Objective Function Research Articles

A Method for Transforming Non-Convex Optimization Problem to Distributed Form

We propose a novel distributed method for non-convex optimization problems with coupling equality and inequality constraints. This method transforms the optimization problem into a specific form to allow distributed implementation of modified gradient descent and Newton’s methods so that they operate as if they were distributed. We demonstrate that for the proposed distributed method: (i) communications are significantly less time-consuming than oracle calls, (ii) its convergence rate is equivalent to the convergence of Newton’s method concerning oracle calls, and (iii) for the cases when oracle calls are more expensive than communication between agents, the transition from a centralized to a distributed paradigm does not significantly affect computational time. The proposed method is applicable when the objective function is twice differentiable and constraints are differentiable, which holds for a wide range of machine learning methods and optimization setups.

Parameter characterization of PEM fuel cell mathematical models using an orthogonal learning-based GOOSE algorithm

In this paper, a new method is designed to effectively determine the parameters of proton exchange membrane fuel cells (PEMFCs), i.e., ξ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\xi }_{1}$$\\end{document}, ξ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\xi }_{2}$$\\end{document}, ξ3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\xi }_{3}$$\\end{document}, ξ4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\xi }_{4}$$\\end{document}, RC\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${R}_{\ ext{C}}$$\\end{document}, λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda$$\\end{document}, and b\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$b$$\\end{document}. The fuel cells (FCs) involve multiple variable quantities with complex non-linear behaviours, demanding accurate modelling to ensure optimal operation. An accurate model of these FCs is essential to evaluate their performance accurately. Furthermore, the design of the FCs significantly impacts simulation studies, which are crucial for various technological applications. This study proposed an improved parameter estimation procedure for PEMFCs by using the GOOSE algorithm, which was inspired by the adaptive behaviours found in geese during their relaxing and foraging times. The orthogonal learning mechanism improves the performance of the original GOOSE algorithm. This FC model uses the root mean squared error as the objective function for optimizing the unknown parameters. In order to validate the proposed algorithm, a number of experiments using various datasets were conducted and compared the outcomes with different state-of-the-art algorithms. The outcomes indicate that the proposed GOOSE algorithm not only produced promising results but also exhibited superior performance in comparison to other similar algorithms. This approach demonstrates the ability of the GOOSE algorithm to simulate complex systems and enhances the robustness and adaptability of the simulation tool by integrating essential behaviours into the computational framework. The proposed strategy facilitates the development of more accurate and effective advancements in the utilization of FCs.

Multi-objective topology optimization of macro structure and microtubule network structure for self-healing material

Abstract Self-healing materials possess the capability to promptly repair minor damages occurring during service, thereby effectively preventing safety accidents. This paper investigates a multi-objective topology optimization method for the macro structure and microtubule network of self-healing materials around pure epoxy resin materials, aiming to enhance the damage healing capability of the microtubule network while meeting the mechanical performance requirements of the macro structure. By introducing the design variables of macro structure and microtubule network, the corresponding topological description functions are established respectively. And study applies logical operations and post-processing techniques to generate an embedded microtubule network structure description. The objective functions include the flexibility of the macro structure, the along-travel head loss, and the total length of the microtubule network, with material volume serving as a constraint. In order to determine the head loss of the three-dimensional microtubule network structure, a Hardy-Cross method based on flow initialization and loop search is proposed. Multi-objective topology optimization is designed based on moving morphable components algorithm, enumeration method and Pareto principle. Develop iterative termination conditions by assessing the disparity between Pareto solution sets in each generation, thereby ensuring algorithm convergence. The numerical example of the Messerschmitt–Bölkow–Blohm (MBB) beamyields a flexibility of 0.059 without a carrier and 0.0728 with a carrier the macrostructural flexibility without a carrier is 81.0% compared to with a carrier, and the macrostructural profiles and the overall flexibility of the MBB beams with/without a carrier are close to each other. This method serves as a reference for optimizing large-scale self-healing structures.

Design and Comparison of Fractional-Order Controllers in Flotation Cell Banks and Flotation Columns Used in Copper Extraction Processes

This work explores efficiency improvements in the copper flotation stage, a complex nonlinear, multivariable process subject to numerous perturbations. The primary objective is to design a fractional-order PID (FOPID) control strategy and a fractional-order model reference adaptive control (FOMRAC) system. The parameters for these controllers are optimized using the particle swarm optimization (PSO) algorithm with an objective function tailored to the control goals. This study employs models of both a bank series of five flotation cells and a flotation column. Their performance results are compared against traditional controllers, such as an integer-order PID and MRAC. The findings reveal that fractional-order controllers offer notable advantages over their integer-order counterparts, showing improved performance metrics with minimal changes to the existing control framework. This research highlights the effectiveness of fractional control in enhancing flotation processes and introduces a novel application of fractional control techniques in this area.

Golden eagle optimized fractional-order PI controller design for a PFC SEPIC converter in EV charging

The power factor correction converter is the function of the front-end converter, followed by the DC–DC converter of the electric vehicle charger. It improves the power factor and regulates the output voltage and current. This research article proposes the Golden Eagle optimization for fractional order PI (FOPI) controller for Single Ended Primary Inductor Converter (SEPIC) power factor correction. The Golden eagle optimization is based on its knowledge of hunting tactics at various degrees of spiral trajectories to catch the prey. The FOPI controller has a broad range of controller parameters that provide better control and performance of the converter. The tuning of the parameters of the FOPI controller is optimized in Golden Eagle Optimization, and the Integral Absolute error with Integral Square error is used for the objective function. The optimized parameters of FOPI compare with the conventional PI controller performance. The SEPIC converter is designed and derived from the state space model by state space averaging, and the reduced model is obtained through the moment matching method. This system is tested under MATLAB/SIMULINK, and simulation results show improved settling time, fast dynamic response, reduction of inrush current, less harmonic distortion, and stability.

Multi-Objective Route Planning Model for Ocean-Going Ships Based on Bidirectional A-Star Algorithm Considering Meteorological Risk and IMO Guidelines

In this study, a new route planning model is proposed to help ocean-going ships avoid dangerous weather conditions and ensure safe ship navigation. First, we integrate ocean-going ship vulnerability into the study of the influence of meteorological and oceanic factors on navigational risk. A multi-layer fuzzy comprehensive evaluation model for weather risk assessment is established. A multi-objective nonlinear route planning model is then constructed by comprehensively considering the challenges of fuel consumption, risk, and time during ship navigation. The International Maritime Organization (IMO) guidelines are highlighted as constraints in the calculations, and wind, wave, and calm water resistance to ships in the latest ITTC method is added to the fuel consumption and sailing time in the objective function. Finally, considering the large amount of data required for ocean voyages, the bidirectional A* algorithm is applied to solve the model and reduce the planning time. Furthermore, our model is applied to the case of an accident reported in the Singapore Maritime Investigation Report, and the results show that the model-planned route is very close to the original planned route using the Towing Manual, with an average fit of 98.22%, and the overall meteorological risk of the model-planned route is 11.19% smaller than the original route; our model can therefore be used to plan a safer route for the vessel. In addition, the importance of risk assessments and the IMO guidelines as well as the efficiency of the bidirectional A* algorithm were analyzed and discussed. The results show that the model effectively lowers the meteorological risk, is more efficient than the traditional route planning algorithm, and is 86.82% faster than the Dijkstra algorithm and 49.16% faster than the A* algorithm.

Three-Layer Artificial Neural Network for Pricing Multi-Asset European Option

This paper studies an artificial neural network (ANN) for multi-asset European options. Firstly, a simple three-layer ANN-3 is established with undetermined weights and bias. Secondly, the time–space discrete PDE of the multi-asset option is given and the corresponding discrete data are fed into the ANN-3. Then, using least squares error as the objective function, the weights and bias of ANN-3 are trained well. Numerical examples are carried out to confirm the stability, accuracy and efficiency. Experiments show the ANN’s relative error is about 0.8%. This method can be extended into multi-layer ANN-q(q>3) and extended into American options.

Biodiversity and Possible Bio-Indicators of Mediterranean Temporary Ponds in Southern Apulia, Italy

Mediterranean Temporary Ponds (MTPs) represent a priority habitat according to Directive 92/43/EEC (Natura 2000 code: 3170*). These are very shallow water habitats only seasonally flooded, with a flora mainly composed of Mediterranean therophytic and geophytic species. Its extreme seasonality and small size make this habitat highly vulnerable and hard to manage. In recent Italian monitoring campaigns, the conservation status of MTP 3170* was considered inadequate. In Apulia, where the habitat is considered as “the most vulnerable type”, 73 sites were censused, with a total coverage of about 10,000 m2. The present work refers to the monitoring for three years of a total of 16 habitat 3170* sites, with the aim of better describing faunal indicator species for this priority habitat. A total of 158 taxa of flora and 103 of fauna were identified from 54 floristic and 44 faunistic samplings in total, with a robust updating of the listed biodiversity. For the first time a group of faunal species is proposed as an indicator of the habitat MTP 3170*. The conservation status, assessed on the basis of structural and functional criteria, gave a satisfactory status for seven sites and an unsatisfactory one (variously rated as inadequate or bad) for nine.

Impact of stacking sequence on burst pressure in glass/epoxy Type IV composite overwrapped pressure vessels for CNG storage

The shift from metallic to composite pressure vessels for storing compressed natural gas (CNG) is driven by the goal of reducing environmental impact by using lighter higher-performing structures. This work focuses on enhancing the internal pressure strength of a type IV composite overwrapped pressure vessel (COPV) by optimising the stacking sequence of the overwrapping composite layers. Parametric finite element (FE) models are developed to reveal symmetry effects. In these models, both the thickness build-up and fibre angle variation at the turnaround zones are accurately modelled. Subsequently, the stacking sequence is optimised with the objective function of maximising burst strength. The parametric modelling demonstrates that representing the COPV as an axisymmetric continuum reduces computational costs in 5400× while yielding results comparable to full 3D continuum models. Experimental burst tests are carried out to validate the numerical predictions, and the difference in pressure between them is 12.6 %.

BO-GRNN: Machine Learning for Underwater Acoustic Source Localization

This article proposes a Bayesian optimization-based generalized regression neural network (BO-GRNN) framework for underwater source localization in shallow water environment. Specially, by leveraging kernel regression analysis, BO-GRNN fits a conditional probability density function using training and test samples to infer the source location of unseen test samples. Furthermore, BO-GRNN models the objective function as a Gaussian process (GP) based on observed samples, utilizing an acquisition function to sample around the best objective function values and high variance regions. By evaluating these samples and updating the GP iteratively, the optimal hyperparameters are identified, ultimately improving the training efficiency and localization performance of the GRNN. Simulation experiments and error analysis validate this framework’s effectiveness. Further validation using SWellEx-96 experimental data demonstrates the superior localization performance of this framework, even with limited data snapshots. This framework estimates the range and depth of underwater sources within a 10 km range, achieving mean absolute errors of less than 0.15 km for range and 0.2 m for depth. Compared to GRNN, this framework demonstrates a 5 times improvement in training efficiency. These advantages make BO-GRNN an efficient, accurate, and robust solution for underwater source localization.

A reliability-based design optimization strategy using quantile surrogates by improved PC-kriging

In recent years, the surrogate-assisted reliability-based design optimization (RBDO) methods have been continuously developed, and numerous advanced optimization strategies have boosted efficiency and accuracy. However, ensuring sufficient accuracy and feasibility at the optimal is still a challenge. In order to achieve a well-balanced between efficiency, accuracy, and optimal feasibility, in this work, a RBDO strategy using quantile surrogates by improved PC-Kriging model is proposed. The novelty of the proposed method lies in the following main aspects: Firstly, an improved learning function has been developed to significantly enhance the convergence efficiency during the construction of the PC-Kriging model. Secondly, in the RBDO analysis process, a novel "MP+EI" combination point addition strategy is adopted to enhance the approximation of the surrogate model to the optimum of the objective function. It can further improve optimization efficiency and accuracy. On the basis of the rough probability constrained surrogate model established by the global enrichment strategy, a local refinement strategy is introduced to guarantee the accuracy of the quantile evaluation of the probability constrained surrogate model for each iteration solution during the optimization process. Finally, the proposed method is validated by three typical RBDO test examples and one engineering application example.

Two-Stage Robust Optimization of Integrated Energy Systems Considering Uncertainty in Carbon Source Load

Integrated Energy Systems (IESs) interconnect various energy networks to achieve coordinated planning and optimized operation among heterogeneous energy subsystems, making them a hot topic in current energy research. However, with the high integration of renewable energy sources, their fluctuation characteristics introduce uncertainties to the entire system, including the corresponding indirect carbon emissions from electricity. To address these issues, this paper constructs a two-stage, three-layer robust optimization operation model for IESs from day-ahead to intra-day. The model analyzes the uncertainties in carbon emission intensity at grid-connected nodes, as well as the uncertainty characteristics of photovoltaic, wind turbine, and cooling, heating, and electricity loads, expressed using polyhedral uncertainty sets. It standardizes the modeling of internal equipment in the IES, introduces carbon emission trading mechanisms, and constructs a low-carbon economic model, transforming the objective function and constraints into a compact form. The column-and-constraint generation algorithm is applied to transform the three-layer model into a single-layer main problem and a two-layer subproblem for iterative solution. The Karush–Kuhn–Tucker (KKT) condition is used to convert the two-layer subproblem into a linear programming model. A case study conducted on a park shows that while the introduction of uncertainty optimization increases system costs and carbon emissions compared to deterministic optimization, the scheduling strategy is more stable, significantly reducing the impact of uncertainties on the system. Moreover, the proposed strategy reduces total costs by 5.03% and carbon emissions by 1.25% compared to scenarios considering only source load uncertainty, fully verifying that the proposed method improves the economic and low-carbon performance of the system.

A modified inertial proximal alternating direction method of multipliers with dual-relaxed term for structured nonconvex and nonsmooth problem

In this research, we introduce a novel optimization algorithm termed the dual-relaxed inertial alternating direction method of multipliers (DR-IADM), tailored for handling nonconvex and nonsmooth problems. These problems are characterized by an objective function that is a composite of three elements: a smooth composite function combined with a linear operator, a nonsmooth function, and a mixed function of two variables. To facilitate the iterative process, we adopt a straightforward parameter selection approach, integrate inertial components within each subproblem, and introduce two relaxed terms to refine the dual variable update step. Within a set of reasonable assumptions, we establish the boundedness of the sequence generated by our DR-IADM algorithm. Furthermore, leveraging the Kurdyka–Łojasiewicz (KŁ) property, we demonstrate the global convergence of the proposed method. To validate the practicality and efficacy of our algorithm, we present numerical experiments that corroborate its performance. In summary, our contribution lies in proposing DR-IADM for a specific class of optimization problems, proving its convergence properties, and supporting the theoretical claims with numerical evidence.

An extended goal programming approach with piecewise penalty functions for uncertain supplier-material selection problem in cardboard box manufacturing systems

In this study a real case multi-objective material and supplier selection problem in cardboard box production industries is studied. This problem for the first time optimizes the objective functions such as total wastage amounts remained from all raw sheets, total costs of the system including purchasing cost and transportation cost (including fixed and variable costs) of the raw sheets, and total overplus of produced cardboard boxes. To be closer to the real situations, as a novelty, the problem is formulated in belief-degree-based uncertain environment with normal distribution where this type of uncertainty applies the ideas of experts. A solution approach including two steps is proposed to solve the problem. In the first step, the proposed uncertain formulation is converted to a crisp form using a typical chance constrained programming scheme. In the second step, a new goal programming approach containing a piecewise penalty function is developed in order to solve the obtained multi-objective crisp formulation. In this approach, based on the ideas of experts, multiple goals are considered with different penalty values. A case study from cardboard box industries is considered to evaluate the proposed formulations and solution approach. According to the obtained results, the proposed solution approach is compared to similar approaches of the literature and its efficiency is studied.

A structurally informed data assimilation approach for nonlinear partial differential equations

Ensemble-based Kalman filtering data assimilation is often used to combine available observations with numerical simulations to obtain statistically accurate and reliable state representations in dynamical systems. However, it is well known that the commonly used Gaussian distribution assumption introduces biases for state variables that admit discontinuous profiles, which are prevalent in nonlinear partial differential equations. This investigation designs a new structurally informed prior that exploits statistical information from the simulated state variables. In particular, based on the second moment information of the state variable gradient, we construct a new weighting matrix for the numerical simulation contribution in the data assimilation objective function. This replaces the typical prior covariance matrix used for this purpose. We further adapt our weighting matrix to include information in discontinuity regions via a clustering technique. Our numerical experiments demonstrate that this new approach yields more accurate estimates than those obtained using standard ensemble-based Kalman filtering on shallow water equations, even when it is enhanced with inflation and localization techniques.

Modified Universal Kriging-based clearance error optimization for orthogonal robot

Abstract Aiming at the problem that the clearance error of the traditional tire laser engraving robot is too high, and thus the resulting positioning accuracy cannot meet the accuracy requirements needed for engraving at this stage, this paper takes the homemade orthogonal robot as the object of research, and researches the optimization method of the positioning error of the orthogonal robot based on the modified Kriging. In this paper, we use spinor theory to construct a robot clearance error model, establish an objective function with orthogonal robot clearance error as the optimization objective, and use a particle swarm algorithm to modify universal kriging (MUK) to optimize the positioning error. The optimization method is simulated and analyzed, and MUK is selected for comparison with other optimization methods. The simulation results show that 52% relative reduces the number of iterations of the MUK to the universal kriging model (UK), 79% relative to the particle Swarm Optimization (PSO), and 81% relative to the ordinary kriging model (OK), which makes the algorithm better applicable in comparison. At the same time, a tire laser engraving orthogonal robot experimental platform was built for physical verification experiments, and the experimental results show that the maximum error before and after the optimization of the integrated error of orthogonal robot positioning accuracy has been reduced by 89.91%; the average positioning error has been reduced by 86.90%; and the root-mean-square error has been reduced by 83.01%, which further proves the practicability of the proposed method.

Robust data-driven predictive control for unknown linear time-invariant systems

This paper presents a new robust data-driven predictive control scheme for unknown linear time-invariant systems by using input-state-output or input–output data based on whether the state is measurable. To remove the need for the persistently exciting (PE) condition of a sufficiently high order on pre-collected data, a set containing all systems capable of generating such data is constructed. Then, at each time step, an upper bound on a given objective function is derived for all systems in the set, and a feedback controller is designed to minimize this bound. The optimal control gain at each time step is determined by solving a set of linear matrix inequalities. We prove that if the synthesis problem is feasible at the initial time step, it remains feasible for all future time steps. Unlike current data-driven predictive control schemes based on behavioral system theory, our approach requires less stringent conditions for the pre-collected data, facilitating easier implementation. The effectiveness of our proposed methods is demonstrated through application to an unknown and unstable batch reactor.

Parallel fractional dominance MOEAs for feature subset selection in big data

In this paper, we solve the feature subset selection (FSS) problem with three objective functions namely, cardinality, area under receiver operating characteristic curve (AUC) and Matthews correlation coefficient (MCC) using novel multi-objective evolutionary algorithms (MOEAs). MOEAs often encounter poor convergence due to the increase in non-dominated solutions and getting entrapped in the local optima. This situation worsens when dealing with large, voluminous big and high-dimensional datasets. To address these challenges, we propose parallel, fractional dominance-based MOEAs for FSS under Spark. Further, to improve the exploitation of MOEAs, we introduce a novel batch opposition-based learning (BOP) along with a cardinality constraint on the opposite solution. Accordingly, we propose two variants, namely, BOP1 and BOP2. In BOP1, a single neighbour is randomly chosen in the opposite solution space, whereas in BOP2, a group of randomly chosen neighbours in the opposite solution space. In either case, the opposite solutions are evaluated to improve the exploitation capability of the underlying MOEAs. We observe that in terms of mean optimal objective function values and across all datasets, the proposed BOP2 variant of parallel fractional dominance-based algorithms emerges as the top performer in obtaining efficient solutions. Further, we introduce a novel metric, namely the ratio of hypervolume (HV) and inverted generated distance (IGD), HV/IGD, that combines both diversity and convergence. With respect to the mean HV/IGD computed over 20 runs and Formula 1 racing, the BOP1 variants of fractional dominance-based MOEAs outperformed other algorithms.

A multi-level multi-product supply chain network design of vegetables products considering costs of quality: A case study.

Effective logistics management is crucial for the distribution of perishable agricultural products to ensure they reach customers in high-quality condition. This research examines an integrated, multi-echelon supply chain for perishable agricultural goods. The supply chain consists of four stages: supply, processing, storage, and customers. This study investigates the quality-related costs associated with product perishability to maximize supply chain profitability. Key factors considered include the network design, location of processing and distribution centers, the ability to process raw products to minimize post-harvest quality degradation, the option to sell the excess produce to a secondary market due to unpredictable yields, and the decision not to fulfill demand from distant customers where significant quality loss and price drops would be involved, instead diverting those products to the aforementioned secondary market. Quantitative methods and linear mathematical programming are employed to model and validate the proposed supply chain using actual data from a real-world case study on vegetable supply chains. The main contribution of this research is the incorporation of quality costs into the objective function, which allows the supply chain to prioritize meeting nearby customers' demands with minimal quality loss over serving distant customers where high quality loss is unavoidable. Additionally, deploying a faster transportation fleet can significantly improve the overall profitability of the perishable product supply chain.

Weight-adaptive augmented Lagrange multiplier sequential convex programming for nonlinear trajectory optimization

Sequential convex programming has become a prominent research focus in aerospace trajectory optimization due to its high computational efficiency. One critical challenge in sequential convex programming is the infeasibility of subproblems, which significantly impacts its stability. To address this issue, certain constraints must be relaxed using relaxation variables, which are then incorporated into the objective function as a penalty term. However, the choice of penalty weight greatly influences the optimality and convergence of sequential convex programming. Therefore, this paper proposes a weight-adaptive augmented Lagrange multiplier sequential convex programming algorithm to tackle the challenge. First, this paper analyzes the difficulty of determining an appropriate penalty weight from the perspective of its penalty effect. Based on this analysis, an augmented penalty term is introduced, and the update rule for the augmented Lagrange multiplier is formulated using the Karush-Kuhn-Tucker conditions. Subsequently, a method with adaptive weight adjustment is designed based on the change in the penalty term to further improve the algorithm's solving speed. Finally, the proposed method is validated through numerical simulations in a Mars-powered landing problem and a small glide vehicle reentry problem. The results show that the proposed method is 20.59 % and 25.16 % faster compared to the general sequential convex programming algorithm, respectively. Furthermore, the proposed method is demonstrated to be robust to the initial choice of penalty weight.