Objective Function Research Articles

Improved Bacterial Foraging Optimization Algorithm with Machine Learning-Driven Short-Term Electricity Load Forecasting: A Case Study in Peninsular Malaysia

Accurate electricity demand forecasting is crucial for ensuring the sustainability and reliability of power systems. Least square support vector machines (LSSVM) are well suited to handle complex non-linear power load series. However, the less optimal regularization parameter and the Gaussian kernel function in the LSSVM model have contributed to flawed forecasting accuracy and random generalization ability. Thus, these parameters of LSSVM need to be chosen appropriately using intelligent optimization algorithms. This study proposes a new hybrid model based on the LSSVM optimized by the improved bacterial foraging optimization algorithm (IBFOA) for forecasting the short-term daily electricity load in Peninsular Malaysia. The IBFOA based on the sine cosine equation addresses the limitations of fixed chemotaxis constants in the original bacterial foraging optimization algorithm (BFOA), enhancing its exploration and exploitation capabilities. Finally, the load forecasting model based on LSSVM-IBFOA is constructed using mean absolute percentage error (MAPE) as the objective function. The comparative analysis demonstrates the model, achieving the highest determination coefficient (R2) of 0.9880 and significantly reducing the average MAPE value by 28.36%, 27.72%, and 5.47% compared to the deep neural network (DNN), LSSVM, and LSSVM-BFOA, respectively. Additionally, IBFOA exhibits faster convergence times compared to BFOA, highlighting the practicality of LSSVM-IBFOA for short-term load forecasting.

In‐Silico Optimization of a Bi‐Enzymatic Reactor for Mannitol Production Using Pareto‐Optimal Fronts

AbstractFor multi‐enzymatic cases, the determination of the batch reactor (BR) optimal operating policy often translates into a difficult multi‐objective problem. Exemplification is made here for the enzymatic reduction of D‐fructose to mannitol by using the mannitol dehydrogenase (MDH) enzyme and nicotinamide adenine dinucleotide (NADH) cofactor, with in situ regeneration of NADH at the expense of formate degradation by using the FDH enzyme. This paper presents an original rule to in silico generate the problem solution, by using the Pareto optimal‐front approach with accounting for pairs of competing economic goals and constraints. The optimal BR is then compared to an optimal fed‐BR (FBR), or a series of equal BRs (SeqBR). As proved, the Pareto‐optimal front alternative is an advantageous option, compared to the classical nonlinear programming technique, being simple to apply, by considering pairs of opposite objective functions. In the present case study, the Pareto‐optimal BR operating mode predicts an M‐productivity 1.5x better than those of an optimized FBR, with comparable enzymes consumption. The MDH consumption of this Pareto‐optimal BR is 10x smaller than an optimal SeqBR, and 130x smaller vs. heuristic (sub)optimal BR.

Masticatory function between chewing with and without wearing clear thermoplastic appliances.

Clear thermoplastic materials are used in a variety of oral appliances. In some situations, patients may wear clear thermoplastic appliances while eating. However, the effect of wearing clear thermoplastic appliances on chewing efficiency is unknown. This study aimed to evaluate the differences in masticatory function between chewing with and without wearing clear thermoplastic retainers over a 6-month period, and its associated factors. Thirty patients who received upper and lower clear retainers after debonding fixed appliances were examined for objective and subjective masticatory function at retainer delivery (T0), 3-month (T1), and 6-month follow-ups (T2) in two conditions: with and without wearing their retainers while chewing. The objective method used multiple sieves reported as the median particle size (MPS). The food intake ability (FIA) test served as the subjective method. Paired t-test was used to compare the outcomes between chewing with and without retainers at each evaluation time point. Repeated measures ANOVA followed by the Bonferroni post-hoc test was used to compare the outcomes between the three evaluating time points. Multivariable linear regression analysis was performed to assess whether age, sex, and extraction/non-extraction was associated with these effects. The MPS was significantly greater when chewing with retainers compared with chewing without them at T0 (P < 0.05), however, it was not significantly greater at T1 and T2 (P > 0.05). The MPS with and without the retainers tend to decrease between T0, T1 and T2. In particular, the MPS while chewing with retainers significantly decreased between T0 and T1. The total FIA score and FIA subscores for hard and soft food were significantly lower when eating with retainers at T0 (P < 0.05), however, all FIA scores when eating without retainers did not significantly change between T0, T1, and T2. No significantly associated factors were identified. Wearing clear retainers while chewing decreases objective and subjective masticatory function immediately after completing orthodontic treatment. However, this significantly improved to levels comparable to chewing without retainers after a 3-month follow-up. Age, sex, and extraction/non-extraction treatment were not associated with the difference in masticatory function while chewing with and without wearing the retainers.

Optimization of trigonometric polynomials with crystallographic symmetry and spectral bounds for set avoiding graphs

AbstractWe provide a new approach to the optimization of trigonometric polynomials with crystallographic symmetry. This approach widens the bridge between trigonometric and polynomial optimization. The trigonometric polynomials considered are supported on weight lattices associated to crystallographic root systems and are assumed invariant under the associated reflection group. On one hand the invariance allows us to rewrite the objective function in terms of generalized Chebyshev polynomials of the generalized cosines; On the other hand the generalized cosines parameterize a compact basic semi algebraic set, this latter being given by an explicit polynomial matrix inequality. The initial problem thus boils down to a polynomial optimization problem that is straightforwardly written in terms of generalized Chebyshev polynomials. The minimum is to be computed by a converging sequence of lower bounds as given by a hierarchy of relaxations based on the Hol–Scherer Positivstellensatz and indexed by the weighted degree associated to the root system. This new method for trigonometric optimization was motivated by its application to estimate the spectral bound on the chromatic number of set avoiding graphs. We examine cases of the literature where the avoided set affords crystallographic symmetry. In some cases we obtain new analytic proofs for sharp bounds on the chromatic number while in others we compute new lower bounds numerically.

Optimization algorithm of UAV‐assisted federated learning resources based on wireless energy harvesting

AbstractFederated learning (FL) enhances data privacy and security by enabling multiple terminals to collaboratively train a global model without transferring data to a central location. To tackle the challenges of limited terrestrial communication resources and device energy in FL, a UAV‐assisted federated learning and energy collection resource optimization algorithm is proposed. This algorithm harvests the working energy of FL terminals from radio frequency signals transmitted by the UAV via wireless energy collection. Considering energy causality and limited resources, we formulate a problem aimed at minimizing the completion time of FL training tasks. As this problem is NP‐hard, we initially employ a greedy algorithm to optimize the UAV's position, then decompose it into three sub‐problems: computing resources, power control, and bandwidth allocation. We derive and establish the optimization objective function for computing resources as convex, obtaining the expression for optimal computing resources. The Brent algorithm, based on golden section interpolation, iteratively solves the power distribution sub‐problem, while the Lagrange‐quasi‐Newton algorithm optimizes bandwidth allocation for each user. Simulation results demonstrate that the proposed algorithm effectively reduces training completion time while maintaining training quality.

Optimal power dispatch in microgrids using mixed-integer linear programming

Abstract As greenhouse gases emissions continue to rise, society is actively seeking methods to reduce them. Microgrids (MGs), which predominantly consist of renewable energy sources, play a significant role in achieving this objective. This paper proposes an optimized methodology for power dispatch in MGs using mixed-integer linear programming (MILP). The MGs include photovoltaic systems, wind turbines, biogas (BG) generators, battery energy storage systems (BESS), electric vehicles (EV), and loads. The model features an objective function focused on cost minimization, power balance, and the necessary limits and constraints for the system’s safe operation. Real-time pricing is employed for energy transactions between the MGs and the main grid. The results demonstrate a cost-efficient operation for the proposed system comprising two MGs and the main grid. During periods of negative power balance, the demand was met by discharging the BESS, EV’s battery, or purchasing energy from the grid. The BESS was charged when energy prices were low and discharged during peak demand periods and high energy prices. The intermittent nature of renewable sources necessitates an efficient management system to ensure reliable operation. Additionally, storage systems help mitigate the variability in generation. The BG generator was another crucial component for power supply due to its flexibility. Integrating these components into the system improved reliability and ensured a secure and balanced operation.

PangeBlocks: customized construction of pangenome graphs via maximal blocks

BackgroundThe construction of a pangenome graph is a fundamental task in pangenomics. A natural theoretical question is how to formalize the computational problem of building an optimal pangenome graph, making explicit the underlying optimization criterion and the set of feasible solutions. Current approaches build a pangenome graph with some heuristics, without assuming some explicit optimization criteria. Thus it is unclear how a specific optimization criterion affects the graph topology and downstream analysis, like read mapping and variant calling.ResultsIn this paper, by leveraging the notion of maximal block in a Multiple Sequence Alignment (MSA), we reframe the pangenome graph construction problem as an exact cover problem on blocks called Minimum Weighted Block Cover (MWBC). Then we propose an Integer Linear Programming (ILP) formulation for the MWBC problem that allows us to study the most natural objective functions for building a graph. We provide an implementation of the ILP approach for solving the MWBC and we evaluate it on SARS-CoV-2 complete genomes, showing how different objective functions lead to pangenome graphs that have different properties, hinting that the specific downstream task can drive the graph construction phase.ConclusionWe show that a customized construction of a pangenome graph based on selecting objective functions has a direct impact on the resulting graphs. In particular, our formalization of the MWBC problem, based on finding an optimal subset of blocks covering an MSA, paves the way to novel practical approaches to graph representations of an MSA where the user can guide the construction.

Assessment Tools for Masticatory Function in Periodontitis Patients: A Scoping Review.

The aim of this scoping review was to map the available evidence on assessment tools for masticatory function for periodontitis patients. It also aimed to examine the methodology of masticatory function assessment and to identify the elements of subjective masticatory function evaluation for periodontitis patients reported in the literature. A scoping review was conducted following the methodological guidance for the conduct of scoping reviews. Embase, MEDLINE, Web of Science, and Scopus were systematically searched for published studies in English reporting objective or subjective masticatory function assessment in periodontitis patients. Forty-five studies were included in the analysis. The identified assessment tools for masticatory function were summarized using the terminology described by the recent consensus. Heterogeneity was observed in the approach of assessment, the type(s) and design of assessment tools, and the methods of measurement employed. Most studies utilized only one assessment tool. Seven studies reported composite objective assessment and five studies utilized assessment tools for both objective and subjective masticatory function. Items from the included instruments for subjective masticatory function were analyzed and categorized into seven potentially clinically relevant elements of subjective masticatory function evaluation. Unclear reporting on validation status was found in all included instruments for subjective masticatory function. Variable methodologies have been reported to assess masticatory function in periodontitis patients. Future research is needed to discern the clinical utility of these assessment tools for masticatory function in periodontitis patients.

Optimal performance design of bat algorithm: An adaptive multi‐stage structure

AbstractThe bat algorithm (BA) is a metaheuristic algorithm for global optimisation that simulates the echolocation behaviour of bats with varying pulse rates of emission and loudness, which can be used to find the globally optimal solutions for various optimisation problems. Knowing the recent criticises of the originality of equations, the principle of BA is concise and easy to implement, and its mathematical structure can be seen as a hybrid particle swarm with simulated annealing. In this research, the authors focus on the performance optimisation of BA as a solver rather than discussing its originality issues. In terms of operation effect, BA has an acceptable convergence speed. However, due to the low proportion of time used to explore the search space, it is easy to converge prematurely and fall into the local optima. The authors propose an adaptive multi‐stage bat algorithm (AMSBA). By tuning the algorithm's focus at three different stages of the search process, AMSBA can achieve a better balance between exploration and exploitation and improve its exploration ability by enhancing its performance in escaping local optima as well as maintaining a certain convergence speed. Therefore, AMSBA can achieve solutions with better quality. A convergence analysis was conducted to demonstrate the global convergence of AMSBA. The authors also perform simulation experiments on 30 benchmark functions from IEEE CEC 2017 as the objective functions and compare AMSBA with some original and improved swarm‐based algorithms. The results verify the effectiveness and superiority of AMSBA. AMSBA is also compared with eight representative optimisation algorithms on 10 benchmark functions derived from IEEE CEC 2020, while this experiment is carried out on five different dimensions of the objective functions respectively. A balance and diversity analysis was performed on AMSBA to demonstrate its improvement over the original BA in terms of balance. AMSBA was also applied to the multi‐threshold image segmentation of Citrus Macular disease, which is a bacterial infection that causes lesions on citrus trees. The segmentation results were analysed by comparing each comparative algorithm's peak signal‐to‐noise ratio, structural similarity index and feature similarity index. The results show that the proposed BA‐based algorithm has apparent advantages, and it can effectively segment the disease spots from citrus leaves when the segmentation threshold is at a low level. Based on a comprehensive study, the authors think the proposed optimiser has mitigated the main drawbacks of the BA, and it can be utilised as an effective optimisation tool.

Study of Isotropic Stellar Models via Durgapal-Lake Solutions in Rastall System

Abstract This manuscript is dealt with the influences of Rastall factor on the physical aspects of isotropic celestial models. In this scenario, both the ideal fluid distribution and static spherically symmetry are taken into consideration. In specifically, the Durgapal-Lake solutions are taken into consideration to analyze the different characteristics of several specific compact star models like Her X-1, Vela X-1, LMC X-4 and RXJ 1856-37. Due to its innovative combination of two methodologies, this solution is a significant advancement on Durgapal-Fuloria and Lake's previous ansatz in enormous crucial eras. Using observed estimates of radii and masses of certain specific star objects, the undefined parameters in Durgapal-Lake ansatz are derived by matching conditions. The consistency of the adopted solutions is examined through the visual interpretation of matter constituents, equation of state factor, energy conditions, mass function and stability criteria corresponding to distinct choices of Rastall factor. The radially symmetric graphs of matter variables as well as the mass function are also displayed. Moreover, We present the graphical analysis for vanishing Rastall factor. It is concluded that in the context of Rastall theory, the stars under examination exhibit stable compositions with Durgapal-Lake solution, while in the context of general relativity, they exhibit instability.

Deephive: A Reinforcement Learning Approach for Automated Discovery of Swarm-Based Optimization Policies

We present an approach for designing swarm-based optimizers for the global optimization of expensive black-box functions. In the proposed approach, the problem of finding efficient optimizers is framed as a reinforcement learning problem, where the goal is to find optimization policies that require a few function evaluations to converge to the global optimum. The state of each particle within the swarm is defined as its current position and function value within a design space, and the particles learn to take favorable actions that maximize the reward, which is based on the final value of the objective function. The proposed approach is tested on 50 benchmark optimization functions and compared to the performance of other global optimization strategies. Furthermore, the generalization capabilities of the trained particles on the four categories of optimization benchmark functions are investigated. The results show superior performance compared to the other optimizers, desired scaling when the dimension of the functions is varied, and acceptable performance even when applied to unseen functions. On a broader scale, the results show promise for the rapid development of domain-specific optimizers.

Optimization of Rotary Drilling Rig Mast Structure Based on Multi-Dimensional Improved Salp Swarm Algorithm

The mast is a critical component of rotary drilling rigs, which has a cross-section consisting of a rectangular shape formed by two web plates and two flange plates. Structural optimization of the mast is necessary to address the issue of excessive weight. The shortcomings of the traditional structural optimization algorithms are summarized as follows: the optimized steel plate thickness is a non-integer, where rounding upwards may increase the cost to a certain extent, but it can ensure the safety of the structure; rounding downwards its load carrying capacity may not satisfy the requirements, and thus a novel Salp Swarm Algorithm is proposed to solve the optimization problem. First, this study improves the initialization and update strategy in the traditional Salp Swarm Algorithm. In order to obtain a solution for engineering, an innovative multi-dimensional running comparison is carried out. Secondly, the optimization model of rotary drilling rigs is established based on the division of the working conditions. The objective function of the optimization model is to minimize the weight of the mast while considering the constraints of strength, stiffness, stability, and welding process. Finally, the proposed optimization algorithm and the established optimization model are applied to optimize the design of the mast for a rotary drilling rig. The empirical results demonstrate that the weight of the mast has been reduced by 20%. In addition, the Improved Salp Swarm Algorithm exhibits higher solution quality, faster iteration capability, and extreme stability in optimizing welded box sections compared to the conventional algorithm. The example shows that the Improved Salp Swarm Algorithm is applicable to the optimization problem of box sections.

Robustifying Conditional Portfolio Decisions via Optimal Transport

We propose a data-driven portfolio selection model that integrates side information, conditional estimation, and robustness using the framework of distributionally robust optimization. Conditioning on the observed side information, the portfolio manager solves an allocation problem that minimizes the worst-case conditional risk-return tradeoff, subject to all possible perturbations of the covariate-return probability distribution in an optimal transport ambiguity set. Despite the nonlinearity of the objective function in the probability measure, we show that the distributionally robust portfolio allocation with a side information problem can be reformulated as a finite-dimensional optimization problem. If portfolio decisions are made based on either the mean-variance or the mean-conditional value-at-risk criterion, the reformulation can be further simplified to second-order or semidefinite cone programs. Empirical studies in the U.S. equity market demonstrate the advantage of our integrative framework against other benchmarks. Funding: The material in this paper is based on work supported by the Air Force Office of Scientific Research [Award FA9550-20-1-0397]. Additional support is gratefully acknowledged from the National Science Foundation [Grants 1915967, 1820942, and 1838676], the Natural Sciences and Engineering Research Council of Canada [Grant RGPIN-2016-05208], and the China Merchant Bank. V. A. Nguyen gratefully acknowledges the generous support from the Chinese University of Hong Kong [Improvement on Competitiveness in Hiring New Faculties Funding Scheme] and the Chinese University of Hong Kong [Direct Grant 4055191]. S. Wang is partially supported by the National Natural Science Foundation of China [Grant 72371022]. Finally, this research was enabled in part by support provided by Compute Canada. Supplemental Material: The computer code and data that support the findings of this study and the online appendix are available within this article’s supplemental material at https://doi.org/10.1287/opre.2021.0243 .

Capacity Optimization of Wind–Solar–Storage Multi-Power Microgrid Based on Two-Layer Model and an Improved Snake Optimization Algorithm

A two-layer optimization model and an improved snake optimization algorithm (ISOA) are proposed to solve the capacity optimization problem of wind–solar–storage multi-power microgrids in the whole life cycle. In the upper optimization model, the wind–solar–storage capacity optimization model is established. It takes wind–solar power supply and storage capacity as decision variables and the construction cost of the whole life cycle as the objective function. At the lower level, the optimal scheduling model is established, considering the output characteristics of various types of power supplies and energy storage, microgrid sales, and purchases of power as constraints. At the same time, the model considers constraints, such as the power balance, the operating state of the energy storage system, the power sales and purchases, and the network fluctuations, to ensure the system operates efficiently. Taking a microgrid in South China as an application scenario, the model is solved and the optimal capacity allocation scheme of the microgrid is obtained. Meanwhile, the demand response mechanism and the influence of planning years are introduced to further optimize the configuration scheme, and the impact of different rigid–flexible load ratios and various planning horizons on microgrid capacity optimization is analyzed, respectively, by a numerical example. The comparison shows that the ISOA has better optimization performance in solving the proposed two-layer model.

Euler Kernel Mapping for Hyperspectral Image Clustering via Self-Paced Learning

Clustering, as a classical unsupervised artificial intelligence technology, is commonly employed for hyperspectral image clustering tasks. However, most existing clustering methods designed for remote sensing tasks aim to solve a non-convex objective function, which can be optimized iteratively, beginning with random initializations. Consequently, during the learning phase of the clustering model, it may easily fall into bad local optimal solutions and finally hurt the clustering performance. Additionally, prevailing approaches often exhibit limitations in capturing the intricate structures inherent in hyperspectral images and are very sensitive to noise and outliers that widely exist in remote sensing data. To address these issues, we proposed a novel Euler kernel mapping for hyperspectral image clustering via self-paced learning (EKM-SPL). EKM-SPL first employs self-paced learning to learn the clustering model in a meaningful order by progressing samples from easy to complex, which can help to remove bad local optimal solutions. Secondly, a probabilistic soft weighting scheme is employed to measure complexity across the data sample, which makes the optimization process more reasonable. Thirdly, in order to more accurately characterize the intricate structure of hyperspectral images, Euler kernel mapping is used to convert the original data into a reproduced kernel Hilbert space, where the nonlinearly inseparable clusters may become linearly separable. Moreover, we innovatively integrate the coordinate descent technique into the optimization algorithm to circumvent the computational inefficiencies and information loss typically associated with conventional kernel methods. Extensive experiments conducted on classic benchmark hyperspectral image datasets illustrate the effectiveness and superiority of our proposed model.

A novel micromixer based on coastal fractal for manufacturing controllable size liposome

The traditional lipid preparation methods are complex, time-consuming, and consume a large amount of reagents, increasing costs and difficulties. Although microfluidic technology is considered a promising solution, achieving controllable liposome production with a simple and inexpensive microfluidic mixing device remains an important problem. This paper presents a wall-type micro-mixer based on coastal zone fractals. Four parameters related to the geometric shape of the coastline fractal in the microchannel are used as design variables, and the mixing index is the objective function. Single-objective optimization numerical analysis of the primary wall-type fractal baffle micromixer under four Reynolds numbers conditions yields the optimal structural configuration. Visualization experiments verify the correctness and accuracy of the numerical simulation, and the optimized mixer is used to produce liposomes. The results show that the micro-mixer with the optimal double-sidewall cross arrangement enhances chaotic convection and improves mixing efficiency. At Re = 0.1 and Re = 100, the mixing efficiency reaches 99%, 50.44% higher than the reference design. By changing the relative flow rates of lipid and aqueous solutions, microfluidic blank liposomes with a particle size of 165.12 ± 11.6 nm and a polydispersity index of 0.35± are obtained. This wall-type fractal micro-mixer has broad application prospects due to its high mixing efficiency.

Automated Layout Design of Hydraulic Components With Constraints on Flow Channels

Abstract The lightweight design of hydraulic equipment has always been of vital interest. Additive manufacturing (AM) technology can meet the manufacturing requirements of heteroideus and lightweight hydraulic equipment. However, traditional layout optimization often cannot satisfy the functional constraints of hydraulic components. This article proposes a design method of function-based automatic layout optimization for hydraulic components to solve this problem. The proposed method combines multi-component layout optimization with flow-up channel path planning and uses the triangular mesh model of hydraulic components directly as layout units. The spatial pose of the layout unit is used as the gene sequence for a genetic algorithm (GA). To meet the functional constraints, this study also proposes a fast, accurate collision detection algorithm for irregular 3D models and the generating strategy for follow-up flow channels. Here, the volume of the layout units, the total centroid radius of the layout plan, the length of flow channels, and the pressure loss are taken as the objective functions, and an automatic layout optimization algorithm for hydraulic components is developed. By optimizing the initial layout plan of an aviation electro-hydrostatic actuator (EHA), the characteristic volume of the optimized layout is reduced by 30.68% and the total length of the flow channels is decreased by 39.53%, demonstrating the efficiency of this method for lightweight hydraulic equipment design.

Simulation model calibration with dynamic stratification and adaptive sampling

ABSTRACT Calibrating simulation models that take large quantities of multi-dimensional data as input is a hard simulation optimization problem. Existing adaptive sampling strategies offer a methodological solution. However, they may not sufficiently reduce the computational cost for estimation and solution algorithm’s progress within a limited budget due to extreme noise levels and heteroscedasticity of system responses. We propose integrating stratification with adaptive sampling for the purpose of efficiency in optimization. Stratification can exploit local dependence in the simulation inputs and outputs. Yet, the state-of-the-art does not provide a full capability to adaptively stratify the data as different solution alternatives are evaluated. We devise two procedures for data-driven calibration problems that involve a large dataset with multiple covariates to calibrate models within a fixed overall simulation budget. The first approach dynamically stratifies the input data using binary trees, while the second approach uses closed-form solutions based on linearity assumptions between the objective function and concomitant variables. We find that dynamical adjustment of stratification structure accelerates optimization and reduces run-to-run variability in generated solutions. Our case study for calibrating a wind power simulation model, widely used in the wind industry, using the proposed stratified adaptive sampling, shows better-calibrated parameters under a limited budget.

A Novel k-Means Framework via Constrained Relaxation and Spectral Rotation.

Owing to its simplicity, the traditional k - means (Lloyd heuristic) clustering method plays a vital role in a variety of machine-learning applications. Disappointingly, the Lloyd heuristic is prone to local minima. In this article, we propose k - mRSR, which converts the sum-of-squared error (SSE) (Lloyd) into a combinatorial optimization problem and incorporates a relaxed trace maximization term and an improved spectral rotation term. The main advantage of k - mRSR is that it only needs to solve the membership matrix instead of computing the cluster centers in each iteration. Furthermore, we present a nonredundant coordinate descent method that brings the discrete solution infinitely close to the scaled partition matrix. Two novel findings from the experiments are that k - mRSR can further decrease (increase) the objective function values of the k - means obtained by Lloyd (CD), while Lloyd (CD) cannot decrease (increase) the objective function obtained by k - mRSR. In addition, the results of extensive experiments on 15 datasets indicate that k - mRSR outperforms both Lloyd and CD in terms of the objective function value and outperforms other state-of-the-art methods in terms of clustering performance.

Boundary value problem of calculating ray characteristics of ocean waves reflected from coastline

A direct variational method for solving the problem of reflection of ocean waves ray paths from the coastline with given positions of the source and the point of registration is considered. It is shown that the original boundary value problem can be reduced to the direct search of stationary points of the functional. Information about the objective function in the area of solutions of the ray problem allows us to construct a systematic procedure for finding minima and saddle points. A feature of the proposed approach is the optimization of the ray reflection point along a given coastline.