Direct Search Method Research Articles

The initialization of initial generating equipment maintenance schedules while a heuristic method is used

Currently, heuristic methods based on iterative changing of feasible solutions set provide a perspective tool for generation equipment maintenance scheduling in power systems. Wherein effectiveness of a heuristic method depends significantly on the initial set of possible schedules or in other words quality of the method initialization. In this case, a widely used methodology of building the initial array of solutions on the basis of pseudorandom uniform generation of control variables seems to be only palliative way to access the problem.
 This paper proposes alternative initialization procedure drawing on the example of generating units maintenance planning with heuristic differential evolution method. The principle of this method is to get initial set of solutions utilizing normal probability distribution to generate pseudorandom deviations from the suboptimal maintenance schedule which is to be preliminarily formed using directed search method. Following this approach allows to improve probabilistic characteristics of resultant maintenance schedule in particular to decrease median value of an objective function and its coefficient of variation, and to maximize probability to get the combination of units outage moments completely suiting operational constraints.

Parametric Optimization & Design of Pelton Turbine Wheel for Hydraulic Efficiency Improvement

Impulse turbine is one of the oldest methods of water power extraction. Pelton turbine is one amongst them. There is notable increase in utilization of renewable energy sources and hydropower in developed countries. The potential energy of water in hilly areas can fulfill the demands of the rural areas and can also contribute to grids. The power output of a turbine can be increased from the same input by improvising its efficiency. This paper aims to design the Pelton turbine wheel to improve its hydraulic efficiency. Parametric optimization of a Pelton turbine wheel is achieved by two methods: direct search method and efficiency optimization method. The effect of bucket exit angle, specific speed, speed ratio and nozzle opening on hydraulic efficiency is described here. The hydraulic efficiency of the turbine will be higher at the complete nozzle opening and at a higher specific speed. The maximum hydraulic efficiency is achieved for the specific speed 20.77. The theoretical hydraulic efficiency 93.9 percentage is achieved for 15 degrees of exit angle at 0.49-speed ratio.

An incentive mechanism based on a Stackelberg game for mobile crowdsensing systems with budget constraint

The adoption of smart device technologies is steadily increasing. Most of the smart devices in use today have built-in sensors which measure motion, direction, and various environmental conditions. Sensors are able to provide raw data with different quality and accuracy. A large group of smart devices forms a mobile crowdsensing system which is capable of sensing, collecting and sharing the environmental data to perform large scale sensing jobs. This paper aims to study and design an incentive mechanism for a mobile crowdsensing system based on a one-leader multi-follower Stackelberg game. A platform provider, as proponent of the sensing job, will act as the leader, while the mobile users will act as the followers. The final goal is to devise an efficient mechanism able to motivate the smart device users to participate in the sensing activity. Different from existing approaches, we propose a centralized method where the platform provider can estimate users’ parameters very efficiently sending and receiving a few messages. We formulate the optimization problem on the platform provider side as a mixed integer nonlinear program with time constraints for each job and a budget constraint. Finally, a heuristic algorithm based on the derivative-free directional direct search method is designed to solve the platform optimization problem and achieve a close-to-optimal solution for the game. Results show that our Stackelberg game solution is much more scalable than the approach proposed in the work by other authors Zhan et al. (2018) as we can decrease the average number of messages by a factor between 53 to 80 and the average running time between 23 and 650 times. Furthermore, we compared our heuristic algorithm with BARON, a state of the art commercial tool for mixed integer global optimization, to solve the platform optimization problem. Results demonstrated that our proposed algorithm converges to a near-optimal solution much faster especially in large scale systems.

An adaptive direct multisearch method for black-box multi-objective optimization

At present, black-box and simulation-based optimization problems with multiple objective functions are becoming increasingly common in the engineering context. In many cases, the functional relationships that define the objective and constraints are only known as black-boxes, cannot be differentiated accurately, and may be subject to unexpected failures. Directional direct search techniques, in particular the direct multisearch (DMS) methodology, may be applied to identify Pareto fronts for such problems. In this work, we propose a mechanism for adaptively selecting search directions in the DMS framework, with the goal of reducing the number of black-box evaluations required during the optimization. Our method relies on the concept of simplex derivatives in order to define search directions that are optimal for a local, linear model of the objective function. We provide a detailed description of the resulting algorithm and offer several practical recommendations for efficiently solving the associated subproblems. The overall performance in an academic context is assessed via a standard benchmark. Through a realistic case study, involving the bi-objective design optimization of a mechatronic quarter-car suspension, the performance of the novel method in a multidisciplinary engineering setting is tested. The results show that our method is competitive with standard implementations of DMS and other state-of-the-art multi-objective direct search methods.

Nonlinear fitting of multi-compartmental data using Hooke and Jeeves direct search method

Abstract Compartmental modelling refers to modelling the transport of substances in a system consisting of multiple compartments, which is characterized by the transfer rates among the relevant compartments. In a generalized compartmental system, recycling of substances among the compartments is allowed. Compartmental modelling is a generic technique which is needed in many branches of applied physics. The most challenging task is to determine the transfer rates. The present work described the use of the Hooke and Jeeves (HJ) derivative free direct search method in determining the transfer rates to construct a multi-compartmental model with recycling among the compartments. The use of a direct search method ensures the applicability of the model to functions which are not continuous or differentiable. The present model was successfully validated using previously reported experimental data for distribution of trace elements in four compartments in an animal.

Frugal Optimization for Cost-related Hyperparameters

The increasing demand for democratizing machine learning algorithms calls for hyperparameter optimization (HPO) solutions at low cost. Many machine learning algorithms have hyperparameters which can cause a large variation in the training cost. But this effect is largely ignored in existing HPO methods, which are incapable to properly control cost during the optimization process. To address this problem, we develop a new cost-frugal HPO solution. The core of our solution is a simple but new randomized direct-search method, for which we provide theoretical guarantees on the convergence rate and the total cost incurred to achieve convergence. We provide strong empirical results in comparison with state-of-the-art HPO methods on large AutoML benchmarks.

Data-Driven Optimization of Piezoelectric Energy Harvesters via Pattern Search Algorithm.

A data-driven optimization strategy based on a generalized pattern search (GPS) algorithm is proposed to automatically optimize piezoelectric energy harvesters (PEHs). As a direct search method, GPS can iteratively solve the derivative-free optimization problem. Taking the finite element method (FEM) as the solver and the GPS algorithm as the optimizer, the automatic interaction between the solver and optimizer ensures optimization with minimum human efforts, saving designers’ time and performing a more precise exploration in the parameter space to obtain better results. When employing it for the optimization of PEHs, the optimal length and thickness of PZT were 6.0 mm and 4.6 µm, respectively. Compared with reported high-output PEHs, this optimal structure showed an increase of 371% in output power, an improvement by 1000% in normalized power density, and a reduction of 254% in resonant frequency. Furthermore, Spearman’s rank correlation coefficient was calculated for evaluating the correlation among geometric parameters and output performance such as resonant frequency and output power, which provides a data-based perspective on the design and optimization of PEHs.

AN M/M/1/N QUEUE WITH WORKING BREAKDOWNS AND VACATIONS-AN APPROACH TO COST ANALYSIS

In this paper, we consider an M/M/1/N queue with working breakdown and two types of server vacations. Sensitivity analysis of the model is performed to aissue with the point of streamlining service rates by limiting the normal expense per unit time using the direct search method.

Cost Analysis of M/M/1/N queue with working breakdowns and a two-phase services

In this article, we analyze the finite capacity Markovian queueing model with working breakdowns and a two-phase service. The server provides the primary service as well as the secondary elective service. The cost optimization problem is to find out the minimum cost per unit time by a direct search method to optimize service rates.

A Systematic Compact Model Parameter Calibration with Adaptive Pattern Search Algorithm

A systematic device-model calibration (extraction) methodology has been proposed to reduce parameter calibration time of advanced compact model for modern nano-scale semiconductor devices. The adaptive pattern search algorithm is a variant of the direct search method, which explore in the parameter space with adaptive searching step and direction. It is very straightforward, but powerful, in high dimensional optimization problem since adaptive step and direction are decided by simple computation. The proposed method iterates less but shows superior accuracy over the conventional method. It is possible to be applied to a behavioral or empirical model correspond to emerging devices, such as tunneling field-effect transistor (TFET) and negative capacitance field-effect transistor (NCFET) due to its universality in parameter calibration for the model accuracy.

Generalised Pattern Search Based on Covariance Matrix Diagonalisation

Pattern Search is a family of gradient-free direct search methods for numerical optimisation problems. The characterising feature of pattern search methods is the use of multiple directions spanning the problem domain to sample new candidate solutions. These directions compose a matrix of potential search moves, that is the pattern. Although some fundamental studies theoretically indicate that various directions can be used, the selection of the search directions remains an unaddressed problem. The present article proposes a procedure for selecting the directions that guarantee high convergence/high performance of pattern search. The proposed procedure consists of a fitness landscape analysis to characterise the geometry of the problem by sampling points and selecting those whose objective function values are below a threshold. The eigenvectors of the covariance matrix of this distribution are then used as search directions for the pattern search. Numerical results show that the proposed method systematically outperforms its standard counterpart and is competitive with modern complex direct search and metaheuristic methods.

Оценка противопожарных расстояний между объектами различного назначения с Учетом охлаждения наружных поверхностей

Проведена серия расчетов по оценке температуры при пожаре в режиме, регулируемом вентиляцией, поскольку при этом достигается максимальное значение температуры. Исследуется радиационный и конвективный теплообмен с наружными поверхностями стен помещения. Рассмотрен случай пожара в помещениях складского типа. The application of local variation method for solving fire safety problems with variational formulation (continuous or finite-dimensional) makes it possible to significantly simplify the procedure of mathematical calculations for fire modeling and calculation tasks in the field of fire safety. The choice of the local variation method as a method of direct search for the conditional optimization problem (if there are restrictions) is associated with the absence of the need to calculate the derivatives of the target function (gradient and Hesse matrix). This saves the user from the tedious procedure of getting analytical or finite-difference expressions for the specified derivatives. This choice seems also to be due to the task of non-differentiable optimization. The article provides a justification for choosing the local variation method as one of the alternating-variable descent methods, developed for problems of thermal conductivity, control, etc. Specific features in the implementation of this algorithm are noted, as well as test examples of its application on several well-known test examples representing poorly conditioned “functionals” of small dimension. There are considered the methods of numerical implementation either by Hooke-Jeeves, which is one of the methods of coordinate descent developed for unconditional optimization problems, or by Nelder - Mead, one of the most effective methods if the number of variables does not exceed 6. Example researches showed that regularization contributes to the convergence of the method of local variations. The most effective one is a stabilizer that uses an exact solution. Some predominance of negative test results on the use of regularization is explained by insufficient conditionality of the examples used. This circumstance indicates the need for further search for optimal possibilities of using the method of local variations for applied problems. If it is necessary other stabilizers, not considered in the article, can be used. For the purpose of optimization of the formulated problem, it seems optimal to design a new type of stabilizers.

Effects of a Battery Energy Storage System on the Operating Schedule of a Renewable Energy-Based Time-of-Use Rate Industrial User under the Demand Bidding Mechanism of Taipower

Due to the increased development of the smart grid, it is becoming crucial to have an efficient energy management system for a time-of-use (TOU) rate industrial user in Taiwan. In this paper, an extension of the direct search method (DSM) is developed to deal with the operating schedule of a TOU rate industrial user under the demand bidding mechanism of Taipower. To maximize the total incentive obtained from the Taiwan Power Company (TPC, namely Taipower), several operational strategies using a battery energy storage system (BESS) are evaluated in the study to perform peak shaving and realize energy conservation. The effectiveness of the proposed DSM algorithm is validated with the TOU rate industrial user of the TPC. Numerical experiments are carried out to provide a favorable indication of whether to invest in a BESS for the renewable energy-based TOU rate industrial user in order to execute the demand bidding program (DBP).

Analysis of Iterative Deepening A* Algorithm

The A* algorithm is a direct search method. It can effectively solve the shortest path problem in static routing networks. It is also an effective method to deal with a large number of search problems. The closer the artificially estimated distance value in the algorithm is to the actual value, the faster the final search speed will be. This article focuses on the heuristic A* algorithm and discusses the problem of out-of-order Rubik’s Cube searching the shortest feasible solution. It discusses the implementation ideas, advantages and disadvantages, and practical application value of the A*(IDA*) algorithm based on iterative deepening. The development of heuristic algorithms presents some personal insights.

Using the directed search method to select the degree of centralization of heat supply schemes in order to ensure the environmental sustainability of a settlement

An algorithm is proposed for the formalized choice of the degree of centralization of the heat supply scheme in order to ensure the environmental sustainability of the settlement. The algorithm is based on a directed search for combinations of centralization and decentralization of the scheme, with the construction of networks of optimal topology for each of the compared options. The calculation begins with a variant of a decentralized heat supply scheme (maximum number of sources), followed by a reduction in heat supply sources by one at each iteration, up to full centralization to achieve ecological balance in the settlement. Taking into account the directed search proposed in the algorithm and excluding non-optimal combinations at each iteration, the number of calculations is significantly reduced compared to a simple enumeration of options. In addition to practical application, the algorithm allows you to study the “heat load density” indicator, which affects the decision on the degree of centralization of the heat supply scheme, and at the same time take into account the uneven distribution of consumers on the ground, their number, the heterogeneity of heat loads and achieving a balance in the environment.

Modified Whale Optimization Algorithm for Solar Cell and PV Module Parameter Identification

The whale optimization algorithm (WOA) is a powerful swarm intelligence method which has been widely used in various fields such as parameter identification of solar cells and PV modules. In order to better balance the exploration and exploitation of WOA, we propose a novel modified WOA (MWOA) in which both the mutation strategy based on Levy flight and a local search mechanism of pattern search are introduced. On the one hand, Levy flight can make the algorithm get rid of the local optimum and avoid stagnation; thus, it is able to prevent the algorithm from losing diversity and to increase the global search capability. On the other hand, pattern search, a direct search method, has not only high convergence rate but also good stability, which can boost the local optimization ability of the WOA. Therefore, the combination of these two mechanisms can greatly improve the capability of WOA to obtain the best solution. In addition, MWOA may be employed to estimate parameters in single diode model (SDM), double diode model (DDM), and PV modules and to identify unknown parameters of two different types of PV modules under diverse light irradiance and temperature conditions. The analytical results demonstrate the validity and the practicality of MWOA for estimating parameters of solar cells and PV modules.

Cost inspection of a Geo/G/1 retrial model using particle swarm optimization and Genetic algorithm

We have studied a Geo/G/1 retrial model with state-dependent and voluntary services in discrete-time frame. The concept of Bernoulli vacation is also considered wherein, after the execution of each service, the server may hold back for next unit or it may leave for a vacation of arbitrary duration. Priority and impatient units are also analysed. This system is applicable in digital and communication fields where the data is transferred in packets like broadband integrated services digital network (B-ISDN) and all those areas where time is considered as a discrete random variable. All necessary performance indices are established for the developed model. Stochastic decomposition laws for the system size are also provided. The calculated results are verified by performing numerical examples and a detailed illustration. We have further carried out a comparative analysis to obtain minimum cost using direct search method (DSM), Particle Swarm Optimization (PSO) and Genetic Algorithm (GA) techniques.

A model-based optimisation strategy for the start-up of anaerobic co-digestion processes

A model-based optimisation method for the start-up of anaerobic co-digestion is presented. The algorithm optimises the time evolution of the organic loading rate (OLR), the blends of substrates corresponding to each OLR and the hydraulic retention time (HRT). These parameters are calculated based on the prediction of a dynamic co-digestion model, so as to track the methane production flowrate set as reference. The mixtures of substrates and the OLR are optimised sequentially with linear programming and a direct-search method, respectively. Initially, a simulated case-study (treating blends of gelatine, glycerine, and pig manure) is tested to verify the convergence of the algorithm. Then, two consecutive start-up experiments are performed at lab-scale to validate the proposed optimisation method. Blends of sewage sludge and pig manure are treated in a 14-litre CSTR at loading rates of 1.70–2.43 gVS/L d for 79 and 90 days at mesophilic conditions. The optimisation strategy satisfactorily computes the best OLR, blends and HRT for the entire operations, improving the biogas yield over traditional manual operation.

Solution of the inverse kinetics problem for the catalytic cracking process based on a 16-component kinetic model

This article presents a 16-lump model of catalytic cracking of vacuum gasoil, which allows to take into account the quantity and quality of the main products of the process: gasoline, propane-propylene and butane-butylene fractions; and by-products of the process: light and heavy catalytic gasoil. To solve the direct problem of solving a system of differential equations, the Runge-Kutta method of order 4 is used, to solve the inverse problem of choosing the constants of the rate of chemical transformations, the method of direct search for minimizing the function of the sum of deviations of calculated values from experimental ones is applied. Thus, the reaction rate constants were chosen for the developed model.

A two-stage optimization strategy for large-scale oil field development

The optimization of the locations of a large number of wells in an oil or gas field represents a challenging computational problem. This is because the number of optimization variables scales with the maximum number of wells considered. In this work, we develop and test a new two-stage strategy for large-scale oil field optimization problems. In the first stage, wells are constrained to lie in repeated patterns, and the reduced set of optimization variables defines the pattern type and geometry (e.g., well spacing, orientation). For this component of the optimization, we introduce several important modifications, including optimization of the drilling sequence, to an existing well-pattern optimization procedure. The solutions obtained in the first stage are then used to initialize the second stage optimization. In this stage we apply comprehensive field development optimization, where the well location, type (injection or production well), drill/do not drill decision, completion interval for 3D models, and drilling time variables are determined for each well. Pattern geometry is no longer enforced in this stage. Specialized treatments (consistent with actual drilling practice) are introduced for cases where multiple geomodels, used to capture geological uncertainty, are considered. In both stages optimization is achieved using a particle swarm optimization-mesh adaptive direct search (PSO-MADS) method. The two-stage procedure is applied to 2D and 3D models corresponding to different geological scenarios. Both deterministic and geologically uncertain systems are considered. Optimization results using the new procedure are shown to clearly outperform those from the single-stage comprehensive field development optimization approach. Specifically, for the same number of function evaluations, the two-stage treatment provides net present values that exceed those of the single-stage approach by about 15–18% for the cases considered. This suggests that this optimization strategy may indeed lead to improved results in practical problems.