Numerical Optimization Method Research Articles

On Some Works of Boris Teodorovich Polyak on the Convergence of Gradient Methods and Their Development

The paper presents a review of the current state of subgradient and accelerated convex optimization methods, including the cases with the presence of noise and access to various information about the objective function (function value, gradient, stochastic gradient, higher derivatives). For nonconvex problems, the Polyak–Lojasiewicz condition is considered and a review of the main results is given. The behavior of numerical methods in the presence of a sharp minimum is considered. The aim of this review is to show the influence of the works of B.T. Polyak (1935–2023) on gradient optimization methods and their surroundings on the modern development of numerical optimization methods.

Trajectory Planning for Autonomous Vehicle with Numerical Optimization Method

Trajectory Planning for Autonomous Vehicle with Numerical Optimization Method

Research on the potential of integrated vehicle control with closed-loop phase portrait analysis

The phase portrait is an important tool for analyzing vehicle stable region widely utilized in vehicle dynamic control systems. At handling limits, traditional open-loop vehicle stable region constraints will restrict the vehicle control ability, while the closed-loop actuator interventions can help states outside the open-loop stable region converge back to stable origin, which is more suitable for constraining vehicle states under extreme working conditions. This article systematically analyzes the potential of integrated vehicle control with closed-loop phase portrait. Firstly, a vehicle dynamic model considering actuator dynamics and tire transient characteristics is established. Then, a numerical optimal control method for calculating closed-loop controllability region is introduced. The controllability region under different friction coefficients, vehicle speeds, convergence time, vehicle steering response characteristics, actuator time constants, and tire transient characteristics are discussed and analyzed. In terms of different control inputs combinations, this article displays the controllability regions under different front steering angles, rear steering angles, direct yaw moments, and their synergistic performances. Control input laws derived from numerical optimization are summarized in two specific directions. Finally, a comparative analysis of open-loop stable regions and closed-loop controllability regions confirms the superiority of the proposed method. Potential applications of the systematic analysis are also discussed.

Trajectory Planning Design for Parallel Parking of Autonomous Ground Vehicles with Improved Safe Travel Corridor

In this paper, the concept of symmetry is utilized to design the trajectory planning for parallel parking of autonomous ground vehicles—that is, the construction and the solution of the optimization-based trajectory planning approach are symmetrical. Parking is the main factor that troubles most drivers for their daily driving travel, and it can even lead to traffic congestion in severe cases. With the rise of new intelligent and autonomous vehicles, automatic parking seems to have become a trend. Traditional geometric planning methods are less adaptable to parking scenarios, while the parking paths planned by graph search methods may only achieve local optimality. Additionally, significant computational time is often required by numerical optimization methods to find a parking path when a good initial solution is not available. This paper presents a hierarchical trajectory planning approach for high-quality parallel parking of autonomous ground vehicles. The approach begins with a graph search layer to roughly generate an initial solution, which is refined by a numerical optimization layer to produce a high-quality parallel parking trajectory. Considering the high dimensionality and difficulty of finding an optimal solution for the path planning optimization problem, this paper proposes an improved safe travel corridor (I-STC) with the construction of collision constraints isolated from surrounding environmental obstacles. By constructing collision constraints of the I-STC based on the initial solution, the proposed method avoids the complexities and non-differentiability of traditional obstacle avoidance constraints, and simplifies the problem modeling the subsequent numerical optimization process. The simulation results demonstrate that the I-STC is capable of generating parallel parking trajectories with both comfort and safety.

Stages and main tasks of the century-long control theory and system identification development. Part VI. Model predictive control

The article is devoted to one of the relatively new areas in the field of control theory and practice, called model predictive control. The ideas underlying it, the connection with the theory of optimal control and numerical methods for solving optimization problems are described. It should be noted that currently the number of problems solved and control systems developed within the framework of this approach has come out on top in comparison with all others. This is evidenced by numerous publications concerning the theory and practical application of control methods and algorithms using a predictive model. The main distinctive feature of this method is that the control is not synthesized in advance, but is calculated directly during the operation of the system, i.e. at the current time. This made it possible, with such control, to implement original feedback that ensures stabilization of the process when the model is inaccurate and there is uncertainty associated with existing disturbances and measurements. It describes how such feedback is organized based on current measurement data. In this case, there is no need to interpret somehow the nature of the disturbances and errors acting on the system. Much attention in the article is paid to the possibilities that are admitted with this method of control. The model predictive control method seems especially attractive for systems with discrete impulse processes that are well described with using cognitive maps. It is considered an example of cognitive modeling of evolutionary demographic processes in Ukraine and the possibility of control them using a predictive model, when the model contains unstable eigenvalues. When solving problems of predictive control, it is proposed to use trajectory models, which directly follow from the Cauchy formula, in particular for discrete systems, instead of a local description in the state space. This makes it possible to take into account, when solving predictive control problems, how well or poorly controlled, observable or identifiable the system is. Non-standard capabilities of this method with a trajectory description appear during terminal control on a sliding interval with a finite horizon. A number of original formulations of terminal control problems are presented in the article and some of their properties are described.

Deep-Learning-Coupled Numerical Optimization Method for Designing Geometric Structure and Insertion-Withdrawal Force of Press-Fit Connector

Deep-Learning-Coupled Numerical Optimization Method for Designing Geometric Structure and Insertion-Withdrawal Force of Press-Fit Connector

Mechanical Properties of End-of-life Waste Tyres Steel Fibre Reinforced Concrete: RSM-Based Modelling and Optimisation

The influence of fibres on the compressive strength of concrete is complex and is determined by the type, quantity, and characteristics of the fibres utilized in designing and forming the concrete. Designing fibre reinforced concrete (FRC) constituents is challenging and affects the concrete's performance and practicality. This study utilized response surface methodology (RSM) to optimize the properties of concrete containing steel fibre extracted from end-of-life tyres (ELTs). Face-cantered central composite design (FC-CCD) of RSM was used in the design of experiments (DOE) with aspect ratio (10-70) and volume fraction (0.5 % - 1.2 %) as the input variables. Three levels of each variable were used in forming the design matrix. The resulting concretes were then tested for slump and compressive strength at 7 and 28 days of curing. As the aspect ratio and volume fraction increased, slump values decreased, and 7- and 28-day compressive strength increased up to an aspect ratio of 45 and 1.0% volume fraction. Beyond an aspect ratio of 45 and 1.0% volume fraction, a declining trend in compressive strength at all curing ages was observed. The Analysis of Variance (ANOVA) revealed that the variables volume fraction and aspect ratio significantly affect the variability in the FRC models, with all models being statistically significant at the 95% level across all factor levels. A numerical optimization method was used to determine the optimal mix proportions for ELTFRC. The optimum response values were achieved by combining a 1.01% volume fraction and a 33.86 aspect ratio, resulting in a desirability of 0.73.

An Optimal Family of Eighth-Order Methods for Multiple-Roots and Their Complex Dynamics

We present a new family of optimal eighth-order numerical methods for finding the multiple zeros of nonlinear functions. The methodology used for constructing the iterative scheme is based on the approach called the ‘weight factor approach’. This approach ingeniously combines weight functions to enhance convergence properties and stability. An extensive convergence analysis is conducted to prove that the proposed scheme achieves optimal eighth-order convergence, providing a significant improvement in efficiency over lower-order methods. Furthermore, the applicability of these novel methods to some real-world problems is demonstrated, showcasing their superior performance in terms of speed and accuracy. This is illustrated through a series of three examples involving basins of attraction with reflection symmetry, confirming the dominance of the new methods over existing counterparts. The examples highlight not only the robustness and precision of the proposed methods but also their practical utility in solving the complex nonlinear equations encountered in various scientific and engineering domains. Consequently, these eighth-order methods hold great promise for advancing computational techniques in fields that require the resolution of multiple roots with high precision.

Optimization of Alkali Treatment Conditions of Ramie Fabrics Using Box–Behnken Method

Ramie is a widely used plant fiber for making textiles and reinforcement in biodegradable composites. Pretreating cellulosic fibers with alkali before producing composites is increasingly used to enhance adhesion with polymeric resin. In this work, response surface methodology (RSM) based on the Box-Behnken technique was utilized to investigate the impact of independent variables on ramie fabric characteristics and determine the optimal treatment condition. The impact of alkali concentration, treatment time, and temperature on the breaking load and elongation at break of woven ramie fabrics were evaluated using Design-Expert software, which established the design matrix and analyzed the experimental data employing numerical and graphical optimization methods. Moreover, the impact of alkali treatment conditions on the surface morphology, structural change of ramie fabrics, and thermal properties was investigated. Based on the analysis of variance (ANOVA) results, the suggested quadratic models can adequately predict the breaking load and elongation at break of the ramie woven fabrics within the range of conditions applied in this investigation. The RSM revealed that an alkali concentration of 6.12%, a treatment time of 30 min, and a temperature of 39.13°C resulted in an optimum treatment condition with a breaking load of 518.28 N and elongation at break of 23.36%.

Modeling and simulation of optical system error transmission in the laser tracker

The optical system of the laser tracker utilizes plane mirrors to construct a reflective path, reducing its size and weight. However, maintaining the alignment of the laser with the ideal optical axis during its propagation in the optical system poses significant challenges in the design, fabrication, and assembly of the optical system. This paper explores the principle of error propagation during the assembly process of the optical system and improves the accuracy of the output laser through a numerical simulation and optimization methods. A general error model for the optical system is established to understand the principle of error propagation. A Monte Carlo numerical simulation and sensitivity analysis are used to study the influence of various errors on the accuracy of the output laser. The machining errors are optimized using a simulated annealing method to balance the manufacturing difficulty and system accuracy. The assembly process is also optimized to reduce the degrees of freedom and the number of optical parts required, and verified by experiments. The experimental results indicate that the average position error of the output laser is 15.743 µm, and the average angle error is 1.427′′. This study provides what we believe is a novel approach and methodology for the design and alignment of optical systems.

Magnetic-field coils for metastability-exchange optical pumping, spectroscopy, and magnetic resonance of helium.

Optical pumping on the 23S-23P transition (1083nm) of metastable 3He or 4He atoms is used for science and applications. Gas is usually enclosed in elongated cells with lengths ranging from several centimeters to several meters for efficient absorption. Good magnetic-field homogeneity is needed for weak diffusion-induced relaxation and long nuclear magnetic resonance (NMR) signal lifetimes. A compact coil system is designed using a target-oriented numerical optimization method. It provides a suitably uniform field over cell volumes, with characteristics depending on the chosen optimization parameters. Additional pairs of coils can be used to generate transverse field components with contributions to the total field inhomogeneities that are discussed.

Визначення оптимальних норм надійності космічного апарата з урахуванням економічних показників

The goal of this paper is to develop a method for determining the values of system failure rates optimal in terms of net profit maximization with account for spacecraft mass limitations. It is adopted that the spacecraft systems are independent in terms of reliability, each of them can be only in two states (a functioning state or a failure state), and each equipment type of the special complex makes an independent contribution to the overall effect. The paper considers the case where the spacecraft system failure time obeys the exponential law. Use is made of the Lagrange multiplier method and numerical optimization methods. The problem is solved using the mass ? failure rate and cost ? failure rate relationships of the spacecraft systems. For the supporting complex, the dimension of the mathematical simulation problem is reduced to two: a formula is derived to find the optimal failure rates of all the systems of the supporting complex using the optimal failure rates of two systems of it. The variables for the special complex and the supporting complex are separated. Due to the fact that each special complex system makes an independent contribution to the overall effect, the problem for the whole spacecraft is reduced to a system of two equations for the special complex and the supporting complex and ncn equations in one variable for the special complex with two coupling variables between the supporting complex and the special complex where ncn is the number of the special complex systems. The paper presents a numerical-and-analytical method for spacecraft system failure rate optimization where the initial guess is specified indirectly: by specifying the supporting complex mass and probability of no-failure operation. The method may be used in the development of a space hardware design methodology accounting for economic factors. From the optimal values of the spacecraft system failure rates found using the mass ? failure rate and cost ? failure rate relationships, one can find the masses and costs of the spacecraft systems to be used in optimizing the parameters and development cost of the systems. The method is expected to increase the profitability and competitiveness of spacecraft under development.

Compressive instabilities enable cell-induced extreme densification patterns in the fibrous extracellular matrix: Discrete model predictions.

We present a new model and extensive computations that explain the dramatic remodelling undergone by a fibrous collagen extracellular matrix (ECM), when subjected to contractile mechanical forces from embedded cells or cell clusters. This remodelling creates complex patterns, comprising multiple narrow localised bands of severe densification and fiber alignment, extending far into the ECM, often joining distant cells or cell clusters (such as tumours). Most previous models cannot capture this behaviour, as they assume stable mechanical fiber response with stress an increasing function of fiber stretch, and a restriction to small displacements. Our fully nonlinear network model distinguishes between two types of single-fiber nonlinearity: fibers that undergo stable (supercritical) buckling (as in previous work) versus fibers that suffer unstable (subcritical) buckling collapse. The model allows unrestricted, arbitrarily large displacements (geometric nonlinearity). Our assumptions on single-fiber instability are supported by recent simulations and experiments on buckling of individual beams with a hierarchical microstructure, such as collagen fibers. We use simple scenarios to illustrate, for the first time, two distinct compressive-instability mechanisms at work in our model: unstable buckling collapse of single fibers, and snap-through of multiple-fiber groups. The latter is possible even when single fibers are stable. Through simulations of large fiber networks, we show how these instabilities lead to spatially extended patterns of densification, fiber alignment and ECM remodelling induced by cell contraction. Our model is simple, but describes a very complex, multi-stable energy landscape, using sophisticated numerical optimisation methods that overcome the difficulties caused by instabilities in large systems. Our work opens up new ways of understanding the unique biomechanics of fibrous-network ECM, by fully accounting for nonlinearity and associated loss of stability in fiber networks. Our results provide new insights on tumour invasion and metastasis.

Fast Low-Sidelobe Pattern Synthesis Using the Symmetry of Array Geometry.

Array pattern synthesis with low sidelobe levels is widely used in practice. An effective way to incorporate sensor patterns in the design procedure is to use numerical optimization methods. However, the dimension of the optimization variables is very high for large-scale arrays, leading to high computational complexity. Fortunately, sensor arrays used in practice usually have symmetric structures that can be utilized to accelerate the optimization algorithms. This paper studies a fast pattern synthesis method by using the symmetry of array geometry. In this method, the problem of amplitude weighting is formulated as a second-order cone programming (SOCP) problem, in which the dynamic range of the weighting coefficients can also be taken into account. Then, by utilizing the symmetric property of array geometry, the dimension of the optimization problem as well as the number of constraints can be reduced significantly. As a consequence, the computational efficiency is greatly improved. Numerical experiments show that, for a uniform rectangular array (URA) with 1024 sensors, the computational efficiency is improved by a factor of 158, while for a uniform hexagonal array (UHA) with 1261 sensors, the improvement factor is 284.

Analytical robust design optimization for hybrid design variables: An active-learning methodology based on polynomial chaos Kriging

In robust design optimization, statistical moments of performance are widely adopted in formulating robustness metrics. To address the high computational costs stemming from the many-query nature of such optimizations with respect to robustness metrics, analytical formulas of the statistical moments have been developed based on surrogate models. However, existing methods consider random variables as the sole model input, which excludes, from the application scope, problems that also involve deterministic design variables. To remedy this issue, this paper proposes a new Polynomial Chaos Kriging-based methodology for efficient and accurate analytical robust design optimization. The analytical solutions for the statistical moments of performance are developed considering that the Polynomial Chaos Kriging model is established in the augmented space of the deterministic design and random variables. This is achieved by systematically decoupling associations with deterministic input from random input, providing effective solutions even when the orthonormality of the basis function is not applicable in the augmented space. This work also presents an active-learning framework enabling seamless implementation of various numerical optimization methods. Several numerical examples and a practical application illustrate the performance and superiority of the proposed method.

A comparison of the convergence rates of Hestenes’ conjugate Gram-Schmidt method without derivatives with other numerical optimization methods

Abstract This article describes an approach known as the conjugate Gram-Schmidt method for estimating gradients and Hessian using function evaluations and difference quotients, and uses the Gram-Schmidt conjugate direction algorithm to minimize functions and compares it to other techniques for solving ∇f = 0. Comparable minimization algorithms are also used to demonstrate convergence rates using quotient and root convergence factors, as described by Ortega and Rheinbolt to determine the optimal minimization technique to obtain similar results to the Newton method, between the Gram-Schmidt approach and other minimizing approaches. A survey of the existing literature in order to compare Hestenes’ Gram-Schmidt conjugate direction approach without derivative to other minimization methods is conducted and the further analytical and computational details are provided.

On optimal three-impulse Earth–Moon transfers in a four-body model

Within the emerging age of lunar exploration, optimizing transfer trajectories is a fundamental measure toward achieving more economical and efficient lunar missions. This study addresses the possibility of reducing the fuel cost of two-impulse Earth–Moon transfers in a four-body model with the Earth, the Moon, and the Sun as primaries. Lawden’s primer vector theory is applied within this framework to derive a set of necessary conditions for a fuel-optimal trajectory. These conditions are used to identify which trajectories from an existing database could benefit from the insertion of an additional intermediate impulse. More than 10,000 three-impulse transfers are computed with a direct numerical optimization method. These solutions contribute to enriching the database of impulsive trajectories, useful to perform trade-off analyses. While the majority of trajectories exhibit only marginal improvements, a significant breakthrough emerges for transfers featuring an initial gravity assist at the Moon. Implementing a corrective maneuver after the lunar encounter yields substantial reductions in fuel costs.

Optimization of the generalized covariance estimator in noncausal processes

This paper investigates the performance of routinely used optimization algorithms in application to the Generalized Covariance estimator (GCov) for univariate and multivariate mixed causal and noncausal models. The GCov is a semi-parametric estimator with an objective function based on nonlinear autocovariances to identify causal and noncausal orders. When the number and type of nonlinear autocovariances included in the objective function are insufficient/inadequate, or the error density is too close to the Gaussian, identification issues can arise. These issues result in local minima in the objective function, which correspond to parameter values associated with incorrect causal and noncausal orders. Then, depending on the starting point and the optimization algorithm employed, the algorithm can converge to a local minimum. The paper proposes the Simulated Annealing (SA) optimization algorithm as an alternative to conventional numerical optimization methods. The results demonstrate that SA performs well in its application to mixed causal and noncausal models, successfully eliminating the effects of local minima. The proposed approach is illustrated by an empirical study of a bivariate series of commodity prices.

Optimization of cutout characteristics in AISI 1045 steel plates for high-cycle fatigue life extension

Crack growth is a pivotal concern in structural engineering, directly impacting structures’ stability, quality, and overall lifespan. This study addresses this challenge by investigating structural optimization strategies to mitigate fatigue crack growth (FCG) by strategically incorporating stop-drilled cutouts near the crack tip. The research introduces three optimization problems, focusing on determining the optimal cutout position with a fixed radius, the optimal cutout radius with a fixed position, and the optimal cutout shape with a constant position for the cutout’s center. Experimental assessments are conducted on an edge-cracked AISI 1045 steel plate to evaluate the effectiveness of these optimization strategies in enhancing fatigue life. Employing the Extended Finite Element Method (XFEM) for numerical simulations and Particle Swarm Optimization (PSO) for the optimization process, the findings reveal that strategically positioned cutouts substantially reduce the crack growth rate. Optimal cutout placement significantly improves fatigue life, demonstrating an approximate 287% increase. The analysis affirms the accuracy of the numerical model in predicting fatigue life and crack growth paths, validated through comparison with experimental data. In summary, this research underscores the effectiveness of optimization techniques in mitigating fatigue crack growth, providing valuable insights for structural integrity enhancement in structural mechanics.

STRENGTH DESIGN OF REINFORCED CONCRETE SECTIONS BASED ON A NUMERICAL OPTIMIZATION METHOD

Reinforced concrete structures are widely used in construction of civil infrastructures. Classical methods are available for the analysis and design of reinforced concrete members, such as beams or columns, which are typically done by hand or using a spreadsheet program. In this study, an analysis and design approach of reinforced concrete sections is developed based on a numerical optimization method, and applied in accordance with the latest concrete design code. The basic idea of the proposed approach is to derive an optimization formulation in order to determine the minimum steel area required to support a given factored load combinations applied on a concrete section. A rigorously derived analysis and design framework of general beam sections is presented. For practical applications by civil and structural engineers, the readily-available optimization solver in Microsoft Excel spreadsheet is used. To illustrate the applicability of the approach, a numerical example with different analysis and design cases is studied. The proposed analysis and design approach using numerical optimization and spreadsheet solver works very well and is easy to implement in the Excel environment. The analysis and design results compare well with hand calculation results based on the classical method.