Penalty Function Research Articles

A trust-region scheme for constrained multi-objective optimization problems with superlinear convergence property

In this paper, a numerical approximation method is developed to find approximate solutions to a class of constrained multi-objective optimization problems. All the functions of the problem are not necessarily convex functions. At each iteration of the method, a particular type of subproblem is solved using the trust region technique, and the step is evaluated using the notions of actual reduction and predicted reduction. A non-differentiable l ∞ penalty function restricts the constraint violations. An adaptive BFGS update formula is introduced. Global convergence of the proposed algorithm is established under the Mangasarian-Fromovitz constraint qualification and some mild assumptions. Furthermore, it is justified that the proposed algorithm displays a super-linear convergence rate. Numerical results are provided to show the efficiency of the algorithm in the quality of the approximated Pareto front.

Inferring dynamical models from time-series biological data using an interpretable machine learning method based on weighted expression trees

The growing time-series data make it possible to glimpse the hidden dynamics in various fields. However, developing a computational toolbox with high interpretability to unveil the interaction dynamics from data remains a crucial challenge. Here, we propose a new computational approach called automated dynamical model inference based on expression trees (ADMIET), in which the machine learning algorithm, the numerical integration of ordinary differential equations and the interpretability from prior knowledge are embedded into the symbolic learning scheme to establish a general framework for revealing the hidden dynamics in time-series data. ADMIET takes full advantage of both machine learning algorithm and expression tree. Firstly, we translate the prior knowledge into constraints on the structure of expression tree, reducing the search space and enhancing the interpretability. Secondly, we utilize the proposed adaptive penalty function to ensure the convergence of gradient descent algorithm and the selection of the symbols. Compared to gene expression programming, ADMIET exhibits its remarkable capability in function fitting with higher accuracy and broader applicability. Moreover, ADMIET can better fit parameters in nonlinear forms compared to regression methods. Furthermore, we apply ADMIET to two typical biological systems and one real data with different prior knowledge to infer the dynamical equations. The results indicate that ADMIET can not only discover the interaction relationships but also provide accurate estimates of the parameters in the equations. These results demonstrate ADMIET’s superiority in revealing interpretable dynamics from time-series biological data.

Error estimates of unilateral piezoelectric contact problem in a curved and smooth boundary domain

We study the linear finite element approximation of piezoelectric unilateral contact problem in a curved and smooth boundary domain. The unilateral contact conditions will be weakly imposed by the penalty method. We derive error estimates which depend on the penalty parameter and the mesh size. In fact, under regularity of the solution, we prove some convergence rate results.

Spectral element-finite element modeling and dynamic analysis of a fluid-delivering cracked pipe subjected to both pulsation and base excitations

Previous studies on cracked pipes have predominantly focused on single excitation, but in practical engineering, pipe systems often experience a combined effect of fluid pulsation and base excitations, which can potentially trigger more complex dynamic behaviors. Moreover, the solid finite element (FE) models are generally adopted to simulate the breathing effect and stress singularity, but the solution efficiency is unacceptable. Therefore, this study presents a spectral element-finite element (SE-FE) method to construct the dynamic models of the fluid-delivering cracked pipes (FDCPs), which combines the efficiency of spectral element (SE) with the adaptability of FE. Furthermore, a fluid pulsation model is constructed using measured data. Specifically, dynamic models of intact and cracked pipe segments are constructed using SE and FE, respectively, and interface coupling is achieved by the penalty function method. Subsequently, an experimental system for fluid pulsation excitation is established, and then a fluid pulsation model is constructed. Finally, the dynamic behaviors of a FDCP under compound excitation are analyzed. The results show that the fluid pulsation in the FDCP is higher than that in the intact pipe due to the breathing effect. Moreover, the beat vibration will be triggered when the pulsation frequency approaches the base excitation frequency. This study can provide a better understanding for the dynamic behaviors of FDCP, thereby providing a reference for crack damage detection.

Approaches to numerical analysis of optimal control with linear phase constraints on the example of the assets and liabilities management by a bank

This paper presents the research of approaches to numerically compute the solution to an optimal control problem with phase constraints. Such models are typical for economic modeling, but can also be found in a variety of applications. The formal description of planning the assets and liabilities by the management of a bank within internal goals and regulatory requirements may be formalized as an optimal control problem with constraints. Solving such models analytically is rarely possible, but numerical experiments give the idea of the properties of the solution and the prospective dynamics of the bank’s balance sheet. Two types of methods are available to solve this problem numerically: indirect and direct methods. We present both approaches in this paper and test them on the example where the analytical solution is found explicitly. The indirect method in the form of the maximum principle, is used for the version of the model with penalty functions and for the version of the problem with one phase constraint. The corresponding system of ordinary differential equations combined with specific equations for the dual variables are solved by the shooting method. The direct method is applied to the model formulated in the discrete time. It is shown that such version of the model has the global solution if we reformulate it as the optimal control problem with mixed constraints. This paper provides the results of computation using both the direct and indirect methods and the results of numerical experiments.

Innovative air supply management in fuel cells: An economic predictive control approach without pre-measured OER

Existing fuel cell air supply control methods improve the economic performance of Proton Exchange Membrane Fuel Cells (PEMFC) under steady-state conditions through oxygen excess ratio curve, while ignoring the transient process of the air supply system under varying operating conditions. Furthermore, the oxygen excess ratio curve may exhibit deviations due to fuel cell aging and measurement inaccuracies, which would further diminish the economic performance derived from the control process. Not relying on pre-measuring the oxygen excess ratio curve, an EMPC (Economic Model Predictive Control) method is proposed for fuel cell air supply systems. It incorporates an economic cost function reflecting the economic objectives of the air supply system. By dynamically optimizing the parasitic power of the air compressor, this method enhances the overall economic performance by achieving performance optimization for both transient and steady-state processes of the fuel cell system. The asymptotic stability of PEMFC is achieved through the utilization of terminal penalty functions and terminal domain constraints. Stability and convergence analysis is carried out by employing auxiliary optimization problems and the Lyapunov method. In transient processes, EMPC method can achieve a maximum 4.97 % improvement in the economic performance of PEMFC. A hardware in loop (HIL) experiment was finally conducted to show the real-time capacity of the proposed EMPC method.

Dynamic analysis of cracked pipe elbows: Numerical and experimental studies

This study aims to investigate the dynamic behaviors of fluid-delivering cracked pipe elbows (FDCPEs), thereby providing theoretical and methodological guidance for vibration-based crack detection in pipe elbows. Specifically, an efficient spectral element-finite element (SE-FE) based modeling method for the FDCPEs is proposed firstly, which considers fluid-structure coupling and breathing effect. For the SE-FE method, the spectral element (SE) and finite element (FE) are constructed to simulate the healthy straight segments and cracked elbows of the pipe, respectively, and a penalty function method is adopted to simulate interface coupling and breathing effects. Furthermore, the computational accuracy and efficiency of the SE-FE method are verified through numerical and experimental studies. Finally, the effects of the elbow and crack parameters on the dynamic behaviors of FDCPE are analyzed. The results indicate that changes in the bending radius and angle may trigger frequency veering behaviors, leading to in-plane and out-plane vibration mode switching. Moreover, the even-mode harmonics of the stress responses can effectively detect cracks in the elbow. Most importantly, the strength of nonlinearity at different measuring locations can be used to predict the crack location and size.

Mesh antenna cable net form finding optimization by an improved NSGA-II

The tension uniformity and surface accuracy of mesh antenna cable net are two well-accepted evaluation criteria for form finding. Traditional form finding techniques often rely on single-objective optimization and weighting factors. Although the Non-dominated Sorting Genetic Algorithm II (NSGA-II) can address the limitation, there are still challenges of wide distribution and insufficient precision of solutions. In the present work, a novel multi-objective optimization method based on an improved NSGA-II is introduced to conduct mesh antenna cable net form finding optimization. Based on the preference region (PR), a self-adaptive penalty function is proposed. The function is designed to steer the optimization process toward the PR, and enhancing the precision of the solutions. The extent of penalization for an individual solution depends on its positioning relative to the PR and is modified by a penalty coefficient that adapts dynamically to the population number within the PR. Additionally, a form finding method considering flexible frames is developed. The initial population for the flexible frame optimization is sourced from results of rigid frames, with the maximum truss node position error as the convergence criterion. Benchmark tests and case studies confirm that the improved NSGA-II enhances distributivity and convergence, while also providing form-finding results with higher surface accuracy and more uniform tension distribution.

Evaluation of Boundary Stiffnesses for Beams with Generally Restrained Edges: Theory and Vibration Experiments

An effective methodology is proposed to evaluate the boundary stiffnesses of beams with generally restrained edges. The boundary conditions at the ends of the beams are implemented by the penalty method. The unified Jacobi polynomials are introduced to represent the assumed mode shape functions of the beam. Hamilton’s principle with the assumed mode method is employed to derive the equation of motion of the beam with generally restrained boundaries. Subsequently, a novel approach is proposed to evaluate the boundary stiffnesses of the beam using the measured natural frequencies of the beam by vibration experiments. The accuracy of this method is verified by comparing the results with the previously published literature, the finite element method (FEM), and the vibration experiments. The effects of the types and values of the boundary spring stiffnesses on the vibration properties of the beam are also discussed. The calculation method in this research can provide an effective prediction technique for the boundary stiffnesses of engineering structures with generally restrained edges.

Sound field reconstruction using improved ℓ1-norm and the Cauchy penalty method

Sound field reconstruction using improved ℓ1-norm and the Cauchy penalty method

Lattice Boltzmann method for the linear complementarity problem arising from American option pricing

In this paper, a lattice Boltzmann method is proposed for solving the linear complementarity problem (LCP) arising in single and multi-asset American put option pricing. The LCP for American option is a variable coefficient parabolic model defined on an unbounded domain. Initially, using the far field estimate method and the penalty method respectively, the LCP could be reformulated into a nonlinear parabolic partial differential equation on a bounded domain. To construct a unified lattice Boltzmann model for the option pricing problems, the above transformation equations are rewritten into an equivalent divergence form. Then, through the incorporation of an amending function into the evolution equation, which assists in recovering the source term and eliminating the error term, the lattice Boltzmann model with spatial second-order accuracy is constructed. Finally, the present model is validated using numerical simulations, and the numerical results agree well with the option values obtained by existing methods, which indicates that the present lattice Boltzmann model is efficient for solving the American put option pricing problem.

Hybrid strategy of flexibility regulation and economic energy management in the power system including renewable energy hubs based on coordination of transmission and distribution system operators

Energy management of energy hubs in the power system impacts both transmission and distribution networks and their operations. Hence, to enhance efficiency, there should be bilateral coordination between distribution and transmission networks. Moreover, thanks to including power sources, storage, and responsive loads, hubs can earn financial benefits from the energy and ancillary services market. Therefore, this paper presents the operation of distribution and transmission networks in the presence of flexible-renewable energy hubs based on transmission system operator (TSO)-distribution system operator (DSO) coordination according to the market clearing price modeling in energy and flexibility markets. The proposed scheme minimizes the sum of the expected operating costs of the networks and hubs minus the expected flexibility profit of the hubs. The scheme is constrained to optimal power flow equations of the networks, flexibility limit of the networks, and flexibility-operation model of renewable energy hubs located in transmission and distribution networks. The unscented transformation method is used to model the uncertainties of load, renewable sources, and stationary storage (aggregation of electric vehicles). Energy price and flexibility are among the dual variables of the problem. To simultaneously calculate these variables and the main variables of the problem, an integrated model of the suggested scheme based on the penalty function technique is extracted. Eventually, numerical results validate the capability of the scheme in enhancing the operation, economic, and flexibility status of the proposed scheme, in which renewable sources along with storage and responsive load in the energy hub can obtain 100 % flexibility conditions for the networks. They help enhance the economic and operation indices of distribution and transmission networks by about 26 %-44 % and 22 %-33 % compared to the case without hubs and renewable sources.

Hybrid weakly over-penalised symmetric interior penalty method on anisotropic meshes

Hybrid weakly over-penalised symmetric interior penalty method on anisotropic meshes

Sustainable optimization of micro-milling machining parameters considering reliability assessment

Micro-milling is a crucial machining technology used for complex and precise 3D parts and plays a crucial role in modern manufacturing. In this work, to comprehensively evaluate the safety, economic feasibility, and environmental friendliness of the micro-milling processing process, a sustainable optimization framework including reliability assessment is designed. Firstly, modeling and experimental measurements are conducted on sustainability evaluation standards such as tool life, machining process, machining surface quality, material removal rate, machining cost, and carbon emissions. Subsequently, the influence of physical information such as observation/measurement data and processing errors is analyzed, and a Bayesian update method is proposed and integrated with neural networks to precisely inversion the probability distribution of parameters within the optimization cycle under conditions of uncertainty. Finally, combined with the overall sustainability indicators, the multi-objective optimization process is refined through penalty functions and weighting methods, aiming to achieve the best balance between processing performance and sustainability. The simulation and experimental results have verified the superiority of the developed framework, and the present method provides valuable guidance for sustainable micromanufacturing.

A bond-augmented stabilized method for numerical oscillations in non-ordinary state-based peridynamics

Non-ordinary state-based peridynamics has been widely used due to its capability to combine non-local theory with conventional material models. However, it suffers from numerical oscillations in calculations. The fundamental reason is that the approximate deformation gradient causes the non-unique mapping between deformation states and force states. To address this issue, the present work utilizes the penalty function method to reformulate the deformation gradient for each individual bond. The force state corresponding to a single bond is then constructed based on this newly developed bond-augmented deformation gradient in the framework of least action principle. This bond-augmented force state has a strong dependence on the deformation of that particular bond, thus well resolving the non-unique mapping issue. The resulting bond-augmented stabilized method does not introduce an additional local bond-associated horizon or extra force state termed as that in the previous studies and does not require tedious parameter adjustments. Numerical results show that the proposed bond-augmented stabilized method can significantly remove numerical oscillations and have good accuracy and convergence. Moreover, the proposed bond-augmented stabilized method is also proved to readily capture crack paths.

Optimal pipe-sizing design of water distribution networks using modified Rao-II algorithm

ABSTRACT Several evolutionary algorithms (EAs) have been suggested in the last three decades for the least-cost design of water distribution networks (WDNs). EAs generally worked well to identify the global/near-global optimal solutions for small- to moderate-size networks in a reasonable time and computational effort. However, their applications to large-size networks are still challenging due to large computational effort. Recent developments in EAs are towards parameter-less techniques that avoid fine-tuning of case-specific parameters to reduce the computational effort. Further, several self-adaptive penalties and search-space reduction methodologies have been suggested to reduce the computational effort. A fast, efficient, and parameter-less Rao-II algorithm has been used earlier with penalty-based approaches for the optimal design of WDNs. In this study, the application of a Rao-II algorithm is further explored with three self-adaptive penalty approaches to compare the convergence characteristics. The Rao-II algorithm is observed to converge at an infeasible solution in cases that the applied penalty to an infeasible solution is not so substantial to make it inferior to the feasible solutions. Modifications are suggested to improve the Rao-II algorithm, named the modified Rao-II algorithm. The modified Rao-II algorithm with the self-adaptive penalty methods resulted in better solutions than those obtained earlier.

Permeability of large‐scale fractures with ununiform proppant distributions in coalbed methane development

AbstractThe coalbed methane (CBM) productivity is directly determined by the fracture permeability during hydraulic fracturing, which is regulated by the distribution of proppants. The proppant may be unevenly distributed in the fracture because of variables like the architecture of the fracture and the characteristics of the sand‐carrying fluid. This study used two types of random functions to produce different ununiform distributions of proppant clusters in large‐scale fractures, with the aim of investigating the effect of these distributions on the overall permeability of the fracture. A model of fluid‐structure coupling is proposed. The closure of large‐scale fractures under in‐situ stress is analyzed using solid mechanics and the penalty function; the CBM flowing in proppant clusters and the high‐speed channel between them is simulated using Darcy's law and the Navier–Stokes equation, respectively; and the overall permeability of fractures is computed using the fluid pressure drop throughout the fracture and the fluid flowing velocity in the fracture's outlet. Since most CBM flows along high‐speed channels between the proppant clusters, the simulated findings show that the overall permeability of fractures with an uneven distribution of proppant clusters is significantly higher than that of the proppant cluster itself. As CBM becomes more discretely distributed, the proportion of CBM flowing within the proppant cluster continuously drops. As the permeability of the proppant cluster increases, the volume ratio of proppant clusters decreases, and the distribution of proppant clusters becomes more discrete, the overall permeability of the fracture continuously increases.

Selecting the number of factors in approximate factor models using group variable regularization

. We propose a novel method for the estimation of the number of factors in approximate factor models. The model is based on a penalized maximum likelihood approach incorporating an adaptive hierarchical lasso penalty function that enables setting entire columns of the factor loadings matrix to zero, which corresponds to removing uninformative factors. Additionally, the model is capable of estimating weak factors by allowing for sparsity in the non zero columns. We prove that the proposed estimator consistently estimates the true number of factors. Simulation experiments reveal superior selection accuracies for our method in finite samples over existing approaches, especially for data with cross-sectional and serial correlations. In an empirical application on a large macroeconomic dataset, we show that the mean squared forecast errors of an approximate factor model are lower if the number of included factors is selected by our method.

Optimal Allocation of Hybrid Energy Storage Capacity Based on ISSA-Optimized VMD Parameters

To address the issue where the grid integration of renewable energy field stations may exacerbate the power fluctuation in tie-line agreements and jeopardize safe grid operation, we propose a hybrid energy storage system (HESS) capacity allocation optimization method based on variational mode decomposition (VMD) and a multi-strategy improved salp swarm algorithm (ISSA). From typical wind load power and contact line agreement power, the HESS power is obtained. VMD decomposes this power into high- and low-frequency power, respectively, for the super capacitor and the Li-ion battery. Considering charging and discharging power and state of charge (SOC) constraints, an optimization model minimizing the system equivalent annual value cost is established. ISSA optimizes the best decomposition layer K and penalty coefficients α in VMD. The optimal cut-off point and corresponding energy storage allocation scheme are analyzed. A simulation and analysis on MATLAB show that the proposed ISSA-VMD HESS capacity allocation scheme saves 7.53% in costs compared to an empirical mode decomposition (EMD) scheme, proving the method’s effectiveness and superiority.

Effect of constraint relaxation on dynamic critical phenomena in minimum vertex cover problem

The effects of constraint relaxation on dynamic critical phenomena in the Minimum Vertex Cover (MVC) problem on Erdős-Rényi random graphs are investigated using Markov chain Monte Carlo simulations. Following our previous work that revealed the reduction of the critical temperature by constraint relaxation based on the penalty function method, this study focuses on investigating the critical properties of the relaxation time along its phase boundary. It is found that the dynamical correlation function of MVC with respect to the problem size and the constraint strength follows a universal scaling function. The analysis shows that the relaxation time decreases as the constraints are relaxed. This decrease is more pronounced for the critical amplitude than for the critical exponent, and this result is interpreted in terms of the system's microscopic energy barriers due to the constraint relaxation.