Mixed-integer Non-convex Optimization Problem Research Articles

Optimization of Clustering and Trajectory for Minimizing Age of Information in Unmanned Aerial Vehicle-Assisted Mobile Edge Computing Network.

With the development of the Internet of Things (IoT) technology, massive amounts of sensor data in applications such as fire monitoring need to be transmitted to edge servers for timely processing. However, there is an energy-hole phenomenon in transmitting data only through terrestrial multi-hop networks. In this study, we focus on the data collection task in an unmanned aerial vehicle (UAV)-assisted mobile edge computing (MEC) network, where a UAV is deployed as the mobile data collector for the ground sensor nodes (SNs) to ensure high information freshness. Meanwhile, the UAV is equipped with an edge server for data caching. We first establish a rigorous mathematical model in which the age of information (AoI) is used as a measure of information freshness, related to both the data collection time and the UAV's flight time. Then a mixed-integer non-convex optimization problem is formulated to minimize the peak AoI of the collected data. To solve the problem efficiently, we propose an iterative two-step algorithm named the AoI-minimized association and trajectory planning (AoI-MATP) algorithm. In each iteration, the optimal SN-collection point (CP) associations and CP locations for the parameter ε are first obtained by the affinity propagation clustering algorithm. The optimal UAV trajectory is found using an improved elite genetic algorithm. Simulation results show that based on the optimized ε, the AoI-MATP algorithm can achieve a balance between data collection time and flight time, reducing the peak AoI of the collected data.

Robust Beamforming Design for IRS-Aided Secure Communication Systems Under Complete Imperfect CSI

We investigate the performance of an intelligent reflecting surface (IRS) aided secure communication system which operates in a complete imperfect channel state information (CSI) regime. By jointly considering the imperfect CSI of both IRS-Bob and IRS-Eve channels, we propose a robust transmission scheme which is based upon a joint optimization of the continuous transmit beamforming at Alice and the discrete phase shifts at the IRS. To deal with this mixed integer non-convex optimization problem, novel lower- and upper-bounding techniques are used to transform it in a more tractable form for which a solution based upon the derivation of a quantization-based alternative optimization (AO) algorithm is presented. More specifically, on the one hand, a successive convex approximation (SCA) algorithm is proposed to optimize the continuous transmit beamforming. On the other hand, by applying a SCA, penalty convex-concave procedure (PCCP), and quantization-based algorithm, the discrete phase shifts are obtained. Various performance evaluation results obtained by means of computer simulations have shown that the proposed robust transmission scheme guarantees a minimum quality of service (QoS) while simultaneously limits Eve's maximum eavesdropping rate.

Energy-Efficient Collaborative Multi-Access Edge Computing via Deep Reinforcement Learning

The joint problem of task offloading, collaborative computing and resource allocation for Multi-access Edge Computing (MEC) is a challenging issue. In this paper, splitting computing tasks at MEC servers through collaboration among MEC servers and a cloud server, we investigate the joint problem of collaborative task offloading and resource allocation. A collaborative task offloading, computing resource allocation, and subcarrier and power allocation problem in MEC is formulated. The goal is to minimize the total energy consumption of the MEC system while satisfying a delay constraint. The formulated problem is a non-convex mixed-integer optimization problem. In order to solve the problem, we propose a Deep Reinforcement Learning (DRL) based bi-level optimization framework. The task offloading decision, computing collaboration decision, and power and subcarriers allocation subproblems are solved at the upper-level, while the computing resource allocation subproblem is solved at the lower-level. We combine Dueling-Deep-Q-Network (DQN), Double-DQN and add adaptive parameter space Noise (D <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$^{2}$</tex-math></inline-formula> N-DQN) to improve DRL performance in MEC. Simulation results demonstrate that the proposed algorithm achieves near-optimal performance in Energy Efficiency (EE) and task completion rate compared to other DRL-based approaches and other benchmark schemes under various network parameter settings.

Low complexity joint spectrum resource and power allocation for ultra dense networks

In this paper, we propose a low complexity spectrum resource allocation scheme cross the access points (APs) for the ultra dense networks (UDNs), in which all the APs are divided into several AP groups (APGs) and the total bandwidth is divided into several narrow band spectrum resources and each spectrum resource is allocated to APGs independently to decrease the interference among the cells. Furthermore, we investigate the joint spectrum and power allocation problem in UDNs to maximize the overall throughput. The problem is formulated as a mixed-integer nonconvex optimization (MINCP) problem which is difficult to solve in general. The joint optimization problem is decomposed into two subproblems in terms of the spectrum allocation and power allocation respectively. For the spectrum allocation, we model it as a auction problem and a combinatorial auction approach is proposed to tackle it. In addition, the DC programming method is adopted to optimize the power allocation subproblem. To decrease the signaling and computational overhead, we propose a distributed algorithm based on the Lagrangian dual method. Simulation results illustrate that the proposed algorithm can effectively improve the system throughput.

Joint Trajectory and Pick-Up Design for UAV-Assisted Item Delivery Under No-Fly Zone Constraints

This paper explores an item delivery scenario using an unmanned aerial vehicle (UAV). In this scenario, in the presence of multiple no-fly zones, an UAV starts from the point of departure, collects the items from ground users, and finally delivers the collected items to the destination. We mathematically characterize the item delivery scenario. Then, we formulate a problem of item value maximization to jointly optimize the UAV trajectory and item pick-up design. The formulated problem is a challenging mixed-integer non-convex optimization problem. Through some equivalent transformations, we convert the original problem into an equivalent problem with a difference-of-convex structure. Using the penalty convex-concave procedure method, we devise a low-complexity algorithm with promising performance. The numerical results demonstrate the superiority of the proposed algorithm.

Joint Communication and Computation Offloading for Ultra-Reliable and Low-Latency With Multi-Tier Computing

In this paper, we study joint communication and computation offloading (JCCO) for hierarchical edge-cloud systems with ultra-reliable and low latency communications (URLLC). We aim to minimize the end-to-end (e2e) latency of computational tasks among multiple industrial Internet of Things (IIoT) devices by jointly optimizing offloading probabilities, processing rates, user association policies and power control subject to their service delay and energy consumption requirements as well as queueing stability conditions. The formulated JCCO problem belongs to a difficult class of mixed-integer non-convex optimization problem, making it computationally intractable. In addition, a strong coupling between binary and continuous variables and the large size of hierarchical edge-cloud systems make the problem even more challenging to solve optimally. To address these challenges, we first decompose the original problem into two subproblems based on the unique structure of the underlying problem and leverage the alternating optimization (AO) approach to solve them in an iterative fashion by developing newly convex approximate functions. To speed up optimal user association searching, we incorporate a penalty function into the objective function to resolve uncertainties of a binary nature. Two sub-optimal designs for given user association policies based on channel conditions and random user associations are also investigated to serve as state-of-the-art benchmarks. Numerical results are provided to demonstrate the effectiveness of the proposed algorithms in terms of the e2e latency and convergence speed.

Trajectory Optimization and Power Allocation for Cell-Free Satellite-UAV Internet of Things

Massive access outside the coverage of terrestrial cellular networks will be the main feature of the sixth generation (6G) networks. In order to cope with it, the cognitive satellite-UAV network (CSUN) has drawn a lot of attentions. In this paper, we investigate the UAV trajectory optimization and power allocation for the cell-free CSUN consisting of one satellite and a swarm of UAVs. Indeed, due to the on-board energy constraints of UAVs, both the trajectory optimization and the power allocation can significantly save the energy to improve the energy efficiency. The joint trajectory optimization and power allocation problem is formulated as a mixed-integer non-convex optimization problem which is extremely difficult to solve. In order to reduce the computational complexity, we decompose the original optimization problem into two subproblems in terms of the trajectory optimization and power allocation. For the trajectory optimization subproblem, we model it as a Traveling Salesman Problem (TSP), and the PSO is adopted to solve it. When the trajectory variables are fixed, the power allocation subproblem is still difficult to tackle due to its non-convexity and large scale. Firstly, we present a kind of centralized algorithm in which the DC (difference of two convex functions) algorithm is applied to optimize it, and then a distributed algorithm based on auxiliary variables is proposed to reduce the signaling overhead and computational complexity. The simulation results demonstrate the effectiveness of the proposed joint trajectory optimization and power allocation algorithm for the cell-free CSUN.

IRS-User Association in IRS-Aided MISO Wireless Networks: Convex Optimization and Machine Learning Approaches

This paper concentrates on the problem of associating an intelligent reflecting surface (IRS) to multiple users in a multiple-input single-output (MISO) downlink wireless communication network. The main objective of the paper is to maximize the sum-rate of all users by solving the joint optimization problem of the IRS-user association, IRS reflection, and BS beamforming, formulated as a non-convex mixed-integer optimization problem. The variable separation and relaxation are used to transform the problem into three convex sub-problems, which are alternatively solved through the convex optimization (CO) method. The major drawback of the proposed CO-based algorithm is high computational complexity. Thus, we make use of machine learning (ML) to tackle this problem. To this end, first, we convert the optimization problem into a regression problem. Then, we solve it with feed-forward neural networks (FNNs), trained by CO-based generated data. Simulation results show that the proposed ML-based algorithm has a performance equivalent to the CO-based algorithm, but with less computation complexity due to its offline training procedure.

Deep Reinforcement Learning for Resource Allocation in Multi-Band and Hybrid OMA-NOMA Wireless Networks

Exploiting the advantages of both non-orthogonal multiple access technique and millimeter-wave communications requires joint efficient resource allocation techniques toward satisfying the stringent requirements of future mobile communication systems. This paper focuses on a multi-band (i.e., millimeter-wave band and sub-6 GHz band) wireless network where both orthogonal and non-orthogonal multiple access techniques coexist. A joint optimization of user association, transmit power allocation, sub-channel assignment, and multiple access technique selection is investigated to maximize the down-link sum-rate under a minimum rate requirement per user and power constraints. The problem is formulated as a non-convex mixed-integer optimization problem; then, it is proved to be <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {NP}$ </tex-math></inline-formula> -hard. First, simple greedy and meta-heuristic solutions are proposed. Then, since model-based approaches have generally a high computational complexity, model-free centralized and distributed approaches based on deep reinforcement learning technique are proposed. The latter are based on multiple parallel deep neural networks to generate resource allocation solutions. The proposed approaches are evaluated and compared. Simulation results corroborate the high performance offered by the proposed solutions for stationary and mobile users. They also highlight the benefits of employing hybrid orthogonal and non-orthogonal multiple access scheme in multi-band systems in terms of down-link sum-rate and user fairness.

Caching and 3D Deployment Strategy for Scalable Videos in Cache-Enabled Multi-UAV Networks

Unleashing the potential of unmanned aerial vehicles (UAVs) to play the role of caching devices for furnishing layered video services to ground users (GUs) holds tremendous promise. However, it is crucial to strike a balance between the quality and the diversity of video content when deciding the placement strategies of video caching. In this study, we examine the feasibility of implementing a layered caching scheme utilizing scalable video coding (SVC) in three-dimensional (3D) multi-UAV networks with cache capabilities. The objective is to investigate the minimization of the cumulative video access delay experienced by all GUs through the joint design of layered scalable video caching, user-UAV association, and 3D UAV deployment. This approach enables the network to offer video services across a range of quality levels. We have formulated an intricate mixed-integer non-convex optimization problem which poses a significant challenge to solve directly. Then we decompose the formulated problem into three interrelated sub-problems and put forth specific algorithms designed to address each component, i.e., layered caching strategy and UAV's two-dimensional (2D) deployment optimization with penalty based difference-of-convex (penalty-DC) framework, user-UAV association optimization with bipartite graph based linear programming (LP) relaxation, as well as UAV altitude optimization with successive convex approximation (SCA) method. Then we introduce a highly efficient iterative algorithm that employs an alternate optimization approach for three sub-problems until convergence. Our simulation results demonstrate that our proposed algorithm surpasses other benchmark schemes significantly in terms of mitigating video access delay.

Robust Ellipse Fitting Based on Maximum Correntropy Criterion With Variable Center

The presence of radically irregular data points (RIDPs), which are referred to as the subset of measurements that represents no or little information, can significantly degrade the performance of ellipse fitting methods. We develop an ellipse fitting method that is robust to RIDPs based on the maximum correntropy criterion with variable center (MCC-VC), where an adaptable Laplacian kernel is used. For single ellipse fitting, we formulate a non-convex optimization problem and divide it into two subproblems, one to estimate the kernel bandwidth and the other the kernel center. We design sufficiently accurate convex approximation to each subproblem that will lead to computationally efficient closed-form solutions. The two subproblems are solved in an alternate manner until convergence is reached. We also investigate coupled ellipses fitting. While there exist multiple ellipses fitting methods in the literature, we develop a coupled ellipses fitting method by exploiting the underlying special structure, where the associations between the data points and ellipses are absent in the problem. The proposed method first introduces an association vector for each data point and then formulates a non-convex mixed-integer optimization problem to establish the data associations, which is approximately solved by relaxing it into a second-order cone program. Using the estimated data associations, we then extend the proposed single ellipse fitting method to accomplish the final coupled ellipses fitting. The proposed method is shown to perform significantly better than the existing methods using both simulated data and real images.

FlyingLoRa: Towards energy efficient data collection in UAV-assisted LoRa networks

Energy efficiency is an increasingly crucial consideration due to the battery-powered end devices in LoRa networks. With the adoption of chirp spread spectrum modulation, end devices far apart from the gateway have to use a high spreading factor and transmit power to send uplink packets, which causes longer transmission time and higher energy consumption compared to the end devices near the gateway, and further causes unfairness issues on energy efficiency. To tackle this problem, we investigate a novel energy-efficient data collection scheme named FlyingLoRa, where an unmanned aerial vehicle carrying one gateway is dispatched to harvest packets from end devices. The goal of FlyingLoRa is to minimize the energy consumption of end devices for packet transmission by jointly optimizing the 3D UAV trajectory, scheduling strategies, and transmission parameters of end devices. We formulate our design as a mixed-integer non-convex optimization problem and propose an efficient iterative algorithm to find a sub-optimal solution. The proposed approach is numerically simulated and the results show that FlyingLoRa improves energy efficiency by 16.13× on average compared with the existing fixed gateway schemes. In addition, we realize FlyingLoRa in real scenarios and evaluate its performance by presenting the actual packet reception rate (PRR) and transmission energy consumption, which shows its effectiveness.

Alternative Optimization for Secrecy Throughput Maximization in UAV-Aided NOMA Networks

With the rapid boost of wireless networks, it is challenging to guarantee the secure transmission and massive connectivity in the process of data collection. This letter considers an unmanned aerial vehicle (UAV)-aided non-orthogonal multiple access (NOMA) communication network, where a UAV is deployed to collect the data from sensing devices (SDs) in the NOMA manner subject to the eavesdropping attack, and a group of auxiliary devices (ADs) are deployed to provide the cooperative jamming to the eavesdropper. We aim to maximize the total secrecy throughput by jointly optimizing SDs’ transmit power and ADs’ scheduling decisions. We first formulate this problem as a mixed-integer non-convex optimization problem, which turns out to be mathematically intractable due to the non-convexity. Then, an alternative optimization algorithm is developed to find a suboptimal solution. Specifically, the successive convex approximation approach and the cross-entropy technique are used to solve the two sub-problems decomposed from the optimization problem, respectively. Numerical results demonstrate that the proposed algorithm can achieve considerable performance compared with the benchmark schemes.

UAV-Assisted Fair Communication for Mobile Networks: A Multi-Agent Deep Reinforcement Learning Approach

Unmanned Aerial Vehicles (UAVs) can be employed as low-altitude aerial base stations (UAV-BSs) to provide communication services for ground users (GUs). However, most existing works mainly focus on optimizing coverage and maximizing throughput, without considering the fairness of the GUs in communication services. This may result in certain GUs being underserviced by UAV-BSs in pursuit of maximum throughput. In this paper, we study the problem of UAV-assisted communication with the consideration of user fairness. We first design a Ratio Fair (RF) metric by weighting fairness and throughput to evaluate the tradeoff between fairness and communication efficiency when UAV-BSs serve GUs. The problem is formulated as a mixed-integer non-convex optimization problem based on the RF metric and we propose a UAV-Assisted Fair Communication (UAFC) algorithm based on multi-agent deep reinforcement learning to maximize the fair throughput of the system. The UAFC algorithm comprehensively considers fair throughput, UAV-BSs coverage, and flight status to design a reasonable reward function. In addition, the UAFC algorithm establishes an information sharing mechanism based on gated functions by sharing neural networks, which effectively reduces the distributed decision-making uncertainty of UAV-BSs. To reduce the impact of state dimension imbalance on the convergence of the algorithm, we design a new state decomposing and coupling actor network architecture. Simulation results show that the proposed UAFC algorithm increases fair throughput by 5.62%, 26.57% and fair index by 1.99%, 13.82% compared to the MATD3 and MADDPG algorithms, respectively. Meanwhile, UAFC can also meet energy consumption limitation and network connectivity requirement.

Fairness-aware resource allocation for D2D-enabled IoT in NOMA-based cellular networks with mutual successive interference cancellation

Device-to-device (D2D) communications and non-orthogonal multiple access (NOMA) are promising technologies to meet the growing demand for IoT-connected devices. However, they bring about new challenges including the co-channel interference, that can limit the performance improvement. To manage the co-channel interference, we address the problem of joint power allocation and sub-channel assignment for D2D-enabled IoT devices (IoTDs) underlaying a NOMA-based cellular network, in which the successive interference cancellation (SIC) decoding is enabled at the level of IoTDs and cellular user equipment (CUE)to increase the number of connected devices and the capacity. This problem is modeled as a mixed-integer nonconvex optimization problem which includes the concept of fairness with respect to the data rates of IoTDs. To solve the problem, a semi-distributed algorithm is developed, which is of polynomial time complexity. The proposed algorithm leverages the successive convex approximation and a heuristic approach. Evaluation results demonstrate the efficiency of the proposed scheme with respect to the sum rate, fairness, access rate and computational complexity.

JDAPCOO: Resource Scheduling and Energy Efficiency Optimization in 5G and Satellite Converged Networks for Power Transmission and Distribution Scenarios.

Along with the continuous revolution of energy production and energy consumption structures, the information data of smart grids have exploded, and effective solutions are urgently needed to solve the problem of power devices resource scheduling and energy efficiency optimization. In this paper, we propose a fifth generation (5G) and satellite converged network architecture for power transmission and distribution scenarios, where power transmission and distribution devices (PDs) can choose to forward power data to a cloud server data center via ground networks or space-based networks for power grid regulation and control. We propose a Joint Device Association and Power Control Online Optimization (JDAPCOO) algorithm to maximize the long-term system energy efficiency while guaranteeing the minimum transmission rate requirement of PDs. Since the formulated issue is a mixed integer nonconvex optimization problem with high complexity, we decompose the original problem into two subproblems, i.e., device association and power control, which are solved using a genetic algorithm and improved simulated annealing algorithm, respectively. Numerical simulation results show that when the number of PDs is 50, the proposed algorithm can improve the system energy efficiency by 105%, 545.05% and 835.26%, respectively, compared with the equal power allocation algorithm, random power allocation algorithm and random device association algorithm.

Joint Resource Allocation Optimization for the Wireless Backhaul Link of User-Centric Cell-Free NOMA Networks

In this paper, we investigate the downlink resource allocation problem for the user-centric ultra-dense networks (UUDN) which can provide high area throughput density. UUDN can effectively improve the user experience since one user can be served by more than one access points (APs) simultaneously. However, the ultra-dense node deployment will also result in serious co-channel interference which will degrade the performance of the networks. Nonorthogonal multiple access (NOMA) is introduced to reduce the interference caused by the network densification. We consider the resource allocation for both the wireless backhaul downlink and the user access downlink at the same time in UUDN with NOMA, which involves the APs grouping and power allocation. The resource allocation problem is formulated as a mixed integer nonconvex optimization problem due to the combinatorial nature of APs grouping. We first decompose the problem into the APs grouping subproblem and power allocation subproblem to reduce the computational complexity. For the APs grouping subproblem, we model it as a bipartite graph maximum weight matching problem, and the Hungarian algorithm is used to solve it. When the APs grouping is fixed, the original problem is still difficult to tackle since the power allocation is a max-min optimization problem. The KKT conditions are applied to optimize the power allocation subproblem. Simulation results illustrate that the proposed resource allocation algorithm can effectively improve the system throughput of UUDN.

Network Energy-Efficiency Maximization in UAV-Enabled Air–Ground-Integrated Deployment

The unmanned aerial vehicle (UAV) technique combined with the cellular network is a promising and reliable technique to provide a low-cost, flexible, and easy-to-expandable network. In this article, we propose a UAV-assisted cooperative transmission network (UCTN), where UAVs and the base station (BS) transmit data jointly through the cooodination of the software-defined network (SDN). As the UAV has limited on-board energy, we aim to maximize the network energy efficiency (EE) by jointly optimizing the UAV–user association, UAV location, BS resource allocation, and the load allocation between the two systems. The formulated problem is a mixed-integer nonconvex optimization problem, which is intractable. To solve it, alternative optimization (AO) and Dinkelbach’s method are adopted to recast the original problem into a tractable form. After that, the corresponding problem is divided into three subproblems and can be solved iteratively. An energy-constrained coalition game is proposed to solve the UAV–user association problem, and the UAV hovering point is carefully designed to balance the coalition serving time. Furthermore, the successive convex approximation (SCA) technique is adopted to optimize the BS transmit power and bandwidth allocation. Numerical experiments demonstrate that the proposed algorithm achieves significant performance gains over other benchmark schemes.

Locating facilities under competition and market expansion: Formulation, optimization, and implications

As the ongoing battle between brick‐and‐mortar stores and e‐commerce shops escalates, managers of the former are becoming more cautious regarding their strategic store site selection and decoration decisions, particularly if foreseeable competition from rival companies exists. This paper investigates a bilevel competitive facility location (BCFL) problem, where two companies, a leader and a follower, plan to enter a market sequentially. Each company has a budget to open and design facilities. The goal is to maximize expected revenue that is forecasted through a discrete choice model. To reflect a practical environment, we further consider a situation with elastic demand, explaining the market expansion effect when customers are offered better service due to the opening of new facilities. We formulate the problem as a nonlinear 0‐1 bilevel program. Because of the bilevel structure and the market expansion effect, this problem is so challenging that we are unaware of any exact algorithms in the literature. Motivated by this gap, we develop an exact framework that leverages the state‐of‐the‐art value‐function–based approach. However, this framework requires solving a mixed‐integer nonconvex optimization problem (MINOP) at each iteration, which is computationally prohibitive even for medium‐scale instances. To mitigate the intractability, we propose a new framework that avoids MINOP and tackles instances with hundreds of location variables. Finally, we conduct extensive computational studies to show the efficiency and effectiveness of our method as well as provide insightful guidance for managers to have win–win/dominate outcomes and choose an appropriate market size function when dealing with expansion decisions in chained business operations.

Voltage Restoration After Unforeseen Disturbances in Weakly Observable Distribution Systems

Following an unforeseen disturbance in a power system, the system state should be known to extract the emergency voltage restoration strategy. Such a condition does not usually hold in distribution systems due to the lack of enough measurements. Here, the joint probability density function ( <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pdf</small> ) of uncertain parameters is extracted from historical data. This <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pdf</small> is updated to comply with the measurements. A set of samples is then generated that best models the updated <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pdf</small> . These data reconciliation and sampling techniques enable the functioning of the emergency voltage control ( <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">evc</small> ) with measurement scarcity. They are designed to be fast enough to meet the quasi-real-time requirements of the intended application. The <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">evc</small> problem is cast as a stochastic programming problem. The resultant problem is a mixed-integer non-convex (and hence, NP-hard) optimization problem. It is solved using five distinct <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">sm</small> s based on relaxation/approximation of the non-convex constraints. The results show that a novel combination of relaxation and approximation aimed at respectively mitigating the emergency under- and over-voltages outperforms the others. The propounded stochastic <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">evc</small> is validated through numerical studies.