Optimal Problem Research Articles

Optimal Learning and Management of Threatened Species

Amid an unprecedented loss of biodiversity, a pressing issue is how to improve the efficiency of conservation with limited resources and information. Collecting data on species with a small population is costly and time consuming, and many high-stakes decisions need to be made based on limited data. We develop a partially observable Markov decision processes model with unknown parameters to jointly optimize the information collection and protection efforts for threatened species. The model takes into account uncertainties about the state, detectability, and dynamics of the species, and it adaptively adjusts the efforts of surveying and protection in real time. Although the standard formulation is intractable, we exploit the structure of ecological problems to identify a hybrid belief state in low dimensions, and we reformulate the stochastic dynamic program as a piecewise deterministic optimal control problem. This enables us to obtain structural insights into the optimal policy (some are in closed form) and find a near-optimal approximate policy with performance guarantee. In certain situations, areas where the species has never been found may be more likely to contain the species than areas where it has been previously found. We also conduct a case study on the conservation of the Hainan gibbon, the rarest primate species, in which we extend the model to optimize the spatiotemporal allocation of limited resources. This paper was accepted by Chung Piaw Teo, optimization. Funding: This work was supported by the Natural Sciences and Engineering Research Council of Canada [Grant RGPIN-2019-05671]. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2023.01753 .

Modeling the Effects of Temperature in Enzymatic Biodiesel Synthesis of Jatropha Oil: An Optimal Control Approach.

Biodiesel, an alternative to diesel, is produced through the enzymatic transesterification of vegetable oil or animal fats. Enzymatic transesterification of oil is gaining more importance, as this method possesses no such disadvantages that are associated with the chemical process. Temperature is the most important factor in enzymatic transesterification for biodiesel synthesis. In this study, a mathematical model is developed to understand the effects of temperature on the enzymatic transesterification of Jatropha curcas oil. Reaction rates are expressed as a function of temperature using an appropriate function, and the effects of temperature on the overall enzymatic system are investigated using the mathematical model. A suitable temperature is determined using the mathematical model, keeping the other reaction conditions unchanged that gives maximum yield. Furthermore, an optimal control problem (OCP) is formulated to identify the temperature control strategy that maximizes the biodiesel yield while minimizing production costs. Simulated outcomes obtained from the mathematical model are analogized with the experimental results to make them acceptable. The optimal profile of temperature is determined from the developed OCP, which maximizes the biodiesel yield.

A numerical study of transformed mixed-integer optimal control problems

Time transformation is a ubiquitous tool in theoretical sciences, especially in physics. It can also be used to transform switched optimal control problems into control problems with a fixed switching order and purely continuous decisions. This approach is known either as enhanced time transformation, time-scaling, or switching time optimization (STO) for mixed-integer optimal control. The approach is well understood and used widely due to its many favorable properties. Recently, several extensions and algorithmic improvements have been proposed. We use an alternative formulation, the partial outer convexification (POC), to study convergence properties of (STO). We introduce the open-source software package ampl_mintoc (Sager et al., czeile/ampl_mintoc: Math programming c release, 2024, https://doi.org/10.5281/zenodo.12520490). It is based on AMPL, designed for the formulation of mixed-integer optimal control problems, and allows to use almost identical implementations for (STO) and (POC). We discuss and explain our main numerical result: (STO) is likely to result in more local minima for each discretization grid than (POC), but the number of local minima is asymptotically identical for both approaches.

Inclusion of water age in conjunctive optimal operation of water and power networks

ABSTRACT Water distribution systems (WDSs) are vital infrastructures designed to deliver water safely to consumers. This complex system necessitates continuous operational decisions, often optimized for efficiency. WDSs rely on power grids (PGs) to operate pumps and treatment facilities. PGs, likewise crucial, require strategic management to meet demand and environmental standards. Integrating the operation of both systems has garnered attention for its potential cost, energy, and environmental benefits. By leveraging the interconnection, trade-offs can be assessed, leading to improved solutions. Past research primarily focused on cost or carbon emissions as well as on hydraulic and voltage constraints. However, this study examines the impact of power systems on water quality, particularly water age, a key operational concern. Incorporating PGs into the optimal operation problem may alter flow directions, thereby influencing water age. Through mathematical modelling, this study evaluates these effects and applies them to simple case studies, demonstrating the influence of PG operation on water quality. Results demonstrate how tank constraints affect water age and show that a conjunctive operation approach, although beneficial for reduction of cost and energy consumption, can be damaging for water quality.

Interval riccati equation-based and non-probabilistic dynamic reliability-constrained multi-objective optimal vibration control with multi-source uncertainties

Structural vibration control is used to ensure the safety of structures in service and has become an active research area in the past few years. In this study, uncertainties in structural dynamics are considered to propose an interval linear quadratic regulator (LQR) method based on an interval Riccati equation. Multi-objective optimization with non-probabilistic dynamic reliability as a constraint is performed to design an optimal uncertain vibration control system. To alleviate the difficulty of quantifying the probabilistic uncertainty and accurately characterize uncertainty information for small samples, multi-source uncertainties are regarded as interval parameters. The bounds on these uncertainties are determined, and the deterministic and uncertain parts of an interval state-space equation for vibration control are then formulated using an expanded-order matrix and the interval analysis method. Next, the interval perturbation method with a matrix series is used to develop a novel solution method for the linear inhomogeneous time-invariant equation in the interval state-space equation for the vibration control system and is used to accurately predict the interval bounds of the state responses. The conventional LQR method from modern control theory is used to propose a novel optimal control method based on a novel interval Riccati equation using the Lyapunov equation and interval uncertainty propagation. The proposed interval control method can be used to estimate the uncertain bounds of the controlled gain and cost function. A non-probabilistic dynamic reliability model is established to evaluate the reliability index of the system, which can overcome the limitations of the large computational cost of determining the probabilistic reliability. The deterministic controlled cost and uncertain states are considered as two optimization objectives with the proposed reliability constraint. The resulting constrained multi-objective optimal vibration control problem is designed and solved using c-DPEA. A flowchart is presented as a clear overview of the uncertain vibration control method, and the effectiveness of the proposed method is validated using two numerical examples. Different optimal control designs are obtained using different reliability constraints for quantitatively analyzing the safety of the control system.

Compactness results for a Dirichlet energy of nonlocal gradient with applications

AbstractWe prove two compactness results for function spaces with finite Dirichlet energy of half‐space nonlocal gradients. In each of these results, we provide sufficient conditions on a sequence of kernel functions that guarantee the asymptotic compact embedding of the associated nonlocal function spaces into the class of square‐integrable functions. Moreover, we will demonstrate that the sequence of nonlocal function spaces converges in an appropriate sense to a limiting function space. As an application, we prove uniform Poincaré‐type inequalities for sequence of half‐space gradient operators. We also apply the compactness result to demonstrate the convergence of appropriately parameterized nonlocal heterogeneous anisotropic diffusion problems. We will construct asymptotically compatible schemes for these type of problems. Another application concerns the convergence and robust discretization of a nonlocal optimal control problem.

Control problems in the coefficients and the domain for linear elliptic equations

In the present work we are interested in an optimal design problem for a linear elliptic state equation with a homogeneous boundary Dirichlet condition. The control variables correspond to the coefficients of the diffusion term and the open set where the equation is posed. From the application point of view these variables represent the layout of the materials composing the corresponding domain and its shape. We obtain a relaxed formulation of the problem, the optimality conditions, and we provide a numerical algorithm to solve it. Some numerical simulations are also carried out.

Malware containment with immediate response in IoT networks: An optimal control approach

The exponential growth of Internet of Things (IoT) devices has triggered a substantial increase in cyber-attacks targeting these systems. Recent statistics show a surge of over 100 percent in such attacks, underscoring the urgent need for robust cybersecurity measures. When a cyber-attack breaches an IoT network, it initiates the dissemination of malware across the network. However, to counteract this threat, an immediate installation of a new patch becomes imperative. The time frame for developing and deploying the patch can vary significantly, contingent upon the specifics of the cyber-attack. This paper aims to address the challenge of pre-emptively mitigating cyber-attacks prior to the installation of a new patch. The main novelties of our work include: (1) A well-designed node-level model known as Susceptible, Infected High, Infected Low, Recover First, and Recover Complete (SIHILRFRC). It categorizes the infected node states into infected high and infected low, according to the categorization of infection states for IoT devices, to accelerate containment strategies for malware propagation and improve mitigation of cyber-attacks targeting IoT networks by incorporating immediate response within a restricted environment. (2) Development of an optimal immediate response strategy (IRS) by modeling and analyzing the associated optimal control problem. This model aims to enhance the containment of malware propagation across IoT networks by swiftly responding to cyber threats. Finally, several numerical analyses were performed to fully illustrate the main findings. In addition, a dataset has been constructed for experimental purposes to simulate real-world scenarios within IoT networks, particularly in smart home environments.

Network reconfiguration and integration of distributed energy resources in distribution network by novel optimization techniques

Nowadays, electrical load demand in radial distribution networks (RDN) is continuously growing, therefore RDNs are facing some serious challenges like voltage violation and line losses. Since modern RDNs have reconfiguration capabilities, optimal network reconfiguration is preferred over the costly construction of new lines and cables. In addition, the optimal integration of distributed energy resources (DER) helps to decrease above mentioned network challenges. This paper presents an improved version of the radiality maintenance algorithm (IRMA) to solve the optimal reconfiguration problem using a meta-heuristics algorithm. It also proposes two different schemes of algorithms by the blending of a Genetic algorithm (GA) and Teaching learning-based optimization (TLBO) to solve the optimal DER integration problem, where blending enhances the search space of each scheme which helps to find global minima in less iterations and time, while existing algorithms get stuck in local minima with same number of searching agents. The proposed problems are solved by minimizing active power loss by the single objective function and the sum of active power loss, reactive power loss, and voltage deviation index by multi-objective function using the weighted sum method where all objectives are equally dealt with, and their sum is used as single fitness function for the optimization algorithm. Four case studies are simulated using different DER technologies to improve each objective function. The efficiency of the proposed IRMA and two newly developed schemes of optimization algorithms, named GA-TLBO and TLBO-GA, are tested on two IEEE benchmark RDN of 33 and 69 buses. The results validate the efficiency of proposed algorithms that remarkably minimize a single objective from 210.9800 kW in the base case to 58.8768 kW, whereas existing algorithms like GWO and HHO were able to only reduce it to 72.7861 kW and 72.900 kW for IEEE 33 bus RDN, respectively. Similarly, for IEEE 69 bus system, single objective is reduced from base case of 224.9917 kW to 36.2543 kW, while existing algorithms like QOTLBO and QOSIMBO were only able to reduce it to 71.6250 kW and 71 kW, respectively. Furthermore, in multi-objective functions, reactive power losses and voltage deviation are also significantly improved from their base values. After that, the results of improved algorithms are compared with those of existing algorithms in the literature review for a comprehensive evaluation, which proves that proposed algorithm schemes are much more efficient and stable for the proposed problem as well as for standard benchmark optimization functions.

Optimal control of Volterra integral equations of third kind using Krall–Laguerre Polynomials

A novel method has been introduced to dissolve optimal control problems in systems governed by third-kind Volterra integral equations. This method utilizes Krall–Laguerre Polynomials as a basis function for expanding functions. By transforming the optimal control problem governed by third-kind Volterra integral equations (OCVIE3k) into a nonlinear programming problem (NLP), the solution process is significantly simplified. This approach involves converting the original problem into a more manageable form, which can be solved using established optimization techniques. The effectiveness and reliability of this proposed method are evaluated by comparing its outcomes to exact solutions when available.

Two-Level Approach for Simultaneous Component Assignment and Layout Optimization with Applications to Spacecraft Optimal Layout

Optimal layout problems consist in positioning a given number of components in order to minimize an objective function while satisfying geometrical or functional constraints. Such kinds of problems appear in the design process of aerospace systems such as satellite or spacecraft design. These problems are NP-hard, highly constrained and dimensional. This paper describes a two-stage algorithm combining a genetic algorithm and a quasi-physical approach based on a virtual-force system in order to solve multi-container optimal layout problems such as satellite modules. In the proposed approach, a genetic algorithm assigns the components to the containers while a quasi-physical algorithm based on a virtual-force system is developed for positioning the components in the assigned containers. The proposed algorithm is experimented and validated on the satellite module layout problem benchmark. Its global performance is compared with previous algorithms from the literature.

Backward Stochastic Differential Equations with Conditional Reflection and Related Recursive Optimal Control Problems

Backward Stochastic Differential Equations with Conditional Reflection and Related Recursive Optimal Control Problems

A Weak Maximum Principle for Discrete Optimal Control Problems with Mixed Constraints

A Weak Maximum Principle for Discrete Optimal Control Problems with Mixed Constraints

Optimal Fast-charging Strategy for Cylindrical Li-ion Cells

This paper presents an innovative approach to optimize the fast-charging strategy for cylindrical Li-ion NMC 3Ah cells, with a focus on enhancing both charging efficiency and thermal safety. Leveraging the power of Model Predictive Control (MPC), we introduce a cost function that approximates the thermal safety boundary of Li-ion batteries, revealing a relationship between temperature gradient and state of charge. Our proposed approach formulates the fast and safe charging problem as an optimal output regulator problem, incorporating thermal safety margin constraints. To solve the optimization problem, we develop an MPC algorithm. Our charge control structure incorporates an equivalent circuit model coupled with a thermal equation for battery state of charge and temperature estimation. Through numerical validation with real experimental data obtained from testing an NMC 3Ah cylindrical cell, we demonstrate that our approach respects the battery’s electrical and thermal constraints throughout the charging process.

Dynamic analysis and optimal control of a mosquito-borne infectious disease model under the influence of biodiversity dilution effect

In this paper, biodiversity and a saturation treatment rate are introduced into a type of mosquito-borne infectious disease model with an incubation time delay. Through the dynamical analysis of the model, conditions for the existence and stability of disease-free and endemic equilibria are determined. The basic reproduction number of the model is calculated, and the condition for the existence of backward bifurcation is outlined. The study finds that under the dual influence of biodiversity and saturation treatment, the threshold characteristic of the basic reproduction number becomes invalidated. When R0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$R_{0}$\\end{document} is less than 1, the model may exhibit four equilibrium states, with both the disease-free equilibrium and the endemic equilibrium being locally stable. In this scenario, whether the virus will become extinct depends on the initial conditions. The study also finds that when the basic reproduction number R0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$R_{0}$\\end{document} is greater than 1, the stability of the model is influenced by the time delay, with Hopf bifurcation occurring at a specific time delay. In addition, another novel contribution of this paper is the formulation of an optimal control problem that takes into account the minimization of damage caused by humans to biodiversity. Based on the Pontryagin’s maximum principle, the specific characteristics of the optimal control measures are given, and the optimal strategy is derived by comparing five groups of control strategies. The optimal control results highlight the synergistic effect of multiple control measures with biodiversity. Under optimal control, a significant complementary effect between medical inputs and the dilution effect of biodiversity is evident. The findings imply that maintaining high biodiversity levels can decrease the demand for medical resources in mosquito-borne disease control efforts.

Efficient predictive control method for ORC waste heat recovery system based on recurrent neural network

The model predictive control performs well in regulating operating parameters and ensuring system safety when the organic Rankine cycles are operating with variable heat sources. However, traditional nonlinear predictive control based on the organic Rankine cycle mechanism model involves significant computational complexity, making it challenging to quickly find a control solution. This limitation hinders its application in the organic Rankine cycle for rapid response control. To address this issue, a fast model predictive control method is proposed in this work. A recurrent neural network model is well-trained using the input–output data of the organic Rankine cycle process, and it is used as a surrogate model in the design of the model predictive controller for the control of organic Rankine cycle operating parameters. The formulated optimal control problem is then transformed into a mixed integer linear programming problem, which can obtain high-quality and fast solutions during the control process. Through comparison with recurrent neural network-based nonlinear predictive control and pseudo-sequential method-based fast nonlinear predictive control, the results show that the designed controller can effectively accomplish the control of organic Rankine cycle operating parameters with smaller overshoot. Moreover, its average control solution time is shorter by 89.59% and 93.27% respectively while the total net output power of the system during the control process is 0.54% and 1.3% higher than that of the other two controllers. It exhibits superior control performance, even under variable waste heat conditions.

Is Reinforcement Learning Good at American Option Valuation?

This paper investigates algorithms for identifying the optimal policy for pricing American Options. The American Option pricing is reformulated as a Sequential Decision-Making problem with two binary actions (Exercise or Continue), transforming it into an optimal stopping time problem. Both the least square Monte Carlo simulation method (LSM) and Reinforcement Learning (RL)-based methods were utilized to find the optimal policy and, hence, the fair value of the American Put Option. Both Classical Geometric Brownian Motion (GBM) and calibrated Stochastic Volatility models served as the underlying uncertain assets. The novelty of this work lies in two aspects: (1) Applying LSM- and RL-based methods to determine option prices, with a specific focus on analyzing the dynamics of “Decisions” made by each method and comparing final decisions chosen by the LSM and RL methods. (2) Assess how the RL method updates “Decisions” at each batch, revealing the evolution of the decisions during the learning process to achieve optimal policy.

Valuation of guaranteed lifelong withdrawal benefit with the long-term care option

In this paper, under the stochastic interest rate framework, we consider the valuation of a Guaranteed Lifelong Withdrawal Benefit (GLWB) annuity product by explicitly incorporating the health state of the policyholder through the long-term care (LTC) option. The product provides policyholders with protection against longevity risk and market downturns, as well as financial support when facing LTC needs. Within the context of dynamic withdrawals, the valuation of the GLWB annuity with the LTC option is characterized as a stochastic optimal control problem. We introduce a novel bang-bang analysis approach without the usual convexity assumption in literature and prove that the optimal withdrawal strategies for the policyholder are constrained to a finite set. Furthermore, we perform a sensitivity analysis on the price determinants of GLWB annuities with and without the LTC option, and provide economic interpretations. Lastly, we investigate the impact of gender on the optimal withdrawal strategy and the fair fee of the annuity with the LTC option.

Optimal dividends and capital injection: A general Lévy model with extensions to regime-switching models

This paper studies a general Lévy process model of the bail-out optimal dividend problem with an exponential time horizon, and further extends it to the regime-switching model. We first show the optimality of a double barrier strategy in the single-regime setting with a concave terminal payoff function. This is then applied to show the optimality of a Markov-modulated double barrier strategy in the regime-switching model via contraction mapping arguments. We solve these for a general Lévy model with both positive and negative jumps, greatly generalizing the existing results on spectrally one-sided models.

Consensus-Based Model Predictive Control for Active Power and Voltage Regulation in Active Distribution Networks

In this paper, a consensus-based model predictive control (Cb-MPC) scheme is proposed to control the active power and voltage at all nodes in grid-connected active distribution networks (ADNs) with multiple distributed energy resources (DERs). The proposed design methodology is based on a multiple-input multiple-output (MIMO) model of an ADN which accounts for both the internal and external interactions among the control loops of the DERs. To achieve the control objective, each DER unit is equipped with a controller–observer system. In particular, the observer implements the consensus algorithm to estimate the collective system state by exchanging data only with its neighbors. The scope of the controller is to solve the MPC optimal problem based on its collective state estimate, and, due to the presence of an integral term in the control action, it is robust against any unknown scenarios of the ADN, which are represented by uncertainty in the model parameters. The results of numerical simulations validate the effectiveness of the proposed method in the presence of unknown changes in the operating conditions of the ADN and of communication using a sample and hold function.