Dynamic Optimization Problems Research Articles

Dynamic optimization of nonlinear differential game problems using orthogonal collocation with analytical sensitivities

AbstractThis article presents an effective computational method based on the orthogonal collocation on finite element for nonlinear pursuit‐evasion differential game problems. The original problems are transformed into two dynamic optimization problems at first, so that the difficulty of obtaining the solution is reduced. To improve the convergence rate and the efficiency, the sensitivities describing the influence of control and interval parameters on state are derived through the discretized dynamic equations for the resulting nonlinear programming problem. The convergence speed is introduced to measure the performance in the upper level iteration. The main structure and the algorithm of the method are also given. Two demonstrative differential game problems with different scenarios from practice are studied. Compared with the approach without sensitivity information, the proposed method needs less function evaluations and saves at least 68.4% of the computational time. The research results show the effectiveness of proposed approach.

Dynamic controlled variables based dynamic self-optimizing control

Self-optimizing control is a strategy for selecting controlled variables, where the economic objective guides the selection and design of controlled variables, with the expectation that maintaining the controlled variables at constant values can achieve optimization effects, translating the process optimization problem into a process control problem. Currently, self-optimizing control is widely applied to steady-state optimization problems. However, the development of process systems exhibits a trend towards refinement, highlighting the importance of optimizing dynamic processes such as batch processes and grade transitions. This paper formally introduces the self-optimizing control problem for dynamic optimization, termed the dynamic self-optimizing control problem, extending the original definition of self-optimizing control. A novel concept, ”dynamic controlled variables” (DCVs), is proposed, and an implicit control policy is presented based on this concept. The paper theoretically analyzes the advantages and generality of DCVs compared to explicit control strategies and elucidates the relationship between DCVs and traditional controllers. Moreover, this paper puts forth a data-driven approach to designing self-optimizing DCVs, which considers DCV design as a mapping identification problem and employs deep neural networks to parameterize the variables. Three case studies validate the efficacy and superiority of DCVs in approximating multi-valued and discontinuous functions, as well as their application to dynamic optimization problems with non-fixed horizons, which traditional self-optimizing control methods are unable to address.

Regression based stochastic energy management strategy for a grid connected microgrid using Artificial Electric Field Algorithm

Recent decade has witnessed steep technological advancement in the renewable energy sector due to growing concern of climate change and emission cut target. The lack of availability of fossil fuels and the high price of the same have made the use of Renewable Energy Sources (RES) as a promising approach for generating clean and sustainable energy at local level. This persuades the implementation of Microgrid (MG) supported with RES, to cater various types of load demand. However, planning and optimizing a MG operation have become challenging due to the intermittent nature of RES and uncertain loads. To address this issue, this paper introduces a Stochastic Energy Management Strategy (SEMS) for a grid connected MG comprising Photovoltaic, Wind Turbine, Fuel Cell, Micro Turbine, Battery Energy Storage and electrical as well as heat energy demand. The volatilities of RES power output and energy demand are modeled via Gaussian Process Regression (GPR) and Monte Carlo Simulation (MCS) respectively. The problem is framed as non-linear (NL) dynamic stochastic optimization problem and solved using ‘Artificial Electric Field Algorithm (AEFA)’. Here, minimization of expected value of operational and emission cost are taken as objective functions; while various practicality aspects are modeled as constraints in the problem. Further, the efficacy of the proposed approach is validated around the first central moment of the load. Various case studies are performed to estimate the day ahead optimum operating schedule of the Distributed Energy Resources (DER) and to assess the impact of load uncertainties on DER operation and total MG operation cost. Further, the efficacy of the GPR is verified through comparison with MCS based result.

Online dynamic multi-user computation offloading and resource allocation for HAP-assisted MEC: an energy efficient approach

Nowadays, the paradigm of mobile computing is evolving from a centralized cloud model towards Mobile Edge Computing (MEC). In regions without ground communication infrastructure, incorporating aerial edge computing nodes into network emerges as an efficient approach to deliver Artificial Intelligence (AI) services to Ground Devices (GDs). The computation offloading and resource allocation problem within a HAP-assisted MEC system is investigated in this paper. Our goal is to minimize the energy consumption. Considering the randomness and dynamism of the task arrival of GDs and the quality of wireless communication, stochastic optimization techniques are utilized to transform the long-term dynamic optimization problem into a deterministic optimization problem. Subsequently, the problem is further decomposed into three sub-problems which can be solved in parallel. An online Energy Efficient Dynamic Offloading (EEDO) algorithm is proposed to address these problems. Then, we conduct the theoretical performance analysis for EEDO. Finally, we carry out parameter analysis and comparative experiments, demonstrating that the EEDO algorithm can effectively reduce system energy consumption while maintaining the stability of the system.

Multiple landscape measure-based approach for dynamic optimization problems

Many practical decision-making problems involve dynamic scenarios, where the decision variables, conditions and/or parameters of their optimization models change over time. Such problems are known as dynamic optimization problems (DOPs). Although evolutionary algorithms (EAs) have been effective in solving static optimization problems, they face challenges in handling DOPs. To improve the performance of EAs in dealing with DOPs, this paper proposes a new evolutionary framework that uses different landscape measures to analyze problem landscapes and utilizes the information gained from this to improve the searching process. The proposed method is adopted in three EAs to deal with dynamic functions from IEEE-CEC2009 and two real-world problems. According to the experimental results, LIDOA with multi-measure methods improves the performance of GA, jDE and CMA-ES, on average by 6.71%, 3.03% and 7.78% on benchmark problems, respectively.

Dynamic optimization on quantum hardware: Feasibility for a process industry use case

The quest for real-time dynamic optimization solutions in the process industry represents a formidable computational challenge, particularly within the realm of applications like model-predictive control, where rapid and reliable computations are critical. Conventional methods can struggle to surmount the complexities of such tasks. Quantum computing and quantum annealing emerge as avant-garde contenders to transcend conventional computational constraints. We convert a dynamic optimization problem, characterized by an optimization problem with a system of differential–algebraic equations embedded, into a Quadratic Unconstrained Binary Optimization problem, enabling quantum computational approaches. The empirical findings synthesized from classical methods, simulated annealing, quantum annealing via D-Wave’s quantum annealer, and hybrid solver methodologies, illuminate the intricate landscape of computational prowess essential for tackling complex and high-dimensional dynamic optimization problems. Our findings suggest that while quantum annealing is a maturing technology that currently does not outperform state-of-the-art classical solvers, continuous improvements could eventually aid in increasing efficiency within the chemical process industry.

Mix-Zones as an Effective Privacy Enhancing Technique in Mobile and Vehicular Ad-hoc Networks

Intelligent Transportation Systems (ITS) promise significant increases in throughput and reductions in trip delay. ITS makes extensive use of Connected and Autonomous Vehicles (CAV) frequently broadcasting location, speed, and intention information. However, with such extensive communication comes the risk to privacy. Preserving privacy while still exchanging vehicle state information has been recognized as an important problem. Mix zones have emerged as a potentially effective way of protecting user privacy in ITS. CAVs are assigned pseudonyms to mask their identity; a mix zone is an area where CAVs can change their pseudonyms to resist being tracked. In order to be effective, mix zone placement must take account of traffic flows. Also, since a mix zone can degrade throughput, mix zones must be used sparingly. Determining the number and placement of mix zones is a difficult dynamic optimization problem. This paper outlines the various approaches recently taken by researchers to deal with this problem.

Scalable benchmarks and performance measures for dynamic multi-objective optimization

Dynamic multi-objective optimization problems (DMOPs) can be utilized to model certain real-world problems that have a dynamic nature. Algorithms for solving DMOPs can be evaluated and improved by comparing their performance on different benchmarks. However, some existing benchmarks for DMOPs have the limitation of non-uniform weights for decision variables. Additionally, dynamic many-objective optimization problems (DMaOPs) involve more than three objectives, but only a few existing benchmarks can be extended to accommodate DMaOPs. Furthermore, some existing performance measures for DMOPs may not effectively compare the relative performance differences between multiple algorithms or evaluate the search uniformity among different objectives. In this paper, we propose improvements to an existing benchmark for DMOPs by expanding the impact range of decision variables. Moreover, a benchmark framework that can be extended to accommodate DMaOPs is proposed, thus addressing a research gap between the optimization of DMOPs and DMaOPs. Additionally, a set of performance measures for DMOPs are proposed, which can evaluate the relative performance and search uniformity of multi-objective optimization algorithms. By comparing the performance of state-of-the-art and commonly used algorithms on test problems, we can gain a better understanding of the characteristics and strengths and weaknesses of the algorithms and test problems.

Optimal Rotation Age in Fast Growing Plantations: A Dynamical Optimization Problem.

Forest plantations are economically and environmentally relevant, as they play a key role in timber production and carbon capture. It is expected that the future climate change scenario affects forest growth and modify the rotation age for timber production. However, mathematical models on the effect of climate change on the rotation age for timber production remain still limited. We aim to determine the optimal rotation age that maximizes the net economic benefit of timber volume in a negative scenario from the climatic point of view. For this purpose, a bioeconomic optimal control problem was formulated from a system of Ordinary Differential Equations (ODEs) governed by the state variables live biomass volume, intrinsic growth rate, and area affected by fire. Then, four control variables were associated to the system, representing forest management activities, which are felling, thinning, reforestation, and fire prevention. The existence of optimal control solutions was demonstrated, and the solutions of the optimal control problem were also characterized using Pontryagin's Maximum Principle. The solutions of the model were approximated numerically by the Forward-Backward Sweep method. To validate the model, two scenarios were considered: a realistic scenario that represents current forestry activities for the exotic species Pinus radiata D. Don, and a pessimistic scenario, which considers environmental conditions conducive to a higher occurrence of forest fires. The optimal solution that maximizes the net benefit of timber volume consists of a strategy that considers all four control variables simultaneously. For felling and thinning, regardless of the scenario considered, the optimal strategy is to spend on both activities depending on the amount of biomass in the field. Similarly, for reforestation, the optimal strategy is to spend as the forest is harvested. In the case of fire prevention, in the realistic scenario, the optimal strategy consists of reducing the expenses in fire prevention because the incidence of fires is lower, whereas in the pessimistic scenario, the opposite is true. It is concluded that the optimal rotation age that maximizes the net economic benefit of timber volume in P. radiata plantations is 24 and 19 years for the realistic and pessimistic scenarios, respectively. This corroborates that the presence of fires influences the determination of the optimal rotation age, and as a consequence, the net economic benefit.

Performance Analysis of TBR Tire Curing Process

Numerical algorithms and computer programs have been developed to determine optimal cure steps in a tire curing process. A dynamic constrained optimization problem was formulated with the following ingredients: (1) an objective function that measures product quality in terms of final state of cure and temperature history at selected points in a tire; (2) constraints that consist of a process model and temperature limits imposed on cure media; (3) B-spine representation of a time-varying profile of cure media temperature. The optimization problem was solved using the complex algorithm along with a finite element model solver. Numerical simulations were carried out to demonstrate the procedure of determining optimal cure steps for a truck/bus radial tire. © 1999 John Wiley & Sons, Inc. J Apple Polymer Sic 74: 2063–2071, 1999.

Efficient Dynamic Distributed Resource Slicing in 6G Multi-Access Edge Computing Networks With Online ADMM and Message Passing Graph Neural Networks

We consider the problem of resource slicing in the 6 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">th</sup> generation multi-access edge computing (6G-MEC) network. The network includes many non-stationary space-air-ground-sea nodes with dynamic, unstable connections and resources, where any node can be in one of two hidden states: i) reliable – when the node generates/propagates no data errors; ii) unreliable – when the node can generate/propagate random errors. We show that solving this problem is challenging, since it represents a non-deterministic polynomial-time (NP) hard dynamic combinatorial optimization problem depending on the unknown distribution of hidden nodes' states and time-varying parameters (connections and resources of nodes) which can only be observed locally. To tackle these challenges, we develop a new deep learning (DL) model based on the message passing graph neural network (MPNN) to estimate hidden nodes' states in dynamic network environments. We then propose a novel algorithm based on the integration of MPNN-based DL and online alternating direction method of multipliers (ADMM) – extension of the well-known classical “static” ADMM to dynamic settings, where the slicing problem is solved distributedly, in real time, based on local information. We prove that our algorithm converges to a global optimum of our problem with a superior performance even in the highly-dynamic, unreliable scenarios.

Stochastic Economic Lot Scheduling via Self-Attention Based Deep Reinforcement Learning

The Stochastic Economic Lot Scheduling Problem (SELSP) is a difficult dynamic optimization problem with wide industrial applications. Traditional methods such as hyper-heuristics are manually designed based on substantial expert knowledge, which may limit their optimization performance. Recently, Deep Reinforcement Learning (DRL) is shown to be promising in automatically learning scheduling policies for SELSP. However, its performance is still quite far from that of hyper-heuristics, due to the lack of suitable deep models. In this paper, we propose a novel DRL method to learn dynamic scheduling policies for SELSP in an end-to-end fashion. Based on self-attention, our method can effectively extract useful features from raw state information, and is flexible in handling different numbers of products, which is not viable for previous methods. Experiments on a complex biopharmaceutical manufacturing process show that our method outperforms a recent DRL method and state-of-the-art hyper-heuristics. Moreover, the trained policy performs better in environments different from training with demand forecast errors and varying number of products, showing its strong robustness and generalization ability. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —The Stochastic Economic Lot Scheduling Problem (SELSP) is an important problem for manufacturing enterprises, which is to optimally balance the production and inventory so as to minimize the total cost. However, SELSP is very challenging to solve due to the involvement of uncertain factors such as customer demands and machine failures. Traditional methods for solving SELSP, such as heuristic policies and hyper-heuristics, heavily rely on human experiences to design and hence the performance could be limited. This paper proposes a Deep Reinforcement Learning (DRL) based method to automatically learn scheduling policy for solving SELSP, which could alleviate the above limitation through a self-attention based feature extraction mechanism and reward based training. Experimental results on a realistic manufacturing process show that our method can deliver higher revenue than conventional manual policy and an existing DRL based method.

A Novel Skip Orthogonal List for Dynamic Optimal Transport Problem

Optimal transport is a fundamental topic that has attracted a great amount of attention from the optimization community in the past decades. In this paper, we consider an interesting discrete dynamic optimal transport problem: can we efficiently update the optimal transport plan when the weights or the locations of the data points change? This problem is naturally motivated by several applications in machine learning. For example, we often need to compute the optimal transport cost between two different data sets; if some changes happen to a few data points, should we re-compute the high complexity cost function or update the cost by some efficient dynamic data structure? We are aware that several dynamic maximum flow algorithms have been proposed before, however, the research on dynamic minimum cost flow problem is still quite limited, to the best of our knowledge. We propose a novel 2D Skip Orthogonal List together with some dynamic tree techniques. Although our algorithm is based on the conventional simplex method, it can efficiently find the variable to pivot within expected O(1) time, and complete each pivoting operation within expected O(|V|) time where V is the set of all supply and demand nodes. Since dynamic modifications typically do not introduce significant changes, our algorithm requires only a few simplex iterations in practice. So our algorithm is more efficient than re-computing the optimal transport cost that needs at least one traversal over all |E|=O(|V|^2) variables, where |E| denotes the number of edges in the network. Our experiments demonstrate that our algorithm significantly outperforms existing algorithms in the dynamic scenarios.

Research on Dynamic Economic Dispatch Optimization Problem Based on Improved Grey Wolf Algorithm

The dynamic economic dispatch (DED) problem is a typical complex constrained optimization problem with non-smooth, nonlinear, and nonconvex characteristics, especially considering practical situations such as valve point effects and transmission losses, and its objective is to minimize the total fuel costs and total carbon emissions of generating units during the dispatch cycle while satisfying a series of equality and inequality constraints. For the challenging DED problem, a model of a dynamic economic dispatch problem considering fuel costs is first established, and then an improved grey wolf optimization algorithm (IGWO) is proposed, in which the exploitation and exploration capability of the original grey wolf optimization algorithm (GWO) is enhanced by initializing the population with a chaotic algorithm and introducing a nonlinear convergence factor to improve weights. Furthermore, a simple and effective constraint-handling method is proposed for the infeasible solutions. The performance of the IGWO is tested with eight benchmark functions selected and compared with other commonly used algorithms. Finally, the IGWO is utilized for three different scales of DED cases, and compared with existing methods in the literature. The results show that the proposed IGWO has a faster convergence rate and better global optimization capabilities, and effectively reduces the fuel costs of the units, thus proving the effectiveness of IGWO.

Performance analysis of machine learning based prediction models in assessing optimal operation of microgrid under uncertainty

Of late, the exponential rise in the global population is driving higher energy demand. However, the rapid depletion of conventional fossil fuels and growing environmental concerns have prompted the evolution of alternative energy sources. To this end, Microgrid (MG) with Renewable Energy Sources (RES) has emerged as popular means of small-scale localized power grid. However, planning of MG operation poses challenges due to the inherent variability and stochasticity in RES power output and energy demand. On account of this, the present study introduces a Stochastic Energy Management Strategy (SEMS) for a grid-connected MG incorporating Micro-Turbine, Fuel-Cell, RES, Battery Energy Storage, and electrical and heat energy demand. The stochasticity of RES is forecasted through a hybrid prediction model (sARIMA-GRU) and the uncertain demand is estimated via 'Monte Carlo Simulation.' The proposed problem is formulated as a dynamic non-linear stochastic optimization problem. It seeks to minimize the expected value of MG operational cost satisfying the practical constraints. Addressing this, a newly developed ‘Artificial Electric Field Algorithm (AEFA)' is utilized. Several case studies are performed to assess MG operation under varied operating conditions. Moreover, the present study analyses the impact of uncertainty on energy contribution from DER, grid dependency, and MG operation cost. Comparative analysis reveals that sARIMA-GRU outperforms other contemporary prediction models. It is noteworthy that the superior prediction accuracy of sARIMA-GRU leads to lower MG operation costs. Moreover, statistical analysis and convergence confirm the proficiency of applied AEFA over state-of-the-art Grey Wolf Optimization and Firefly Algorithm in solving the proposed problem.

Dynamic constrained evolutionary optimization based on deep Q-network

Dynamic constrained optimization problems (DCOPs) are common and important optimization problems in real-world, which have great difficulty to solve. Dynamic constrained evolutionary algorithms (DCEAs) are widely used methods for solving DCOPs. However, existing DCEAs often struggle with convergence, particularly for DCOPs with drastic dynamic changes or intricate constraints. To address this issue, this paper proposes a novel DCEA called DCEA-DQN, which leverages the powerful perception and decision-making capabilities of Deep Q-Network (DQN). DCEA-DQN integrates two DQNs to enhance its performance. The first DQN is designed to adaptively respond to dynamic changes, enabling effective handling of DCOPs with various types and degrees of changes. It provides a high-quality re-initialized population for subsequent static optimization, resulting in faster and improved convergence. The second DQN is introduced to guide the mutation direction during offspring generation. It steers the population towards better feasible regions or directs it towards the optimal individual within the current feasible region. Moreover, a penalty mechanism is employed to handle constraints during offspring generation. To evaluate the performance of DCEA-DQN, comprehensive empirical studies are conducted using a new test suite called C-GMPB and a dynamic flexible job-shop scheduling problem. The experimental results, using two commonly used metrics EB and EO in the field of DCOPs, demonstrate that DCEA-DQN outperforms six state-of-the-art DCEAs and achieved optimal performance on 80% and 75% of all 24 test problems, respectively. The source code for DCEA-DQN is available at https://github.com/CIA-SZU/YRT.

Process control of chemical dynamic system based on multi-strategy mayfly optimization algorithm

Dynamic optimization problems exist widely in chemical industry, and its operational variables change with the evolution of both space and time. Therefore, dynamic optimization problems have important research significance and challenges. To solve this problem, a multi-strategy mayfly optimization algorithm (MMOA) combined with control variable parameterization method(CVP) is proposed in this paper. MMOA introduces three improvements on the basis of the original algorithm, namely, circle chaos crossover strategy, center wandering strategy and boundary correction strategy. The hybrid strategy can better balance the exploration and exploitation ability of the algorithm. Based on MATLAB simulation environment, MMOA was evaluated. The experimental results show that MMOA has excellent performance in solving precision, convergence speed and stability for the benchmark function. For the six classical chemical dynamic optimization problems, MMOA obtained the performance indexes of 0.61071, 0.4776, 0.57486, 0.73768, 0.11861 and 0.13307, respectively. Compared with the data in the previous literature, MMOA can obtain more accurate control trajectory and better performance indicators. It provides an effective way to solve the dynamic optimization problem.

A data‐driven Bayesian approach for optimal dynamic product transitions

AbstractIn the processing industry, dynamic product transitions are essential for achieving high product quality, minimizing the use of raw materials and energy and reducing production costs. However, optimizing dynamic product transitions is a challenging task due to the complex dynamics of the process and the uncertainty in the measurements. In this article, a data‐driven Bayesian approach for optimal dynamic product transitions is proposed. The proposed approach is based on a dynamic optimization problem that is solved using a Bayesian optimization algorithm. One of the advantages of this approach for process optimization tasks is that it does not require a first‐principles dynamic mathematical model for drawing optimal solutions. The approach is applied to three case studies, and the results are comparable in performance quality with those obtained using a traditional gradient‐based optimization approach. The results show that the proposed approach is able to find optimal transition trajectories that meet the product composition requirements using smooth control actions. The approach is also able to cope with noisy measurements, which is an important feature in real‐world applications. The proposed approach has several advantages over traditional optimization approaches, including being data driven, able to cope with noisy measurements, computationally efficient, and it requires modest computational effort. Complex online optimal control problems can benefit from adopting a data‐driven Bayesian optimization scheme.

Comparative dynamic optimization study of batch hydrothermal liquefaction of two microalgal strains for economic bio-oil production

This work presents dynamic optimization strategies of batch hydrothermal liquefaction of two microalgal species, Aurantiochytrium sp. KRS101 and Nannochloropsis sp. to optimize the reactor temperature profiles. Three dynamic optimization problems are solved to maximize the endpoint biocrude yield, minimize the final time, and minimize the reactor thermal energy. The biocrude maximization and time minimization problems demonstrated 11% and 6.18% increment in the optimal biocrude yields and reduction of 78.2% and 61.66% in batch times compared to the base cases for the microalgae with higher lipid and protein fractions, respectively. The energy minimization problem revealed a significant reduction in the reactor thermal energies to generate the targeted biocrude yields compared to the biocrude maximization. Therefore, the identified optimal temperature trajectories outperformed the conventional fixed temperature profiles and could improve the overall economics of the batch bio-oil production from the algal-based biorefineries by significantly enhancing the reactor performance.

A Novel Dynamic Operation Optimization Method Based on Multiobjective Deep Reinforcement Learning for Steelmaking Process.

This article studies a dynamic operation optimization problem for a steelmaking process. The problem is defined to determine optimal operation parameters that bring smelting process indices close to their desired values. The operation optimization technologies have been applied successfully for endpoint steelmaking, but it is still challenging for the dynamic smelting process because of the high temperature and complex physical and chemical reactions. A framework of deep deterministic policy gradient is applied to solve the dynamic operation optimization problem in the steelmaking process. Then, an energy-informed restricted Boltzmann machine method with physical interpretability is developed to construct the actor and critic networks in reinforcement learning (RL) for dynamic decision-making operations. It can provide a posterior probability for each action to guide training in each state. Furthermore, in terms of the design of neural network (NN) architecture, a multiobjective evolutionary algorithm is used to optimize the model hyperparameters, and a knee solution strategy is designed to balance the model accuracy and complexity of neural networks. Experiments are conducted on real data from a steelmaking production process to verify the practicability of the developed model. The experimental results show the advantages and effectiveness of the proposed method compared with other methods. It can meet the requirements of the specified quality of molten steel.