Approximate Optimal Solution Research Articles

Robust Direction Estimation of Terrestrial Signal via Sparse Non-Uniform Array Reconfiguration under Perturbations

DOA (Direction of Arrival), as an important observation parameter for accurately locating the Signals of Opportunity (SOP), is vital for navigation in GNSS-challenged environments and can be effectively obtained through sparse arrays. In practical application, array perturbations affect the estimation accuracy and stability of DOA, thereby adversely affecting the positioning performance of SOP. Against this backdrop, we propose an approach to reconstruct non-uniform arrays under perturbation conditions, aiming to improve the robustness of DOA estimation in sparse arrays. Firstly, we theoretically derive the mathematical expressions of the Cramér–Rao Bound (CRB) and Spatial Correlation Coefficient (SCC) for the uniform linear array (ULA) with perturbation. Then, we minimize CRB as the objective function to mitigate the adverse effects of array perturbations on DOA estimation, and use SCC as a constraint to suppress sidelobes. By doing this, the non-uniform array reconstruction model is formulated as a high-order 0–1 optimization problem. To effectively solve this nonconvex model, we propose a polynomial-time algorithm, which can converge to the optimal approximate solution of the original model. Finally, through a series of simulation experiments utilizing frequency modulation (FM) signal as an example, the exceptional performance of this method in array reconstruction has been thoroughly validated. Experimental data show that the reconstructed non-uniform array excels in DOA estimation accuracy compared to other sparse arrays, making it particularly suitable for estimating the direction of terrestrial SOP in perturbed environments.

Resource allocation in RISs-assisted UAV-enabled MEC network with computation capacity improvement

Unmanned aerial vehicle (UAV)-enabled mobile edge computing (MEC) networks have recently been considered to be a support for ground MEC networks to enhance their computation capability. However, the line-of-sight (LOS) channels between the UAV and Internet of Things (IoT) devices can be interfered by various obstacles, such as trees and buildings, resulting in a considerable reduction in the capacity of MEC networks. To solve this issue, a system that combines multiple reconfigurable intelligence surfaces (RISs) with a UAV-enabled MEC network is proposed in this study. A UAV equipped with edge servers is treated as an aerial computing platform for IoT devices, and multi-RISs are utilized to simultaneously improve the communication quality between enhanced UAV and IoT devices. To maximize the sum computation bits of the system, a problem that jointly optimizes the time slot partition, local computation frequency, transmit power of the devices, UAV receive beamforming vectors, phase shifts of the RISs, and the trajectory of the UAV is formulated. The problem is a typical nonconvex optimization problem; therefore, we propose a recursive algorithm based on the successive convex approximation (SCA) and block coordinate descent (BCD) technology to find an approximate optimal solution. Simulation results demonstrate the effectiveness of the proposed algorithm compared with various benchmark schemes.

Two-timescale joint service caching and resource allocation for task offloading with edge–cloud cooperation

Task offloading with edge–cloud cooperation has emerged as a pivotal solution for meeting the intricate array of application coupled with dynamically evolving business demand in 6G business scenarios, such as traffic sensing, environmental monitoring, and video surveillance in smart cities. Nonetheless, effectively leveraging heterogeneous edge–cloud network resources for effective task offloading presents substantial challenges. Additionally, the inherent differences in system decision cycles escalate the complexity of the task offloading problem to a new dimension. In this study, we delve into a two-timescale joint service caching and resource allocation optimization for task offloading within edge–cloud cooperation aiming to maximize long-term network performance while adhering to energy constraints. We propose a novel edge–cloud cooperation task offloading scheme that supports both edge–cloud and edge–edge cooperation to effectively balance the edge–cloud and edge–edge loads, promoting the efficient co-utilization of all edge–cloud system resources. Furthermore, we devise an online two-timescale Lyapunov-based joint optimization framework for service caching, task offloading, and computing resource allocation. Our two-timescale decision-making framework can flexibly accommodate the inherent differences in the sensitive decision optimization periods, thereby mitigating the degradation of task offloading performance caused by frequent service caching updates. Finally, theoretical analysis confirms that our proposed algorithm can converge to an approximate optimal solution in polynomial time, and the superiority of our scheme is validated by extensive simulation experiments.

[formula omitted]-perturbation of bilevel optimization problems: An error bound analysis

In this paper, we analyze a perturbed formulation of bilevel optimization problems, which we refer to as δ-perturbed formulation. The δ-perturbed formulation allows to handle the lower level optimization problem efficiently when there are multiple lower level optimal solutions. By using an appropriate perturbation strategy for the optimistic or pessimistic formulation, one can ensure that the optimization problem at the lower level contains only a single (approximate) optimal solution for any given decision at the upper level. The optimistic or the pessimistic bilevel optimal solution can then be efficiently searched for by algorithms that rely on solving the lower level optimization problem multiple times during the solution search procedure. The δ-perturbed formulation is arrived at by adding the upper level objective function to the lower level objective function after multiplying the upper level objective by a small positive/negative δ. We provide a proof that the δ-perturbed formulation is approximately equivalent to the original optimistic or pessimistic formulation and give an error bound for the approximation. We apply this scheme to a class of algorithms that attempts to solve optimistic and pessimistic variants of bilevel optimization problems by repeatedly solving the lower level optimization problem.

Optimizing green splits in high‐dimensional traffic signal control with trust region Bayesian optimization

AbstractCentralized traffic signal control has long been a challenging, high‐dimensional optimization problem. This study establishes a simulation‐based optimization framework and develops a novel optimization algorithm based on trust region Bayesian optimization (TuRBO), which can efficiently obtain an approximate optimal solution to the high‐dimensional traffic signal control problem. Local Gaussian process (GP), trust region, and Thompson sampling are employed in the TuRBO and contribute considerably to performance in terms of computational speed, solution quality, and scalability. Empirical studies are carried out using data from Mudanjiang and Chengdu, China. The performance of TuRBO is compared with that of Bayesian optimization (BO), genetic algorithm and random sampling. The results show that TuRBO converges the fastest because of its ability to balance exploration and exploitation through the trust region and Thompson sampling. Meanwhile, because TuRBO enables more efficient exploitation through the local GP, the solution quality of TuRBO outperforms others significantly. The average waiting time achieved by TuRBO was 2.84% lower than that achieved by BO. Finally, the method has been successfully extended to a large network with 233‐dimensional spaces and 122 signalized intersections, demonstrating that the developed methodology can deal with high‐dimensional traffic signal control effectively for real case applications.

Distributed Multiagent Reinforcement Learning With Action Networks for Dynamic Economic Dispatch.

A new class of distributed multiagent reinforcement learning (MARL) algorithm suitable for problems with coupling constraints is proposed in this article to address the dynamic economic dispatch problem (DEDP) in smart grids. Specifically, the assumption made commonly in most existing results on the DEDP that the cost functions are known and/or convex is removed in this article. A distributed projection optimization algorithm is designed for the generation units to find the feasible power outputs satisfying the coupling constraints. By using a quadratic function to approximate the state-action value function of each generation unit, the approximate optimal solution of the original DEDP can be obtained by solving a convex optimization problem. Then, each action network utilizes a neural network (NN) to learn the relationship between the total power demand and the optimal power output of each generation unit, such that the algorithm obtains the generalization ability to predict the optimal power output distribution on an unseen total power demand. Furthermore, an improved experience replay mechanism is introduced into the action networks to improve the stability of the training process. Finally, the effectiveness and robustness of the proposed MARL algorithm are verified by simulation.

Convex Approach to Real-Time Multiphase Trajectory Optimization for Urban Air Mobility

Electric vertical takeoff and landing technology has emerged as a promising solution to alleviate ground transportation congestion. However, the limited capacity of onboard batteries enforces hard time constraints for vehicle operations in a complex urban environment. Therefore, a constrained, real-time trajectory optimization method is necessary to enable safe and energy-efficient vehicle flight. Unfortunately, most existing trajectory optimization methods suffer from low computational efficiency, unpredictable convergence processes, and strong dependence on a good initial trajectory. In this paper, we employ a successive convex programming approach to solve the multiphase minimum-energy-cost electric vertical takeoff and landing vehicle trajectory optimization problem for urban air mobility applications. The main contribution is the development and validation of two novel convexification methods that find approximate optimal solutions to the multiphase nonlinear trajectory optimization problem in real time by solving a sequence of convex subproblems. Specifically, the first method transforms the original nonlinear problem into a sequence of second-order cone programming problems through a convenient change of variables and lossless convexification, while the second approach achieves a similar goal without the need for control convexification. The resulting convex subproblems can be solved reliably in real time by advanced interior point methods. The proposed methods are demonstrated through numerical simulations of two different urban air mobility scenarios and compared with the solution obtained from GPOPS-II, a state-of-the-art general-purpose optimal control solver. Our proposed sequential convex programming methods can obtain near-optimal solutions with faster computational speed than GPOPS-II.

Exponential stabilization of linear systems with feedback optimization and its application to microgrids control

This paper proposes a feedback-optimization-based control method for linear time-invariant systems, which is aimed to exponentially stabilize the system and, meanwhile, drive the system output to an approximate solution of an optimization problem with convex set constraints and affine inequality constraints. To ensure the exponential stability of the closed-loop system, the original optimization problem is first reformulated into a counterpart that has only convex set constraints. It is shown that the optimal solution of the new optimization problem is an approximate optimal solution of the original problem. Then, based on this new optimization problem, the projected primal–dual gradient dynamics algorithm is used to design the controller. By using the singular perturbation method, sufficient conditions are provided to ensure the exponential stability of the closed-loop system. The proposed method is also applied to microgrid control.

A mathematical programming model integrating waste pickers in urban area recycling practice

Urbanization has resulted in a significant accumulation of waste in densely populated areas. Effective household waste separation practices within urban areas are crucial for advancing the concept of circular economy. The informal system has contributed remarkably to urban material circulation, but it has not been appropriately valued. Through the partnership among the municipal solid waste authorities, condominium building residents, and waste pickers, the latter can be engaged in the community to assist in household waste separation. This initiative would not only enhance their income and reduce probable occupational risks but also augment the recycling rate in the community. Based on the insights from the current waste picker assignment policy in Taichung City, this study has established a mathematical programming model by considering the overall waste separation volume and proximal distance to the neighborhood, facilitating the assignment of individual waste picker to multiple communities within the neighborhood area. Consequently, concerned waste pickers may increase their income and avoid the constraints of picking up recyclables in high-occupational-risk areas. Additionally, it may reduce the volume of unsorted garbage in the community. According to the findings of case studies, the developed model could obtain an optimal solution or acceptable approximate solution in a short timeframe. In contrast to the outcomes from the existing waste picker assignment policy, the application of the developed model could increase the total volume of sorted recyclables, which was directly proportional to the income of waste pickers, by 2.3–2.6 times.

An integration method of a hybrid genetic algorithm and the Levenberg–Marquardt algorithm for ultrasonic testing

The accurate measurement of time-of-flight (TOF) is essential in ultrasonic testing. Further, noise interference is the key factor affecting the measurement accuracy. Therefore, to develop a reliable computational method of TOF for test pieces working in noisy environments, an integration method of a hybrid genetic algorithm and the Levenberg–Marquardt algorithm (GA–LM) for ultrasonic thickness measurement is proposed in the present research. A Gaussian model is first established for an echo signal. Further, the model-based parameter estimation is converted into a nonlinear optimization problem by applying the least square method. As the parameter estimation methods are easily affected by the initial value, an integrating innovation of the GA–LM algorithm is proposed. The initial values of the model parameters are selected by GA to obtain an approximate global optimal solution. Subsequently, this approximate solution is used as the initial value for the LM algorithm to perform iterations. The accurate global optimal solution of the Gaussian model is obtained through these iterations. Finally, the measuring accuracy and robustness of the GA–LM algorithm for TOF computation are verified by both numerical simulation and experiment data

Research on airport apron planning strategy in emergency situations

In events of local war or highly infectious epidemics, airports face the problem of how to park numerous non-missioned aircraft on aprons. In this study, the apron planning problem in emergency situations (APPES) was considered to replan existing parking stands and taxiways, thus determining the numbers and locations of different aircraft types. A mixed-integer programming formulation for the APPES was developed to maximise the total revenue for the apron, where the aircraft was planned to be parked based on factors such as the safety distance and aircraft taxiing method. A two-stage approach based on the discrete particle swarm optimisation algorithm and bottom-left algorithm was designed to solve the APPES. Random instances of different sizes were generated, and the experimental results demonstrated the effectiveness of the proposed approach. The two-stage approach could find optimal solutions to the APPES for 19 of the 20 small instances and could well approximate optimal solutions for the medium and large instances within a reasonable timeframe. Parking priority factor and aircraft taxiing methods sensitivity analyses were also designed to develop a strategy that can adhere to the preferences of decision makers or increase the total revenue.

An adaptive fractional-order regularization primal-dual image denoising algorithm based on non-convex function

In this paper, a novel non-convex fractional-order image denoising model is proposed to suppress the staircase effect produced by the TV model while maintaining a neat contour. The model combines ℓq(0<q<1) quasi-norm and fractional-order regularization, and employs a diffusion coefficient with a faster convergence rate to preserve more image edges and details. Additionally, an adaptive regularization parameter is designed to adjust the denoising performance of the algorithm. To obtain the optimal approximate solution of the model, an enhanced primal-dual algorithm is adopted and the complexity and convergence of the algorithm are theoretically analyzed. Finally, the effectiveness of the proposed method is demonstrated through numerical experiments.

Real-Time Batch Optimization for the Stochastic Container Relocation Problem

The container relocation problem (CRP) is an important factor affecting the operation efficiency of container terminal yards, and it has attracted much attention for decades. The CRP during the pickup operations of import containers is still an intractable problem for two reasons: the first is that the solution efficiency of the algorithms developed in the existing literature cannot meet the real-time operation requirements; the second is that the pre-optimized operation plan cannot cope with the changes in the real-time operation scenarios caused by the uncertainty of the arrival time of external trucks. This paper proposes an optimization method for the real-time operation scenario which aims to solve the most reasonable operation plan quickly according to the arrivals of external trucks, in which a dynamic upper bound of the optimal solution is derived based on the dynamic programming model of the import containers’ CRP, and an approximate optimal solution can be obtained by minimizing this dynamic upper bound. A heuristic algorithm based on three relocation rules is developed to implement this method, considering the adjustment of the pickup sequence of the target containers. Numerical experiments show that (1) when the number of a batch of target containers is less than 10 (excluding target containers that can be directly picked up), the method proposed in this paper can solve the problem quickly to meet the demand of optimizing real-time pickup operations; (2) compared with other outstanding algorithms, the quality of the solutions obtained by this method is also improved; and (3) this method can be applied to the most container terminals for optimizing real-time pickup operations.

A multi-skilled staff scheduling and team configuration optimisation model for artificial intelligence project portfolio considering competence development and innovation-driven

This paper emphasises the pivotal role of enhancing research and development (R&D) staff competence and boosting team innovation efficiency in configuring R&D project teams for artificial intelligence (AI) product development. Diverging from traditional studies primarily focused on time, quality, and cost objectives, this study proposes prioritising the skill increments of staff and team diversity, aligning with the overarching goals of the R&D cycle. Targeting multi-skilled R&D personnel for scheduling and configuration, a three-objective tradeoff optimisation model is established. The nonlinear mixed-integer constrained programming model incorporates a learning effect for calculating employee skill value and employs a heterogeneity efficiency formula for assessing team diversity, particularly emphasising the diversity of R&D personnel research backgrounds. An algorithm based on Non-dominated Sorting Genetic Algorithm-II (NSGA-II) is designed to obtain the approximate Pareto optimal solution. Furthermore, it compares the algorithm with the multi-objective particle swarm optimisation (MOPSO), explores practical decision-making methods for Pareto solutions, and conducts sensitivity analyses on the learning rate and diversity level. Our research is relevant to enterprises seeking to enhance R&D capabilities with a certain degree of homogeneity among R&D employees. This paper exemplifies and validates the proposed model and solution approach using a new AI product from a healthcare company.

A digital economy development index based on an improved hierarchical data envelopment analysis approach

The digital economy is playing an increasingly important role in the global economy. National and international organizations commonly utilize a composite index composed of multi-dimensional indicators to monitor performance, analyse policies, and communicate in the digital economy. This study introduces a hierarchical framework for constructing a Digital Economy Development Index (DEDI). One of the key challenges is determining the attribute weights to be assigned by aggregating the sub-indicators with hierarchical dimensions for DEDI. The current study shows that an H-DEA model can be transformed into a parametric linear programming problem and develops an improved golden section algorithm to search for approximate optimal solutions. The newly developed method is highly robust and provides an alternative procedure to determine the optimal weights for building a DEDI. We established a hierarchical evaluation framework and utilized a new approach to measure the DEDI of 30 provinces in China from 2015 to 2020. The results show that inter-provincial digital economy development in China shows a sequential weakening from the east, centre, and northeast to the west. The East leads the country's digital economy overall, while there is a clear gradient gap between provinces. The central region needs to create a cluster for the development of digital industries. The Northeast should enhance the competitiveness of the industry across the board. The West must strengthen the construction of innovative talent and new infrastructure for the digital economy. These findings can serve as guidelines for designing China's ongoing digital economy development.

Approximate Optimal Solutions for Multiobjective Optimization Problems with Infinite Constraints

Approximate Optimal Solutions for Multiobjective Optimization Problems with Infinite Constraints

Neural Network Methods Based on Efficient Optimization Algorithms for Solving Impulsive Differential Equations

In view of the fact that impulsive differential equations have the discreteness due to the impulse phenomenon, this paper proposes a single hidden layer neural network method based extreme learning machine and a physicsinformed neural network method combined with learning rate attenuation strategy to solve linear impulsive differential equations and nonlinear impulsive differential equations respectively. For the linear impulsive differential equations, firstly, the interval is segmented according to the impulse points, and a single hidden layer neural network model is constructed, the weight parameters of hidden layer are randomly set, the optimal output parameters and solution of the first segment are obtained by the extreme learning machine algorithm, then we calculate the initial value of the second segment according to the jumping equation, and the remaining segments are solved in turn in the same way. Although the single hidden layer neural network method proposed can solve linear equations with high accuracy, it is not suitable for solving nonlinear equations. Therefore, we propose the physics-informed neural network combined with learning rate attenuation strategy to solve the nonlinear impulsive differential equations, then the Adam algorithm and L-BFGS algorithm are combined to find the optimal approximate solution of each segment. Numerical examples show that the single hidden layer neural network method with Legendre polynomials as the activation function and the physics-informed neural network method combined with learning rate attenuation strategy can solve linear and nonlinear impulsive differential equations with higher accuracy.

Research on optimization method for flyby observation mission adapted to satellite on-board computation

In this paper, the optimization problem of orbital transfer strategy for orbital flyby observation missions is studied. A hybrid optimization method is proposed, which is improved to make it more suitable for satellite on-board computing. This new algorithm is designed to solve the initial value sensitivity problem of the sequential quadratic programming algorithm (SQP). It is consisted of the depth-first search algorithm (DFS) and the SQP algorithm and thus has the characteristics of fast convergence, high reliability, and good robustness. With this method, the DFS with a large step size is calculated first, and then the optimal value in the calculation result is used as the initial value of the SQP algorithm for further optimization. This method can obtain the approximate optimal solution available in engineering. The numerical simulation of an orbital transfer optimization problem is set to verify the effectiveness of the new hybrid algorithm. The simulation results compared with the genetic algorithm (GA) show that the proposed hybrid algorithm can effectively reduce the on-board resource occupation when getting similar results and thus can meet the needs of satellite on-board computing.

Optimized convergence of stochastic gradient descent by weighted averaging

Under mild assumptions stochastic gradient methods asymptotically achieve an optimal rate of convergence if the arithmetic mean of all iterates is returned as an approximate optimal solution. However, in the absence of stochastic noise, the arithmetic mean of all iterates converges considerably slower to the optimal solution than the iterates themselves. And also in the presence of noise, when a termination of the stochastic gradient method after a finite number of steps is considered, the arithmetic mean is not necessarily the best possible approximation to the unknown optimal solution. This paper aims at identifying optimal strategies in a particularly simple case, the minimization of a strongly convex function with i. i. d. noise terms and termination after a finite number of steps. Explicit formulas for the stochastic error and the optimality error are derived in dependence of certain parameters of the SGD method. The aim was to choose parameters such that both stochastic error and optimality error are reduced compared to arithmetic averaging. This aim could not be achieved; however, by allowing a slight increase of the stochastic error it was possible to select the parameters such that a significant reduction of the optimality error could be achieved. This reduction of the optimality error has a strong effect on the approximate solution generated by the stochastic gradient method in case that only a moderate number of iterations is used or when the initial error is large. The numerical examples confirm the theoretical results and suggest that a generalization to non-quadratic objective functions may be possible.

An autonomous ore packing system through deep reinforcement learning

In the contemporary era, the limited availability of terrestrial resources has prompted an increasing number of nations to turn their attention towards space, wherein extraterrestrial minerals hold considerable allure for both resource provisioning and scientific inquiry. Numerous nations have initiated the deployment of unmanned operational platforms towards extraterrestrial asteroids with the objective of accomplishing sampling return missions. Given the restricted storage capacity and energy limitations of these platforms, the optimization of mineral placement algorithms assumes paramount importance in enhancing the efficacy of these missions. In this paper, we propose an autonomous ore packing system capable of autonomously measuring ore characteristics and addressing the ore packing optimization problem in extraterrestrial unfamiliar environments. A deep reinforcement learning method that utilizes physical constraints to enhance overall performance was proposed to solve the ore packing problem with uncertainty, while meeting the high demands for real-time performance in the mission. To augment the autonomy and adaptability of our approach, we leverage advanced visual technology to transform the spatial distribution of bin utilization and the characteristics of the ore into a matrix representation. This empowers the robotic system to autonomously perceive bin information and capture essential ore features. The empirical findings substantiate that our algorithm attains a human-level performance in the majority of instances, rendering an approximate optimal solution within a concise timeframe. Additionally, we introduce a novel reinforcement learning training technique known as Maximum Worth Reinforcement Learning (MWRL) to address the optimization conundrum associated with incomplete Markov chains, which outperforms existing approaches in our comparative analysis. Lastly, we validate the efficacy of our algorithm in real-world scenarios by deploying it on a robotic manipulator.