Numerical Optimization Method Research Articles

The Hypervolume Newton Method for Constrained Multi-Objective Optimization Problems

Recently, the Hypervolume Newton Method (HVN) has been proposed as a fast and precise indicator-based method for solving unconstrained bi-objective optimization problems with objective functions. The HVN is defined on the space of (vectorized) fixed cardinality sets of decision space vectors for a given multi-objective optimization problem (MOP) and seeks to maximize the hypervolume indicator adopting the Newton–Raphson method for deterministic numerical optimization. To extend its scope to non-convex optimization problems, the HVN method was hybridized with a multi-objective evolutionary algorithm (MOEA), which resulted in a competitive solver for continuous unconstrained bi-objective optimization problems. In this paper, we extend the HVN to constrained MOPs with in principle any number of objectives. Similar to the original variant, the first- and second-order derivatives of the involved functions have to be given either analytically or numerically. We demonstrate the applicability of the extended HVN on a set of challenging benchmark problems and show that the new method can be readily applied to solve equality constraints with high precision and to some extent also inequalities. We finally use HVN as a local search engine within an MOEA and show the benefit of this hybrid method on several benchmark problems.

An Optimized Dental Implant Model Using Finite Element Analysis and Design of Experiment

Purpose: To develop, analyze, and optimize a dental implant by considering square threads and varying the thread dimensions to obtain an optimal shape. Materials and Methods: For this study, finite element analysis (FEA) and numerical optimization method were integrated to develop a mathematical model. The critical parameters of dental implants were studied, and an optimized shape was obtained using response surface method (RSM) and design of experiment (DOE). The simulated results were then compared to the predicted values under optimal conditions. Results: Using the one-factor RSM design model for the dental implant and a vertical compressive load of 450 N for testing, the optimal depth to width ratio for the thread was 0.7 in order to achieve the minimum von Mises and shear stress. Conclusion: The buttress thread was found to be the optimal shape for achieving the lowest von Mises and shear stress compared to square threads, and the thread parameters were calculated accordingly, with a thread depth 0.45 times the pitch, a width 0.3 times the pitch, and a thread angle of 17 degrees. Also, due to the constant diameter of the implant, common 4-mm diameter abutments can be used interchangeably.

Accelerated and Adaptive Power Scheduling for More Electric Aircraft Via Hybrid Learning

In recent studies, the power scheduling in more electric aircraft (MEA) has been formulated as a mixed-integer quadratic programming problem. Many model-driven methods, such as branch-and-bound algorithms, are advocated to solve it. However, these methods are often prone to considerably high complexities, which makes real-time processing a problem. In this article, a two-stage hybrid learning-based optimization approach is proposed to address this challenging issue. In the first stage, the task of optimizing integer variables is considered as a multilabel classification problem, which is solved by a data-driven method, i.e., ensemble deep neural networks (EDNNs). In the second stage, with the obtained integer solutions, the problem is transformed into a quadratic programming problem that can be quickly solved by <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">model-driven</i> numerical optimization methods. Compared to the state-of-the-art power scheduling algorithms, the model-driven and data-driven methods can compensate each other so that this two-stage hybrid learning-based approach can achieve orders of magnitude speedup in computational time, while guaranteeing the optimal scheduling performance. The structures of EDNNs are well designed so that the proposed approach can adapt to varying operating conditions in MEA. An offline simulation test and an online hardware-in-the-loop test validate the above advantages of the proposed approach.

Comparative Analysis for Evaluating Wind Energy Resources Using Intelligent Optimization Algorithms and Numerical Methods

Comparative Analysis for Evaluating Wind Energy Resources Using Intelligent Optimization Algorithms and Numerical Methods

Web computer model of optimal control of the fish production process in the correctional facilities of the Russian Federation

The paper deals with the problems of ensuring the food security of correctional facilities through the development of fish farms in the correctional system of the Russian Federation. The rational use of fish stocks in inland waters and increasing their productivity is one of the most important and urgent fisheries problems, the complexity and versatility of which require a systematic approach and the use of program methods to solve it. The paper proposes a statement of the problem of fish catching control in the form of an optimal control problem, for the solution of which a numerical optimization method implemented in the software is used. The results of the software tool operation for various values of input parameters are given. The developed software tool can be used to provide decision support in the production of fish products.

Numerical methods to calculate magnetic fields of magnetostriction level converter

This article discusses the method for obtaining a system of finite- difference equations to calculate the magnetic fields of magnetostrictive level transducers (MLT) of the applied type. An optimal numerical method for their solution is presented. The formulated problem of finding the optimal width of the non-magnetic wall of the reservoir of the applied MLT is solved by using the finite-difference approximation of the Maxwell system. The article shows the method for obtaining such equations based on the method of grids and considers in detail the optimal method for their numerical solution. The results of mathematical modeling of the magnetic field of the applied MLT using the developed program make it possible to determine the optimal value of the non-magnetic wall width of the tank of the applied MLT, at which the strength of the longitudinal magnetic field of the permanent magnet will be sufficient to form an ultrasonic torsion wave in medium of magnetostrictive sound line.

Improving the Stability of Intrusion Detection With Causal Deep Learning

Due to factors such as differing distributions of training data and test data, false associations between features and weight associations lead to unstable detection performance and lack of generalization of network intrusion detection systems (NIDSs) based on machine learning (ML). To improve the stability and generalization of NIDSs, a detection system based on causal deep learning is proposed in this paper. First, causal weights were optimized by the propensity score through causal effects, the correlation between causal features and attack labels was increased, and the correlation between false correlation variables was weakened to improve the stability performance. Second, the approximate numerical optimization method of the Tammes problem was used to remove correlations between weights, maintain the independence of causal features, and improve the generalization of the detection system. Last, the feature distribution was disrupted by adding noise to four datasets to simulate different network environments. The results showed that our system can achieve good stability in various network environments where the training and testing datasets are not independently and identically distributed. In particular, after applying binary coding features and causal intervention (CIT) screening features, the average stability of the system improved by more than 10%.

An optimal eighth-order multipoint numerical iterative method to find simple root of scalar nonlinear equations

An optimal eighth-order multipoint numerical iterative method is constructed to find the simple root of scalar nonlinear equations. It is a three-point numerical iterative method that uses three evaluations of func-tion f(¢) associated with a scalar nonlinear equation and one of its deriv-atives f0 (¢). The four functional evaluations are required to achieve the eighth-order convergence. According to Kung-Traub conjecture (KTC), an iterative numerical multipoint method without memory can achieve maximum order of convergence 2n¡1 where n is the total number of func-tion evaluations in a single instance of the method. Therefore, following the KTC, the proposed method in this article is optimal.

A geographic information model for 3-D environmental suitability analysis in railway alignment optimization

Railway alignment design is a complicated problem affected by intricate environmental factors. Although numerous alignment optimization methods have been proposed, a general limitation among them is the lack of a spatial environmental suitability analysis to guide the subsequent alignment search. Consequently, many unfavorable regions in the study area are still searched, which significantly degrades optimization efficiency. To solve this problem, a geographic information model is proposed for evaluating the environmental suitability of railways. Initially, the study area is abstracted as a spatial voxel set and the 3-D reachable ranges of railways are determined. Then, a geographic information model is devised which considers topographic influencing factors (including those affecting structural cost and stability) as well as geologic influencing factors (including landslides and seismic impacts) for different railway structures. Afterward, a 3-D environmental suitability map can be generated using a multi-criteria decision-making approach to combine the considered factors. The map is further integrated into the alignment optimization process based on a 3-D distance transform algorithm. The proposed model and method are applied to two complex realistic railway cases. The results demonstrate that they can considerably improve the search efficiency and also find better alignments compared to the best alternatives obtained manually by experienced human designers and produced by a previous distance transform algorithm as well as a genetic algorithm.

Numerical Optimization of the Capacitors Selection in the MSBA Converter to Reduce the Output Voltage Ripple

DC–DC power electronics converters are widely used in many applications, such as renewable energy systems. The multistage-stacked boost architecture (MSBA) converter is a large voltage gain converter whose PWM scheme may reduce a percentage of the output voltage ripple, taking advantage of the symmetry of the voltage signals in capacitors (they are triangular waveforms) to have a symmetry cancelation. The switching ripple is unavoidable; the correct selection of components can reduce it, but this may result in a large amount of stored energy (larger size). The selection of capacitors influences the output voltage ripple magnitude. This article proposes a design methodology that combines a recently introduced PWM scheme with a numerical optimization method to choose the capacitors for the MSBA converter. The objective is to minimize the output voltage ripple by choosing two capacitors simultaneously while ensuring the constraint of a certain (maximum) amount of stored energy in capacitors is not overpassed. The internal optimization was performed with the differential evolution algorithm. The results demonstrate that the proposed method that includes numerical optimization allows having a very low output voltage ripple with the same stored energy in capacitors compared to the traditional converter. In a design exercise, up to 60% reduction was observed in the output voltage ripple with the same stored energy in capacitors.

Semi-analytical configuration optimization of geocentric gravitational wave observatory

The detection of gravitational waves (GWs) from space is a hot topic in astronomy and space science, as the Space Gravitational Wave Observatory can detect low-frequency GWs from various sources. The long-term stability of the configuration of GW observatories is the key to the success of GWs detection, which challenges the configuration design and optimization. This paper innovatively proposes an semi-analytical double-layer iterative optimization algorithm (DLI) to solve the geocentric configuration optimization problem. First, an iterative method with gradient information of breathing angle, which is compensated by its standard deviation, is introduced as the inner iteration layer. Then, the outer iterative layer executes the initial value update process via the selection of the constructed candidates with smallest arm length variation rate, until the maximum breathing angle variation converges to the preset accuracy. The proposed method is applied to optimize the nominal configuration of TianQin mission. The results shows that DLI can reduce about 99% computation duration compared with the numerical optimization method. Furthermore, the influence of initial configuration elements on stability parameters are analyzed, providing better alternation for geocentric GWs detection missions.

Overview of Numerical Analysis and Optimization Methods for Eddy Current Retarders

Aiming at the analysis and optimization of eddy current retarder, this paper reviews the development of numerical analysis based on finite element method. The characteristics of finite element method and the application of numerical method in the analysis of eddy current retarder are briefly introduced. In terms of algorithm, the different applications of multi-objective genetic algorithm, robustness principle, Taguchi algorithm and Rogowski method in the analysis of important parameters of eddy current retarder are introduced and summarized; In terms of software, the application of commercial software in magnetic field and temperature field is briefly introduced; In industrial practice, the development of numerical simulation methods in the research of braking torque is introduced. On the basis of reviewing the numerical analysis and optimization methods of eddy current retarder, the future of its digital design is prospected.

Special issue on PID control in the information age: Theoretical advances and applications

Special issue on PID control in the information age: Theoretical advances and applications

Time-optimal Dubins trajectory for moving obstacle avoidance

In this paper, the problem of computing the global time-optimal Dubins trajectory between two configurations (two points with prescribed heading angles) while avoiding a moving obstacle by a specified safe distance is discussed. The obstacle is allowed to move on any arbitrary trajectory as long as its trajectory satisfies some regularity constraints. By Pontryagin’s maximum principle, the fundamental segments of the time-optimal trajectory are proposed. Along each segment, the control input is either constant or is expressed in state-feedback form. Thereafter, using the necessary conditions for state-inequality constraints, the underlying geometric and analytical properties that govern the concatenation of these fundamental segments are revealed. All these properties exhibit solely state-dependent relations. This has two-fold applications: (1) such relations are essential in ruling out spurious extremals obtained by numerical optimal control methods, and (2) they enable the formulation of a set of nonlinear equations whose solution gives rise to the exact time-optimal trajectory. Numerical examples are presented, validating and highlighting the application of these properties.

Rod temperature regulation using current and time-delayed feedback

The synthesis problem of control of objects with distributed parameters with feedback is investigated on the example of the process of heating a rod in a furnace. To generate the values of control actions, we propose to use their linear de- pendence on the values of the state at the measurement points, both at the current and previous moments of time. The unknown coefficients involved in this dependence of control on the measured state values are feedback parameters. They are deter- mined by minimizing the objective functional using numerical methods of first-order optimization. Formulas for the gradient of the objective functional with respect to the feedback parameters are obtained. The results of numerical experiments are presented.

Pinch point determination and Multi-Objective optimization for working parameters of an ORC by using numerical analyses optimization method

In parallel with Kyoto, Paris and the green production agreements, the nowadays crucial subject is minimising energy consumption, increasing renewable energy usage and recovering waste heat. When it comes to waste heat, the organic Rankine cycle (ORC) technology comes fore. However, although the presence of many studies on the ORC in the literature, there is still an intense gap related to information on ORC working conditions, pinch point detection and numerical design and analysis methods. In this respect, the present paper suggests a multi-objective optimization model for an ORC by considering many parameters that affect the performance of the ORC like fluid type, thermodynamic properties of the organic fluids, pinch point temperature etc. For the analyses, experimentally recorded exhaust gas parameters for a reheat furnace located in the iron and steel plant were used. By considering all these, all performance-affecting parameters were included through the development of the multi-objective model. For the analyses, two wet types (ethanol, methanol), two isentropic (acetone, butene), and two dry types (cyclohexane, benzene) working fluids were selected to develop the multi-objective design and optimising method. After comprehensive analyses, it was seen that the developed mathematical model was valid for the designing and analysing of the ORC. In addition, it was observed that the system performance increases as the pinch point temperature difference decreases. In addition, it was seen that isentropic fluids have low efficiency in medium-temperature heat sources. Considering the system performance, initial investment cost and payback periods, it was seen that a waste heat recovery plant using Benzene, Methanol and Ethanol as working fluids is more feasible.

High-Bandwidth Tracking Control of Piezoactuated Nanopositioning Stages via Active Modal Control

Due to the lightly damped resonance and intrinsic nonlinearities, it is difficult for the piezoactuated nanopositioning stage to realize high-bandwidth and high-accuracy control. To handle these limitations, in this work, a dual-loop control scheme based on state-feedback-based modal method is designed to both actively damp and stiffen the resonant mode and to suppress the effects of nonlinearities of the piezoactuated nanopositioning stage. In this scheme, the state-feedback-based modal controller is first designed in the inner loop to enlarge both the damping ratio and natural frequency of the first resonant mode. Then, a proportional–integral (PI) controller is utilized in the outer loop for eliminating the tracking errors caused by other disturbances and nonlinearities including hysteresis and creep. To maximize the control bandwidth of system under the proposed dual-loop scheme, an optimization method is thus proposed for simultaneously tuning the parameters of the inner and the outer loop controllers. Finally, to validate the proposed dual-loop control scheme, comparative experiments are carried out on a piezoactuated nanopositioning stage. Results demonstrate that the proposed control scheme improves the bandwidth of the system from 497 Hz (with PI control) and 1543 Hz (with a commonly used positive acceleration, velocity, and position damping control and a PI controller) to 6546 Hz, which is 664 Hz larger than the first resonant frequency of the original system, validating the effectiveness of the proposed dual-loop scheme on high-bandwidth control. Note to Practitioners—The demand of high-bandwidth and high-accuracy piezoactuated nanopositioning stages increases rapidly. However, the lightly damped resonance of the mechanism and the intrinsic nonlinearities of the piezoelectric actuator limit the tracking performance of the stage. A dual-loop control structure is adopted in this work to improve the tracking performance of the nanopositioning stage. Different from most of the vibration control methods proposed in the literature which aimed only at improving the damping ratio, a state-feedback-based modal controller is designed in the inner-loop for improving both the damping ratio and the stiffness of the system. This task is realized by re-placing the resonant poles of the system to the optimized location. The outer-loop controller adopts the high-gain PI control for eliminating the tracking errors. More importantly, in order to realize the high-bandwidth and high-accuracy control, a numerical optimization method is proposed for simultaneously tuning the parameters of the controllers in inner and outer loops. The controller design is simple, and it can be applied to other systems with second or higher order in which the first resonant mode dominates the system dynamics.

Quasi-Developable B-Spline Surface Design with Control Rulings

An intuitive design method is proposed for generating developable ruled B-spline surfaces from a sequence of straight line segments indicating the surface shape. The first and last line segments are enforced to be the head and tail ruling lines of the resulting surface while the interior lines are required to approximate rulings on the resulting surface as much as possible. This manner of developable surface design is conceptually similar to the popular way of the freeform curve and surface design in the CAD community, observing that a developable ruled surface is a single parameter family of straight lines. This new design mode of the developable surface also provides more flexibility than the widely employed way of developable surface design from two boundary curves of the surface. The problem is treated by numerical optimization methods with which a particular level of distance error is allowed. We thus provide an effective tool for creating surfaces with a high degree of developability when the input control rulings do not lie in exact developable surfaces. We consider this ability as the superiority over analytical methods in that it can deal with arbitrary design inputs and find practically useful results.

Stigmatic optical system with corrected third-order spherical aberration for an arbitrary position of the object

Our paper is focused on a problem of analysis and design of a stigmatic optical system that has corrected third-order spherical aberration for an arbitrary position of the object. The relations for Seidel aberration coefficients of the system for an object at infinity that must be satisfied to ensure that the third-order spherical aberration does not depend on the position of the object are given. The method for obtaining design parameters of the initial optical system that can serve as a good starting point for further refinement using numerical optimization methods is proposed. Based on the use of modified formulas for third-order aberration coefficients, this method enables one to decide if the individual members of the optical system can be simple lenses or if these should be more complex elements (cemented doublets, triplets, etc.). As a final result, one obtains the design parameters of the above-mentioned optical system (radii of curvature, optical materials, axial separations between individual elements). The analysis is performed for a thin-lens representation of the system. The transition to the thick-lens optical system then can be done by mathematical methods of numerical optimization using commercially available optical design software. The proposed method is shown on a practical example of calculation of parameters of such an optical system.

The intrinsic quality of proteoglycans, but not collagen fibres, degrades in osteoarthritic cartilage

The function of articular cartilage as a load-bearing connective tissue is derived primarily from a balanced interaction between the swelling proteoglycan (PG) matrix and tension-resistant collagen fibrous network. Such balance is compromised during joint disease such as osteoarthritis (OA) due to degradation to PGs and/or collagens. While the PG degradation is generally thought to be related to a loss of protein abundance, the collagenous degradation is more complex as it can be caused independently by a decrease of collagen content, disorganisation of fibrous structure and softening of individual collagen fibrils. A comprehensive understanding of the initial trajectories of degradation of PGs and collagen network can improve our chance of finding potential therapeutic solutions for OA. Here, we developed geometrically, structurally, and compositionally realistic and sample-specific Finite Element (FE) models under the framework of multiphasic mixture theory, from which the elastic moduli of collagen fibres and the PG load-bearing quality in healthy and diseased cartilages were estimated by numerical optimisation of the multi-step indentation stress relaxation force-time curves. We found the intrinsic quality of collagen fibres, measured by their elastic moduli, to stay constant for healthy and diseased cartilages. Combining with previous findings which show unaltered collagen content during early stages of OA, our results suggest the disorganisation of collagen fibrous network as the first form of collagenous degradation in osteoarthritic cartilage. We also found that PG degradation involves not only a loss of protein abundance, but also the quality of the remaining PGs in generating sufficient osmotic pressure for load bearing. This study sheds light on the mechanism of OA pathogenesis and highlights the restoration of collageneous organisation in cartilage as key medical intervention for OA. Statement of significanceCollagen network in articular cartilage consists of individual fibres that are organised into depth-dependent structure specialised for joint load-bearing and lubrication. During osteoarthritis, the collagen network undergoes mechanical degradation, but it is unclear if a loss of content, disorganisation of fibrous structure, or softening of individual fibres causes this degeneration. Using mechanical indentation, Finite Element modelling, and numerical optimisation methods, we determined that individual fibres did not soften in early disease stage. Together with previous findings showing unaltered collagen content, our results pinpoint the disorganisation of collagen structure as the main culprit for early collagenous degradation in osteoarthritic cartilage. Thus, early restoration in cartilage of collagen organisation, instead of individual fibre quality, may be key to slow osteoarthritis development.