on a hybrid strategy, mixing local and global optimization techniques. Regarding the local optimization scheme, based on the null gradient condition, a novel methodology to solve the resulting non-linear equations is developed. The solving procedure consists of decoupling two subsystems of equations which can be solved separately and following an iterative process. In relation to the global technique, a multi-start method based on a genetic algorithm is implemented. The fitness function incorporated in the genetic algorithm will take as arguments the set of dimensional parameters of the slider-crank mechanism. Several illustrative examples will prove the validity of the proposed optimization methodology, in some cases achieving an even better result compared to mechanisms with a higher number of dimensional parameters, such as the four-bar mechanism or the Watt’s mechanism.

Discrete optimization design for a cabin-skeleton coupling structure of blended-wing-body underwater glider

Objectives The blended-wing-body underwater glider cabin-skeleton coupling structure is optimized by using a data-driven discrete optimization idea. Methods Firstly, a Kriging-assisted discrete global optimization algorithm (KDGO) is proposed for computationally intensive black-box problems. The KDGO uses a novel infill-sampling strategy to capture discrete sample points with better performance. In the infill-sampling strategy, a multi-start method with data mining strategy is introduced, including multi-start optimization, projection, sampling, and selection. Secondly, the parametric skeleton-cabin coupling structure model was established and was analyzed by finite element in the lifting deformation and deep-water pressure conditions. Then, take the float-to-weight ratio and the strength and stability of the skeleton-cabin structure as the as the goal and constraints, respectively. Meanwhile, considering the interference between shape and cabin and the coupling relationship between skeleton and cabin, then establish a discrete optimization mathematical model of the overall coupling structure. Finally, the discrete optimization algorithm and coupling structure simulation are combined to build an overall optimization framework. Results By using KDGO to conduct 200 function evaluations and comparing the optimal feasible point in DoE with the global optimal feasible point after optimization, it was found that the optimized float-to-weight ratio of the coupling structure increased by nearly 40%, and satisfactory results were obtained. Conclusion The research results can provide a reference for the cabin-skeleton coupling structure design of blended-wing-body underwater glider.

Escaping local minima with local derivative-free methods: a numerical investigation

ABSTRACT We investigate the potential of applying a state-of-the-art, local derivative-free solver, Py-BOBYQA to global optimization problems. In particular, we demonstrate the potential of a restarts procedure – as distinct from multistart methods – to allow Py-BOBYQA to escape local minima (where ordinarily it would terminate at the first local minimum found). We also introduce an adaptive variant of restarts which yields improved performance on global optimization problems. As Py-BOBYQA is a model-based trust-region method, we compare largely with other global optimization methods for which (global) models are important, such as Bayesian optimization and response surface methods; we also consider state-of-the-art representative deterministic and stochastic codes, such as DIRECT and CMA-ES. We find numerically that the restarts procedures in Py-BOBYQA are effective at helping it to escape local minima, when compared to using no restarts in Py-BOBYQA. Additionally, we find that Py-BOBYQA with adaptive restarts has comparable performance with global optimization solvers for all accuracy/budget regimes, in both smooth and noisy settings. In particular, Py-BOBYQA variants are best performing for smooth and multiplicative noise problems in high-accuracy regimes. As a by-product, some preliminary conclusions can be drawn on the relative performance of the global solvers we have tested with default settings.

A Novel Sampling Technique for Multistart-Based Methods

The problem of locating the global minimum of a function is a challenging one with application in many problems. A common method to tackle this problem is the so-called multistart method, which is the base method for many modern optimization methods. This article proposes a new sampling technique for multistart-based methods, that utilizes an artificial neural network as an approximator of the original objective function. The proposed sampling technique is tested against uniform sampling on a wide set of well-known benchmark optimization problems from the relevant literature and the results are reported.

Cluster Gauss\u2013Newton method

Parameter estimation problems of mathematical models can often be formulated as nonlinear least squares problems. Typically these problems are solved numerically using iterative methods. The local minimiser obtained using these iterative methods usually depends on the choice of the initial iterate. Thus, the estimated parameter and subsequent analyses using it depend on the choice of the initial iterate. One way to reduce the analysis bias due to the choice of the initial iterate is to repeat the algorithm from multiple initial iterates (i.e. use a multi-start method). However, the procedure can be computationally intensive and is not always used in practice. To overcome this problem, we propose the Cluster Gauss–Newton (CGN) method, an efficient algorithm for finding multiple approximate minimisers of nonlinear-least squares problems. CGN simultaneously solves the nonlinear least squares problem from multiple initial iterates. Then, CGN iteratively improves the approximations from these initial iterates similarly to the Gauss–Newton method. However, it uses a global linear approximation instead of the Jacobian. The global linear approximations are computed collectively among all the iterates to minimise the computational cost associated with the evaluation of the mathematical model. We use physiologically based pharmacokinetic (PBPK) models used in pharmaceutical drug development to demonstrate its use and show that CGN is computationally more efficient and more robust against local minima compared to the standard Levenberg–Marquardt method, as well as state-of-the art multi-start and derivative-free methods.

Evaluating the Risk of Local Optima in Aerodynamic Shape Optimization

A gradient-based multistart method based on a set of 17 to 33 random initial geometries is used to examine the risk associated with multimodality when applying gradient-based optimization to aerodynamic shape optimization. Aerodynamic shape optimization problems typical of detailed, preliminary, and exploratory design are shown to consistently present design spaces with multiple local optima. In the case of detailed design, the risk of converging to a local optimum with performance significantly inferior to that of the best local optimum found is reduced due to the ability of a well-designed initial geometry, which is often available for such problems, to converge to a well-performing local optimum. In problems permitting increased geometric freedom typical of preliminary design, the risk associated with multimodality is much higher. This risk is further exacerbated in exploratory cases where high geometric freedom is combined with limited knowledge of the design space in question and hence greater differences between available initial geometries and the optimal geometry. Therefore, for preliminary and exploratory design, allocating resources toward addressing multimodality can significantly reduce the risk of overlooking a superior optimum.

Development of models for the rational choice and accommodation of people in mobile technical vehicles when evacuating from buildings

A significant growth in the volume of high-rise construction gives special relevance and urgency to the problem of safety of such facilities. Scientifically grounded plans of people’s evacuation, including all sorts of scenarios for people’s evacuation from buildings are developed for such structures. Scenarios include simulations of the motion of human flows along corridors, stairs, using elevators and mobile evacuation vehicles. An unresolved part of the problem is the problem of the rational choice and accommodation of people in stationary and mobile evacuation vehicles.The MIP model of the rational choice and accommodation of people in mobile vehicles of evacuation from buildings was developed. A particular case of the model – optimization of accommodation of people in the emergency evacuation vehicles according to the sequence of people’s arrival from the flow – was explored. The basic features of the model were analyzed: the model of the problem of mixed integer programming with piecewise continuous objective function. The specific features of the model allowed reasonable boiling down the problem to a sequence of sub-problems of accommodation the first objects (people) according to the sequence of their arrival and adapt each of them to the solution employing the multistart method with the application of artificial basis.A three-component model is considered as objects (of human bodies). The model is subject to restrictions that ensure the conditions for "gluing" the model's components into a single complex object. Continuous rotations of the model components with limitations to the turning angles are allowed.The proposed models and the solution methods modified in the present research make it possible to find both the configurations of the optimal-local accommodation of complex objects and the spatial shapes of objects.

Stochastic optimization with adaptive restart: a framework for integrated local and global learning

A common approach to global optimization is to combine local optimization methods with random restarts. Restarts have been used as a performance boosting approach. They can be a means to avoid “slow progress” by exploiting a potentially good solution, and restarts can enable the potential discovery of multiple local solutions, thus improving the overall quality of the returned solution. A multi-start method is a way to integrate local and global approaches; where the global search itself can be used to restart a local search. Bayesian optimization methods aim to find global optima of functions that can only be point-wise evaluated by means of a possibly expensive oracle. We propose the stochastic optimization with adaptive restart (SOAR) framework, that uses the predictive capability of Gaussian process models as a means to adaptively restart local search and intelligently select restart locations with current information. This approach attempts to balance exploitation with exploration of the solution space. We study the asymptotic convergence of SOAR to a global optimum, and empirically evaluate SOAR performance through a specific implementation that uses the Trust Region method as the local search component. Numerical experiments show that the proposed algorithm outperforms existing methodologies over a suite of test problems of varying problem dimension with a finite budget of function evaluations.

On Construction of the External Frankl Nozzle Contour Using Quadratic Curvature

The aim of the article is to develop a method, an algorithm, and appropriate software for constructing the external contour of the Frankl nozzle in the supersonic part using S-shape curves. The method is based on the problem of constructing a curve with the natural parameterization. The curve passes through two given points with the given inclination angles of the tangents and provides the given inclination angle of the tangent at the point with the given abscissa [4]. To control the inflection point of the S-shaped curve, the inclination angle of the tangent at a point with the known abscissa is used. In the case, when the curvature is given by a quadratic function, the system of five nonlinear equations is formulated, among which three equations are integral. The system has five unknown variables – three coefficients of the quadratic function, the total length of the curve and the length of the curve to the point with a known abscissa. The lemma on the relation between solutions of the original and the scalable systems, in which the coordinates of the points are multiplied by the same value, is proved. Due to this lemma, it becomes possible, using the obtained solution of the well-scalable system, to find easily the corresponding solution of a bad-scalable (singular) system. To find a solution to the system, we suggest to use the modification of the r-algorithm [5] solving special problem on minimization of the nonsmooth function (the sum of the modules of the residuals of the system), under controlling of the constraints on unknown lengths, in order to guarantee their feasible values. The algorithm is implemented using the multistart method and the ralgb5a octave function [6]. It finds the best local minimum of nonsmooth function by starting the modification of the r-algorithm from a given number of starting points. The algorithm uses an analytical computation of generalized gradients of the objective function and the trapezoid rule to calculate the integrals. The computational experiment was carried out to design the fragment of supersonic part in the external contour of a Frankl-type nozzle. The efficiency of the algorithm, developed for constructing S-shape curves, is shown.