Dynamic Penalty Function Research Articles

Collaborative optimization of a composite pressure cylindrical shell based on dynamic penalty functions

This article reports the design and optimization of pressure cylindrical shells comprising carbon/epoxy and glass/epoxy composite materials. It presents a method for analyzing the pressure cylindrical shells of composite materials, analyzes the strength and buckling of composite pressure cylindrical shells with different rib numbers, and investigates the influence of thickness and ply angles ([±θ]10 and [903(±θ)2903]) on the critical buckling and strength-failure stresses. An approximate model of composite pressure cylindrical shells and collaborative optimization based on dynamic penalty functions are combined into a multidisciplinary optimization framework of composite pressure cylindrical shells. The optimization aims to maximize the critical buckling stress while minimizing the buoyancy coefficient. At a buoyancy coefficient similar to that of the initial scheme, the critical buckling stresses of the carbon/epoxy and glass/epoxy cylindrical shells increased by 21.85% and 9.5%, respectively, verifying the effectiveness of the collaborative optimization framework based on dynamic penalty functions.

Circulatory system-based optimization algorithm with dynamic penalty function for optimum design of large-scale water distribution networks

Water distribution networks (WDNs) are a crucial component of urban infrastructure, and their optimization plays a vital role in minimizing construction costs. However, existing heuristic algorithms for WDN optimization often rely on specific parameters, limiting their performance and hindering their applicability to diverse network configurations, particularly in large-scale systems. To address this challenge, a novel method called circulatory system-based optimization (CSBO) is proposed, which minimizes the need for algorithm-specific parameters and achieves optimal designs across WDNs of various scale. The impact of a dynamic penalty function tailored to pressure constraints on the optimal design is also investigated. This study encompasses the optimization of the Goyang and combined gravity networks and the large-scale Balerma irrigation and Kang and Lansey networks under identical conditions. The results of the standard and dynamic CSBO approaches prove that the proposed algorithms have a superior performance, particularly in convergence rate and stability, compared to previous methods.

Resilient inequality constrained GNSS kinematic precise point positioning considering the terrain topography

In complex environments, signals are inevitably subject to phenomena such as reflection, refraction, diffraction, and obstruction, which result in significant unmodeled errors like colored noise, residual systematic errors, and other special outliers. Consequently, the achievement of high-precise and high-reliable global navigation satellite systems (GNSS) precise point positioning (PPP) is not a readily solvable problem in complex environments, especially in varying terrain topography. This paper proposed the resilient inequality constrained GNSS kinematic PPP method considering the terrain topography to improve the abnormal positioning results caused by unmodeled errors. Specifically, the proposed method is composed of the adaptive inequality constraint with dynamic penalty function and the timing-varying inequality considering the terrain topography. Two representative experiments including one set of designed data and three sets of daily measured data were conducted. The results show that the proposed method can improve the positioning results resulting from the unmodeled errors while preserving the trend of the original data. Typically, the proposed method decreases the standard deviations by 2.47, 0.25, and 1.46 cm in the U direction of the three real datasets, respectively. Consequently, the proposed method exhibits prospects in precision and reliability for complex environments.

Mathematical Models for the Design of GRID Systems to Solve Resource-Intensive Problems

Artificial neural networks are successfully used to solve a wide variety of scientific and technical problems. The purpose of the study is to increase the efficiency of distributed solutions for problems involving structural-parametric synthesis of neural network models of complex systems based on GRID (geographically disperse computing resources) technology through the integrated application of the apparatus of evolutionary optimization and queuing theory. During the course of the research, the following was obtained: (i) New mathematical models for assessing the performance and reliability of GRID systems; (ii) A new multi-criteria optimization model for designing GRID systems to solve high-resource computing problems; and (iii) A new decision support system for the design of GRID systems using a multi-criteria genetic algorithm. Fonseca and Fleming’s genetic algorithm with a dynamic penalty function was used as a method for solving the stated multi-constrained optimization problem. The developed program system was used to solve the problem of choosing an effective structure of a centralized GRID system that was configured to solve the problem of structural-parametric synthesis of neural network models. To test the proposed approach, a Pareto-optimal configuration of the GRID system was built with the following characteristics: average performance–103.483 GFLOPS, cost–500 rubles per day, availability rate–99.92%, and minimum performance–51 GFLOPS.

A constrained multi-objective deep reinforcement learning approach for temperature field optimization of zinc oxide rotary volatile kiln

In the zinc oxide rotary volatile kiln (ZORVK), an optimal temperature field is essential to balance the strong conflict between zinc recovery rate and carbon emissions. However, the complex and diverse temperature distribution modes make it challenging to quickly obtain optimization results under intricate controllability constraints and multi-conflict production objectives. In this study, a novel constrained multi-objective deep reinforcement learning (CMODRL) approach for temperature field optimization of the ZORVK is proposed. First, an evaluation metric called the uncontrollable factor is designed to quantify the controllability of the temperature field. Then, a dynamic penalty method in deep reinforcement learning (DRL) is proposed to handle the controllability constraint, in which the penalty coefficient is dynamically adjusted according to the training loss. After that, the Chebyshev scalarization function is introduced as an action selection mechanism in DRL. Finally, the CMODRL is developed by integrating the dynamic penalty and Chebyshev scalarization function into the multi-objective deep reinforcement learning (MODRL) framework. As a result, for any given preference between the two production objectives, the proposed method can rapidly get the Pareto-optimal solution fulfilling the constraint. Moreover, the optimization efficiency of the MODRL-based algorithm is forty times higher than that of the multi-objective genetic algorithm, which serves better for practical optimization problems.

Physics-guided mixture density networks for uncertainty quantification

This paper proposes a Physics-guided Mixture Density Network (PgMDN) model for uncertainty quantification of regression-type analysis. It integrates a Mixture Density Network for probabilistic modeling and physics knowledge as regularizations. This model can handle arbitrary distribution of data (e.g., strongly non-Gaussian, multi-mode, and truncated distributions). The physics knowledge from parameters and their partial derivatives is used as equality/ inequality constraints. The training of physics-guided machine learning is formulated as a constrained optimization problem. The constrained optimization problem is transformed to an unconstrained one using a dynamic penalty function algorithm. Thus, the commonly used backpropagation algorithm can be used to train the neural network. With the physics constraints, the required training data size can be reduced, and the overfitting problem can be mitigated. This paper demonstrates the application of the PgMDN using a numerical example, an engineering problem for fatigue stress-life curve estimation, an engineering problem for natural frequency prediction of bridges, and an engineering problem for fatigue life prediction of corroded steel reinforcing bars. Some discussions are given to illustrate the effectiveness of incorporating the physics knowledge when data are sparse, the improvement of the dynamic penalty function method compared with the static method, and the benefits achieved from the distribution mixture compared with a single Gaussian distribution.

A Dynamic Penalty Function Approach for Constraint-Handling in Reinforcement Learning

Abstract Reinforcement learning (RL) is attracting attention as an effective way to solve sequential optimization problems that involve high dimensional state/action space and stochastic uncertainties. Many such problems involve constraints expressed by inequality constraints. This study focuses on using RL to solve constrained optimal control problems. Most RL application studies have dealt with inequality constraints by adding soft penalty terms for violating the constraints to the reward function. However, while training neural networks to learn the value (or Q) function, one can run into computational issues caused by the sharp change in the function value at the constraint boundary due to the large penalty imposed. This difficulty during training can lead to convergence problems and ultimately lead to poor closed-loop performance. To address this issue, this study proposes a dynamic penalty (DP) approach where the penalty factor is gradually and systematically increased during training as the iteration episodes proceed. We first examine the ability of a neural network to represent a value function when uniform, linear, or DP functions are added to prevent constraint violation. The agent trained by a Deep Q Network (DQN) algorithm with the DP function approach was compared with agents with other constant penalty functions in a simple vehicle control problem. Results show that the proposed approach can improve the neural network approximation accuracy and provide faster convergence when close to a solution.

Comparing ordinary ridge and generalized ridge regression results obtained using genetic algorithms for ridge parameter selection

Ridge regression is an alternative to the ordinary least squares method when multicollinearity presents among the regressor variables in multiple linear regression analysis. The selection of the ridge parameter is an important issue to obtain a good performance of the ridge regression. In this article, a new method is proposed for determining the ridge parameter in ridge regression. This method is based on minimizing the statistic measures, which are mean squared error (MSE), mean absolute error (MAE), mean absolute prediction error (MAPE), by using genetic algorithms with the dynamic penalty function and also managing the values of Variance Inflation Factors to be less than or equal to 10. The ordinary ridge regression (ORR) and generalized ridge regression (GRR) are compared by performing a simulation study with many scenarios and a numerical example. Findings show that the GRR provides better (minimum MSE, MAE, and MAPE) results than the ORR.

Improved Cuckoo Search algorithmic variants for constrained nonlinear optimization

Although Cuckoo Search (CS) is a quite new nature-inspired metaheuristic optimization algorithm, it has been extensively used in engineering applications, since it has been proven very efficient in solving complex nonlinear problems. In this paper, efficient modifications have been made to the original CS algorithm to enhance its efficiency and robustness. More specifically, constant parameters of the algorithm, such as the probability of the alien egg being discovered by the host bird and the step size of Levy flights have been dynamically tuned. In addition, static and dynamic penalty functions are introduced within the optimization formulation. Finally, a hybrid optimization approach is developed to combine the advantages of CS with those of Bird Swarm Algorithm (BSA). Benchmark problems, widely used in relevant studies, have been solved and the obtained solutions are compared with those previously reported using the standard CS algorithm and other popular evolutionary optimization techniques (i.e., Genetic Algorithms, Particle Swarm Optimization, etc.).

Truss optimization using eigenvectors of the covariance matrix

In this paper, by combining eigenvectors of the covariance matrix and random normal distribution, a new method has been presented to solve optimization problems. This method has been inspired by CMA-ES method and has been named eigenvectors of covariance matrix (ECM). ECM generates some solutions in each stage; then it assigns a value to all solutions by utilizing a dynamic penalty function. The best solution in each stage is selected as the answer of optimization problem and ECM tries to push towards a better point. According to the penalty function, the solutions with no violation along with the solutions with little violation are considered as good solutions and named desirable data. By utilizing desirable data, a square matrix is formed and its covariance matrix can be calculated. The eigenvectors of the covariance matrix are orthogonal and they show directions of distribution of desirable points that have been chosen before. We can hopefully get a better answer by moving from the best current answer towards the points which are distributed normally and randomly around the weighted combination of eigenvectors. To ensure the validity and accuracy of ECM method, weights of six truss structures have been optimized through this method and the results have been compared with previous studies.

Operation Cost Optimization on an Ultralow Emission System Based on Improved Collaborative Optimization

Due to the lack of an effective overall coordinated treatment method, it is difficult to achieve low cost and efficient removal of pollutants from coal-fired flue gas. This paper establishes a collaborative optimization model for ultralow emission systems, including a system level model of operation cost and three discipline level models for denitration, desulfurization, and dust removal. An improved collaborative optimization method with a dynamic penalty function is proposed to optimize an ultralow emission system. Simulation results show that the improved method achieves better global optimization and effectively reduces the operation cost of the system under ultralow emission constraints.

Path-Based Dynamic User Equilibrium Model with Applications to Strategic Transportation Planning

This study proposes an analytical capacity constrained dynamic traffic assignment (DTA) model along with an efficient path-based algorithm. The model can be applied to analyzing dynamic traffic demand management (TDM) strategies, but its specific feature is allowing for an evaluation of advanced traveler information systems (ATIS) within the strategic transportation planning framework. It is an extension of a former DTA model, where each link is given an infinite capacity and is assumed to be completely traversed inside any time interval it is reach by a path. Thereby, the length of time intervals should be very longer than the link travel times, and the link capacity constraints are overlooked. The paper rests on three key ideas to overcome these restrictions: (1) adding path-link fraction variables to the base model, allowing path flows to spread out over time intervals on long links; (2) uniformly dividing each link into smaller parts (segments), so that each part is more likely to be traversed inside a time interval; (3) imposing a dynamic penalty function on each link, thereby the capacity constraint can be included. The proposed DTA algorithm decomposes the augmented model in terms of origin-destination (OD) pairs and departure time intervals, and utilizes a dynamic column generation technique for generating active paths between the OD pairs. The optimal solution to a one-link network demonstrates that the model is able to approximate the dynamic flow propagation over a link with sensible accuracy. Besides, investigation of the results for a small test network reveals that the algorithm performs very well in computing temporal link flows and queuing delays. Finally, numerical experiments on a real life network indicate that the algorithm converges sufficiently fast and provides path information for each time interval. The network is further used to show the algorithm is capable of assessing a dynamic TDM strategy as well as an ATIS system.

OPTIMIZATION METHOD BASED ON THE SYNTHESIS OF CLONAL SELECTION AND ANNEALING SIMULATION ALGORITHMS

Context. The problem of increasing the efficiency of optimization methods by synthesizing metaheuristics is considered. The object of the research is the process of finding a solution to optimization problems. Objective. The goal of the work is to increase the efficiency of searching for a quasi-optimal solution at the expense of a metaheuristic method based on the synthesis of clonal selection and annealing simulation algorithms. Method. The proposed optimization method improves the clonal selection algorithm by dynamically changing based on the annealing simulation algorithm of the mutation step, the mutation probability, the number of potential solutions to be replaced. This reduces the risk of hitting the local optimum through extensive exploration of the search space at the initial iterations and guarantees convergence due to the focus of the search at the final iterations. The proposed optimization method makes it possible to find a conditional minimum through a dynamic penalty function, the value of which increases with increasing iteration number. The proposed optimization method admits non-binary potential solutions in the mutation operator by using the standard normal distribution instead of the uniform distribution. Results. The proposed optimization method was programmatically implemented using the CUDA parallel processing technology and studied for the problem of finding the conditional minimum of a function, the optimal separation problem of a discrete set, the traveling salesman problem, the backpack problem on their corresponding problem-oriented databases. The results obtained allowed to investigate the dependence of the parameter values on the probability of mutation. Conclusions. The conducted experiments have confirmed the performance of the proposed method and allow us to recommend it for use in practice in solving optimization problems. Prospects for further research are to create intelligent parallel and distributed computer systems for general and special purposes, which use the proposed method for problems of numerical and combinatorial optimization, machine learning and pattern recognition, forecast.

Evaluation of Transportation Network Reliability under Emergency Based on Reserve Capacity

There are differences between the requirements for traffic network for traffic demand in daily and emergency situations. In order to evaluate how the network designed for daily needs can meet the surging demand for emergency evacuation, the concept of emergency reliability and corresponding evaluation method is proposed. This paper constructs a bilevel programming model to describe the proposed problem. The upper level problem takes the maximum reserve capacity multiplier as the optimization objective and considers the influence of reversible lane measures taken under emergency conditions. The lower level model adopts the combined traffic distribution/assignment model with capacity limits, to describe evacuees’ path and shelter choice behavior under emergency conditions and take into account the traits of crowded traffic. An iterative optimization method is proposed to solve the upper level model, and the lower level model is transformed into a UE assignment problem with capacity limits over a network of multiple origins and single destination, by adding a dummy node and several dummy links in the network. Then a dynamic penalty function algorithm is used to solve the problem. In the end, numerical studies and results are provided to demonstrate the rationality of the proposed model and feasibility of the proposed solution algorithms.

Modified PSO algorithm for real-time energy management in grid-connected microgrids

In real-time energy management of a converter-based microgrid, it is difficult to determine optimal operating points of a storage system in order to save costs and minimise energy waste. The complexity arises due to time-varying electricity prices, stochastic energy sources and power demand. Many countries have imposed real-time electricity pricing to efficiently control demand side management. This paper presents a particle swarm optimisation (PSO) for the application of real-time energy management to find optimal battery controls of a community microgrid. The modification of the PSO consists in altering the cost function to better model the battery charging/discharging operations. As optimal control is performed by formulating a cost function, it is suitably analysed and then a dynamic penalty function is proposed in order to obtain the best cost function. Several case studies with different scenarios are conducted to determine the effectiveness of the proposed cost function. The proposed cost function can reduce operational cost by 12% as compared to the original cost function over a time horizon of 96 h. Simulation results reveal the suitability of applying the regularised PSO algorithm with the proposed cost function, which can be adjusted according to the need of the community, for real-time energy management.

Real-time Battery Energy Management for Residential Solar Power System

Abstract An energy storage system is a key element of renewable-based power generation. Its flexible operational capabilities reduce not only the impact of intermittent power generation but also operational costs. In this paper, a dynamic penalty function is proposed to the charging term of the cost function to efficiently manage the battery energy and thereby reducing operational costs. The charging/discharging periods of the battery are effectively controlled based on the solar power generation and residential real-time electricity prices (RRTP). The optimisation problem formulated for the application of real-time energy management is solved with the help of particle swarm optimisation (PSO). It is shown that the proposed cost function can reduce operational costs over a time horizon of 96 hours by 4.2 per cent as compared to the cost function reported in the literature. Simulation studies are carried out to demonstrate the effectiveness of the proposed cost function over the existing cost function.

An Artificial Bee Colony Algorithm Based on Dynamic Penalty and Lévy Flight for Constrained Optimization Problems

Artificial bee colony (ABC) algorithm is one of the most popular intelligence algorithms, which has been widely applied to some unconstrained optimization problems. Many improved versions of ABC algorithm are also used for solving constrained optimization problems (COPs). An artificial bee colony algorithm based on dynamic penalty function and Levy flight (DPLABC) is presented for COPs in this paper. Based on the original ABC algorithm, four modifications are put forward in this newly proposed algorithm: The dynamic penalty method is used to handle the constraints; Levy flight with logistic map is applied in the employed bee phase; according to the selection probability, a further search mechanism which is learned from the best solution and two other neighbor food sources is adopted for onlooker bees; different from pulling back to the upper and lower limits, the new boundary handling mechanism inspired by the best solution is also given to repair the invalid solutions. To validate the performance of DPLABC algorithm, it is tested on 13 constrained benchmark functions from 2006 IEEE Congress on Evolution Computation (CEC2006) and four engineering design problems. The experimental results indicate that DPLABC is competitive with the state-of-the-art algorithms including dynamic difference search algorithm and several improved variants of ABC for solving COPs.

Optimum stacking sequence design of laminates using a hybrid PSO-SA method

Stacking sequence optimization of laminated composite materials subjected to in-plane forces and bending moments for minimum weight is addressed using a hybrid method based on the particle swarm optimization (PSO) and simulated annealing (SA) methods. The proposed hybrid method uses SA as a local search mechanism to improve convergence behavior of PSO. Layer thickness and fiber orientation angle of each ply group were defined as discrete variables and the laminate effective engineering constants and failure stresses were specified as constraints. Moreover, satisfaction of some composite design rules for reducing coupling effects and improving stiffness and strength of the laminate have been taken into consideration. A combination of a specific feasible-guiding strategy and dynamic penalty function was integrated to the solution procedure for handling constraints. Numerical examples are included to demonstrate the performance and effectiveness of the proposed solution procedure. The results of this research show that the proposed PSO-SA hybrid algorithm can reliably and effectively be used for composite weight optimization problems subjected to a variety of constraints.

Speeding Up Topology Optimization of Compliant Mechanisms With a Pseudorigid-Body Model

This paper seeks to speed up the topology optimization using a pseudorigid-body (PRB) model, which allows the kinetostatic equations to be explicitly represented in the form of nonlinear algebraic equations. PRB models can not only accommodate large deformations but more importantly reduce the number of variables compared to beam theory or finite element methods. A symmetric 3R model is developed and used to represent the beams in a compliant mechanism. The design space is divided into rectangular segments, while kinematic and static equations are derived using kinematic loops. The use of the gradient and hessian of the system equations leads to a faster solution process. Integer variables are used for developing the adjacency matrix, which is optimized by a genetic algorithm. Dynamic penalty functions describe the general and case-specific constraints. The effectiveness of the approach is demonstrated with the examples of a displacement inverter and a crimping mechanism. The approach outlined here is also capable of estimating the stress in the mechanism which was validated by comparing against finite element analysis. Future implementations of this method will incorporate other pseudorigid-body models for various types of compliant elements and also try to develop multimaterial designs.

Constrained cohort intelligence using static and dynamic penalty function approach for mechanical components design

Most of the metaheuristics can efficiently solve unconstrained problems; however, their performance may degenerate if the constraints are involved. This paper proposes two constraint handling approaches for an emerging metaheuristic of Cohort Intelligence (CI). More specifically CI with static penalty function approach (SCI) and CI with dynamic penalty function approach (DCI) are proposed. The approaches have been tested by solving several constrained test problems. The performance of the SCI and DCI have been compared with algorithms like GA, PSO, ABC, d-Ds. In addition, as well as three real world problems from mechanical engineering domain with improved solutions. The results were satisfactory and validated the applicability of CI methodology for solving real world problems.