Nonlinear Programming Problem Research Articles

Optimization of the on/off status configuration of control units in an active vibration isolation system—experiment

Active vibration isolation (AVI) is an important measure to attenuate vibration transmission in underwater vehicles and multiple active control units are usually used to attenuate low frequency vibrations induced by power equipment. Under certain operation conditions, however, the pre-installed control units in an AVI system may be redundant. Hence, it is necessary to optimize the on/off status configuration of all the control units. An optimization approach is proposed for configuration optimization of control units and validated by experiments. The criterion of the proposed approach is derived from the vibration responses of the monitoring points. The involved optimization, formulated as a constrained 0–1 nonlinear programming problem, utilizes the on/off status of control units as the input variable and the number of active units as the constraint. Optimized configurations were obtained by solving the nonlinear programming problem and deployed to the experimental system. The test results have demonstrated that vibration of the monitoring points can be effectively suppressed by activating only a portion of control units, and the configurations corresponding to maximum vibration reduction vary with different excitations.

Economic modelling and optimisation of SPRT-based quality control schemes for individual observations

In many industrial processes, it may not be practical to obtain sample sizes larger than one due to various reasons. As a result, practitioners must rely on control charts based on individual observations for statistical process monitoring and control. The sequential probability ratio test (SPRT) chart is an appropriate solution for such situations. Existing literature predominantly delves into the statistical aspects of designing or optimising an SPRT chart, neglecting the economic and associated design aspects. However, past experience suggests that statistical design may not always be optimal from an economic perspective. This paper aims to bridge this research gap by proposing a two-stage Markov chain approach to develop an economic model for the optimal design of the SPRT chart with individual observations. Following the principles of economic-statistical designs, optimal parameters are determined by solving a mixed-integer non-linear programming problem that minimises the long-run cost per item while ensuring compliance with practical constraints. Through an extensive study, we find that the economic performance of the SPRT chart surpasses that of competing individual control charts in many cases. This indicates that the economic-statistical design of the SPRT chart can yield significant economic benefits while maintaining desirable statistical properties. Moreover, enhancements have been introduced to refine the optimal design of the SPRT chart, retaining excellent overall performance across a wide range of shifts. Finally, we provide two industrial examples to illustrate the effectiveness of the proposed SPRT chart.

Geographic coordinate validation and assignment using an edge-constrained layout

Electric grids with buses that are mapped to geographic latitude and longitude are useful for a growing number of applications, such as data visualization, geomagnetically induced current calculations, and multi-energy coupled infrastructure simulations. This paper presents a methodology for validating the quality of geographic coordinates for a power system model, and to assign coordinates to buses with missing or low-quality coordinates. This method takes advantage of geographic indicators already intrinsic to a grid model, such as branch length as implied by impedance and susceptance parameters. The coordinate assignment process uses an approach inspired by graph drawing, that lays out the vertices (buses) and edges (transmission lines), formulated as a nonlinear programming problem with soft edge length constraints. The layout method is very computationally fast and scalable to large power system cases. The method is demonstrated in this paper using a 37-bus test case and a 6717-bus test case, both publicly available, along with a large actual grid model. The results show that, for cases with only a few errors in the coordinates, cases with no coordinates known beforehand, and others in between, this method is able to assign reasonable geographic coordinates to best match known data about the grid.

An SDP relaxation in the complex domain for the efficient coordination of BESS and DGs in single-phase distribution grids while considering reactive power capabilities

This research proposes an efficient energy management system design for medium-voltage distribution networks from a perspective of convex optimization. The exact nonlinear programming problem that represents the operation of distributed energy resources (DERs), such as batteries and photovoltaic sources, is approximated as an equivalent semi-definite programming (SDP) problem. This SDP approximation is achieved in the complex domain using Hermitian matrices, which are the equivalent of positive semi-definite matrices in the real number domain. Two test feeders, one with 27 nodes and the other with 33 nodes, are modified to match the characteristics of Colombian rural and urban networks. The 27-bus grid is adapted to accommodate the average demand and solar generation conditions of the municipality of Capurganá, while those of the city of Medellín are used for the 33-bus grid. One of the main contributions of this research is that it considers a DER operation with variable power factors, which allows for significant improvements compared to the traditional use of a unitary power factor scheme. Numerical comparisons are conducted using metaheuristic optimizers, including parallel versions of the particle swarm optimizer, the vortex search algorithm, and the continuous version of the genetic algorithm. These comparisons demonstrate the effectiveness of the proposed SDP approximation. All simulations are carried out in the MATLAB software (version 2022b), using the disciplined convex optimizer (CVX) and the SDPT3 and SEDUMI solvers.

A Nonlinear Programming Approach to Solving Interval-Valued Intuitionistic Hesitant Noncooperative Fuzzy Matrix Games

Initially, fuzzy sets and intuitionistic fuzzy sets were used to address real-world problems with imprecise data. Eventually, the notion of the hesitant fuzzy set was formulated to handle decision makers’ reluctance to accept asymmetric information. However, in certain scenarios, asymmetric information is gathered in terms of a possible range of acceptance and nonacceptance by players rather than specific values. Furthermore, decision makers exhibit some hesitancy regarding this information. In such a situation, all the aforementioned expansions of fuzzy sets are unable to accurately represent the scenario. The purpose of this article is to present asymmetric information situations in which the range of choices takes into account the hesitancy of players in accepting or not accepting information. To illustrate these problems, we develop matrix games that consider the payoffs of interval-valued intuitionistic hesitant fuzzy elements (IIHFEs). Dealing with these types of fuzzy programming problems requires a significant amount of effort. To solve these matrix games, we formulate two interval-valued intuitionistic hesitant fuzzy programming problems. Preserving the hesitant nature of the payoffs to determine the optimal strategies, these two problems are transformed into two nonlinear programming problems. This transformation involves using mathematical operations for IIHFEs. Here, we construct a novel aggregation operator of IIHFEs, viz., min-max operators of IIHFEs. This operator is suitable for applying the developed methodology. The cogency and applicability of the proposed methodology are verified through a numerical example based on the situation of conflict between hackers and defenders to prevent damage to cybersecurity. To validate the superiority of the proposed model along with the computed results, we provide comparisons with the existing models.

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.

Collaborative Computation Offloading and Resource Management in Space–Air–Ground Integrated Networking: A Deep Reinforcement Learning Approach

With the increasing dissemination of the Internet of Things and 5G, mobile edge computing has become a novel scheme to assist terminal devices in executing computation tasks. To elevate the coverage and computation capability of edge computing, a collaborative computation offloading and resource management architecture was proposed in space–air–ground integrated networking (SAGIN). In this manuscript, we established a novel model considering the computation offloading cost constraints of the communication, computing and cache model in the SAGIN. To be specific, the joint optimization problem of collaborative computation offloading and resource management was modeled as a mixed integer nonlinear programming problem. To address this issue, this paper proposed a computation offloading and resource allocation strategy based on deep reinforcement learning (DRL). Differing from traditional methods, DRL does not need a well-established formulation or previous information, and it is capable of revising the strategy adaptively according to the environment. The simulation results demonstrate the proposed approach can achieve the optimal reward values in the case of different terminal device numbers. Furthermore, this manuscript provided the analysis with variant parameters of the proposed approach.

Sequential multichannel allocation in cache-enabled multicasting small-cell networks

This paper investigates the optimization of multichannel allocation in cache-enabled multicasting small-cell networks (CMSNs). In the traditional fixed channel allocation scheme (FCA), small-cell base stations (SBSs) allocate a fix channel to each local cache popular content. In order to further increasing the utilization of allocated channels, we first propose a sequential multichannel allocation scheme (SMA), in which SBSs allocate a sequential channel set to their cached files dynamically. We formulate the problem of sequential multichannel allocation as two NP-hard non-linear programming problems with two optimization objectives of minimizing the global collision level (GCL) and maximizing the global file-downloading level (GFL), and demonstrate the advantages of the proposed DSCA over FCA. Then, two channel allocation potential games are proposed to achieve GCL and GFL optimization respectively. To achieve Nash equilibria (NEs) of the proposed games, we present a better response based SMA algorithm (BR-SMA) for both two games, and propose an improved better response based SMA algorithm (IBR-SMA) for the game for GCL maximization which have two exploration modes for each decision-maker.

Fourier–Gegenbauer pseudospectral method for solving periodic fractional optimal control problems

This paper introduces a new, highly accurate model for periodic fractional optimal control problems (PFOCPs) based on Riemann–Liouville and Caputo fractional derivatives (FDs) with sliding fixed memory lengths. This new model enables periodic solutions in fractional-order control models by preserving the periodicity of the fractional derivatives. Additionally, we develop a novel numerical solution method based on Fourier and Gegenbauer pseudospectral approaches. Our method simplifies PFOCPs into constrained nonlinear programming problems (NLPs), readily solvable by standard NLP solvers. Some key innovations of this study include: (i) The derivation of a simplified FD calculation formula to transform periodic FD calculations into evaluations of trigonometric Lagrange interpolating polynomial integrals, efficiently computed using Gegenbauer quadratures. (ii) The introduction of the αth-order fractional integration matrix based on Fourier and Gegenbauer pseudospectral approximations, which proves to be highly effective for computing periodic FDs. A priori error analysis is provided to predict the accuracy of the Fourier–Gegenbauer-based FD approximations. Benchmark PFOCP results validate the performance of the proposed pseudospectral method.

An online integrated satellite–terrestrial IoT task offloading and service deployment strategy

As a core component of 6G networking, low earth orbit (LEO) satellite networks are considered a promising solution for providing network services in remote areas and have received considerable attention in recent years. Meanwhile, with the rapid development of IoT terminals and applications, an increasing emphasis has been placed on task offloading for real-time tasks. However, most existing works on task offloading for integrated satellite–terrestrial IoT (IST-IoT) applications are based on the assumption that the global task and network information is known, an assumption that is obviously not met in online real-time task offloading. In addition, current studies rarely consider the tradeoff among the task offloading volume, satellite endurance and wholesale cost of terrestrial resources, which are precisely the holistic issues that satellite network operators must confront. To address such issues, this study considers task offloading, service deployment and resource allocation in highly dynamic online task offloading scenarios with the goal of maximizing the average task offloading volume. Moreover, virtual queues are constructed in this study to account for the impacts of long-term satellite endurance and terrestrial network resource costs on task offloading. We first decouple the original problem to obtain a single-time-slot optimization problem within the Lyapunov optimization framework, and we then propose a 2-stage optimization approach based on deep reinforcement learning (DRL) to solve the complex single-time-slot mixed integer nonlinear programming (MINLP) problem. Simulation experiments demonstrate that our proposed algorithm can yield near-optimal solutions that show superior performance compared to existing results.

Bidirectional Service Function Chain Embedding for Interactive Applications in Mobile Edge networks

Bidirectional service function chain (BSFC) consists of multiple virtual network functions (VNFs). Through VNF deployment and link mapping, BSFCs can be embedded into resource-constrained mobile edge networks to provide low-latency network function services to users participating in interactive applications such as multi-player online games. Data from these users are routed through BSFCs to the edge node where the application is located for interaction and then returned to the users through the BSFCs, thus enabling synchronization among multiple users. However, the edge nodes or links have limited computing or bandwidth resources to serve only a fraction of users simultaneously. Therefore, the embedding decisions among different users can affect each other. In this paper, we propose a novel BSFC embedding strategy for interactive applications with the goal of minimizing computing and bandwidth resources while satisfying users' latency requirements. We first model the BSFC embedding problem as an integer nonlinear programming problem. Then, by closely examining the complexity of the problem, we propose a distributed algorithm based on game theory. We theoretically analyze the properties of the proposed algorithm and show that it can obtain a solution with a worst-case performance bound. Finally, extensive experiments show that the proposed algorithm outperforms several existing algorithms.

NOMA Assisted Two-Tier VR Content Transmission: A Tile-Based Approach for QoE Optimization

Virtual reality (VR) provides users with an immersive and interactive experience through head-mounted devices, which has attracted increasing attention in recent years. Specifically, tile-based VR content transmission provides a promising approach to alleviate the conflict between limited bandwidth and high-performance requirements (e.g., high-resolution and low-delay). However, the tiling pattern affects the encoding efficiency and visual distortion of the VR content. Accounting for this issue, in this paper, a quality of experience (QoE)-aware cost minimization problem is investigated for a tile-based VR content transmission scenario. In particular, an edge server (ES) co-located at a cellular base station (BS) separates its generated VR content into several tiles according to the tiling pattern selection, and a weighted-to-spherically-uniform quality model is used to evaluate the effect of different tiling patterns on QoE. Moreover, to improve the transmission performance between the edge server and VR users (VRUs), unmanned aerial vehicles (UAVs) are leveraged as relay points to provide line of sight channels. Then, we formulate an optimization problem to minimize the sum of weighted total energy consumption and VR content distortion (i.e., QoE-aware cost) by jointly optimizing the tiling pattern selections, the VRUs-UAV grouping, partial computing decisions, and resource allocation. The formulated problem is a mixed integer non-linear programming problem, which is challenging to solve. To address this difficulty, we equivalently decompose the formulated problem into three subproblems and propose corresponding algorithms to solve them, respectively. Numerical results demonstrate that our proposed solution can effectively reduce the QoE-aware cost for VR content transmission in comparison with other baseline algorithms.

Network-level pavement maintenance and rehabilitation planning considering uncertainties using chance-constrained programming

ABSTRACT In practice, given limited funds, to consider multiple strategic goals/objectives that different stakeholders concern, pavement network-level maintenance and rehabilitation (M&R) planning becomes a multi-objective optimisation (MOO) based project selection and budget allocation problem. In an attempt to solve this problem, most agencies established MOO models under the deterministic situation without appropriate consideration of uncertainties. However, ignoring performance uncertainties often leads to unreasonable decisions. To provide more convincing and reliable pavement M&R decisions, this paper proposes a Chance-Constrained Programming (CCP) based MOO method to incorporate performance uncertainties in network-level single period pavement M&R planning. First, a general deterministic MOO model with budget and network performance constraints is established. Then, three commonly-used statistical forms of network-level performance measures are introduced. To incorporate uncertainties, the probability distribution of each form of performance measure is derived. Based on the CCP method, the MOO model is transformed to an equivalent deterministic formulation as a mixed non-linear integer programming (MNLIP) problem. To demonstrate the proposed method, a case study using real data is conducted. The results show that the proposed method can effectively help decision-makers to appropriately incorporate performance uncertainties in conducting network-level pavement M&R planning.

Small-disturbance input-to-state stability of perturbed gradient flows: Applications to LQR problem

This paper studies the effect of perturbations on the gradient flow of a general nonlinear programming problem, where the perturbation may arise from inaccurate gradient estimation in the setting of data-driven optimization. Under suitable conditions on the objective function, the perturbed gradient flow is shown to be small-disturbance input-to-state stable (ISS), which implies that, in the presence of a small-enough perturbation, the trajectories of the perturbed gradient flow must eventually enter a small neighborhood of the optimum. This work was motivated by the question of robustness of direct methods for the linear quadratic regulator problem, and specifically the analysis of the effect of perturbations caused by gradient estimation or round-off errors in policy optimization. We show small-disturbance ISS for three of the most common optimization algorithms: standard gradient flow, natural gradient flow, and Newton gradient flow.

A Multi-Agent RL Algorithm for Dynamic Task Offloading in D2D-MEC Network with Energy Harvesting.

Delay-sensitive task offloading in a device-to-device assisted mobile edge computing (D2D-MEC) system with energy harvesting devices is a critical challenge due to the dynamic load level at edge nodes and the variability in harvested energy. In this paper, we propose a joint dynamic task offloading and CPU frequency control scheme for delay-sensitive tasks in a D2D-MEC system, taking into account the intricacies of multi-slot tasks, characterized by diverse processing speeds and data transmission rates. Our methodology involves meticulous modeling of task arrival and service processes using queuing systems, coupled with the strategic utilization of D2D communication to alleviate edge server load and prevent network congestion effectively. Central to our solution is the formulation of average task delay optimization as a challenging nonlinear integer programming problem, requiring intelligent decision making regarding task offloading for each generated task at active mobile devices and CPU frequency adjustments at discrete time slots. To navigate the intricate landscape of the extensive discrete action space, we design an efficient multi-agent DRL learning algorithm named MAOC, which is based on MAPPO, to minimize the average task delay by dynamically determining task-offloading decisions and CPU frequencies. MAOC operates within a centralized training with decentralized execution (CTDE) framework, empowering individual mobile devices to make decisions autonomously based on their unique system states. Experimental results demonstrate its swift convergence and operational efficiency, and it outperforms other baseline algorithms.

SOLVING FUZZY NONLINEAR PROGRAMMING PROBLEMS WITH EXTERIOR PENALTY FUZZY VALUED FUNCTIONS

In this article, we explore the exterior penalty fuzzy valued function method for Fuzzy Nonlinear Programming Problems (FNLPP). We present methods of penalty fuzzy valued function for solving constrained optimization problems by converting to unconstrained optimization problems. Triangular fuzzy numbers are used to represent the problem’s decision variables and coefficients. Using a new fuzzy arithmetic and fuzzy ranking method on the parametric form of triangular fuzzy numbers, the optimal solution of the FNLPP is found by applying the exterior penalty fuzzy valued function method without changing to its equivalent crisp form. We present a numerical example of the suggested method and compare the results to those produced by existing methods

Dynamic modeling and contraction metric-based tube model predictive control for mineral hydraulic collection

Efficient mineral hydraulic collection is a key issue in the mining of deep-sea minerals, where the collection efficiency is related to the lifted fluid velocity and the mineral particle velocity. In this paper, a robust control contraction metric (RCCM)-based tube model predictive control (Tube-MPC) for efficient mineral hydraulic collection is proposed. Firstly, a simplified control model is designed with empirical formula of hydrodynamic forces to avoid the computational burden in solution of the fluid equations. Secondly, a nonlinear programing problem (NLP) for efficiency optimization is formulated to calculate the nominal control law and the optimal state. To address external disturbance and model mismatch from the simplified model, a robust control contraction metric is utilized to calculate the feedback control law to track the optimal state. Finally, numerical simulations and experiments are conducted to verify the mineral hydraulic collection performance of the control strategy.

SMITS: Social and Mobility aware Intelligent Task Scheduling in Vehicular Fog Computing — A Federated DRL Approach

Vehicular fog computing (VFC) has become an enticing research hot spot to provide resources for the Internet of Vehicles (IoV) application requests. Moreover, with the massive expansion of services in vehicular fog networks such as augmented reality, 3-D gaming, and autonomous driving, the allocation of fog computing resources to IoV applications constraining the deadline and energy consumption of the vehicles is becoming challenging. In this paper, we investigate a task scheduling problem in vehicular fog networks to jointly optimize the success rate of vehicular tasks and energy consumption of vehicles by considering the moving paths of the vehicles and the social relationships among the vehicles. In this problem, we model the social relationship among the vehicles based on their communication patterns and social characteristics to improve the success rate of the tasks. We also incorporate the mobility of vehicles using the Markov renewal process (MRP) technique. Initially, we formulate a mixed integer nonlinear programming (MINLP) problem for the proposed problem and further prove it to be an NP-hard problem. To solve the proposed problem, we design a federated deep reinforcement learning (FDRL) mechanism by converting the problem into a Markov decision process (MDP) problem. We compare our proposed algorithm with existing scheduling approaches, and simulation results show that the proposed algorithm performs efficiently for task scheduling problems. The proposed algorithm achieved a substantial improvement of 12% in the maximization of the success rate and 36% in the minimization of the energy consumption compared to the existing algorithms.

A cloud-edge collaborative computing framework using potential games for space-air-ground integrated IoT

As a critical component of space-air-ground integrated IoT, the aerial network provides highly reliable, low-latency and ubiquitous information services to ground users by virtue of their high mobility, easy deployment and low cost. However, the current computation and resource management model of air-ground integrated networks are insufficient to meet the latency demanding of emerging intelligent services such as autonomous systems, extended reality and haptic feedback. To tackle these challenges, we propose a computation offloading and optimization method based on potential game. First, we construct an cloud-edge collaborative computing model. Secondly, we construct Offloading Decision Objective Functions (ODOF) with the objective of minimum task processing latency and energy consumption. ODOF is proved to be a Mixed Inferior Nonlinear Programming (MINLP) problem, which is hard to solve. ODOF is converted to be a full potential game, and the Nash equilibrium solution exists. Then, a computational resource allocation algorithm based on Karush–Kuhn–Tucker (KKT) conditions is proposed to solve resource allocation problem. On this basis, a distributed game-based computational offloading algorithm is proposed to minimize the offloading cost. Extensive simulation results demonstrate that the convergence performance of the proposed algorithm is reduced by 50%, the convergence time is reduced by 13.3% and the average task processing delay is reduced by 10%.

Simultaneous Minimization of Water Age and Pressure in Water Distribution Systems by Pressure Reducing Valves

AbstractPressure reducing valves (PRVs) are essentially used to reduce operational pressures in water distribution systems (WDSs) to minimize water leakage. However, water age in a WDS is an important variable describing the water quality and should be kept as low as possible. Therefore, the aim of this study is to investigate the possibility and potential of simultaneously minimizing both pressure and water age by using PRVs. To determine the optimal location and setting of PRVs, a mixed-integer nonlinear programming (MINLP) problem is formulated with minimization of the sum of the weighted total water age and pressure as the objective function, where the weighting factor can be defined by the user’s preference. The equality constraints consist of the hydraulic equations and water age functions to describe pressure and water age in the distribution network, while the inequality constraints ensure them in the defined operating ranges, respectively. Applying the proposed approach to two case studies, the results show that both water age and pressure can indeed be significantly reduced by the optimized position and setting of the PRVs.