Lagrangian Duality Research Articles

AoI Optimization in the UAV-Aided Traffic Monitoring Network Under Attack: A Stackelberg Game Viewpoint

Intelligent Vehicle Systems (IVSs) devote to integrating the data sensing, processing, and transmission in the Vehicle to Everything (V2X) scenarios, where the Unnamed Aircraft Vehicle (UAV)-aided traffic monitoring network is one of the most significant applications. Moreover, since the central premise to support the IVS is timely and effectively sensing data processing, Age of Information (AoI) can precisely reflect the timeliness and effectiveness of the communication process in the UAV-aided traffic monitoring network. However, recent researches pay little attention to AoI minimization issue, especially when the malicious attacker attempts to deteriorate the network performance. The accurately modelling of the adversarial relationship between legitimate UAVs and attacker is not fully investigated. To make up this research gap, we start from the Stackelberg game viewpoint to investigate the AoI optimization problem in the UAV-aided traffic monitoring network under attack. Firstly, the system model and three-layer Stackelberg game-based optimization goal are established. Secondly, based on the Backward Induction (BI) analysis, the follower’s data sensing rate, transmission power, and the leader’s attacking power are determined by the Lagrange duality optimization technology successively. Moreover, the sub-gradient update-based optimization technology is used to achieve the Stackelberg Equilibrium (SE). Finally, simulations are performed under various parameters. The evaluation results present better performance of our proposed approach when compared with the typical baselines.

Lagrange duality of vector variational inequalities

Lagrange duality of vector variational inequalities

An anytime algorithm for constrained stochastic shortest path problems with deterministic policies

Sequential decision-making problems arise in every arena of daily life and pose unique challenges for research in decision-theoretic planning. Although there has been a wide variety of research in this field, most of the studies have largely focused on single objective problem without constraints. In many real-world applications, however, it is often desirable to bound certain costs or resources under some predefined level. Constrained stochastic shortest path problem (C-SSP), one of the most well-known mathematical frameworks for stochastic decision-making problems with constraints, can formally model such problems, by incorporating constraints in the model formulation. However, it remains an open challenge to produce a deterministic optimal policy with desirable computation time due to its intrinsic complexity.In this paper, we propose a method that produces an optimal and deterministic policy for a C-SSP based on the Lagrangian duality theory and the heuristic forward search method. To address the intrinsic complexity of C-SSP, the proposed method is designed to have an anytime property. In other words, the proposed algorithm tries to find a feasible but decent solution quickly, then improves the solution incrementally until it converges to a true optimal solution. An extensive experimental evaluation on three problem domains shows that the proposed method outperforms the state-of-the-art methods in terms of the near-optimal solution with an optimality gap of less than 0.1%.

Worst CVaR based energy management for generalized energy storage enabled building-integrated energy systems

Generalized energy storage (GES) can reduce the renewable energy curtailment, carbon emission and operational cost risk in the building-integrated energy system (BIES) energy management when faced with uncertainties of multi-energy loads and PV. In this context, GES integrated with the differentiated building thermal load-based virtual energy storage (VES) is proposed in the energy management model of BIES. Specifically, the differentiated building thermal load VES is introduced by differentiated personal factors (e.g., metabolic rate and clothing insulation) based on the predicted mean vote. Furthermore, a worst conditional value at risk (WCVaR) energy management method is developed to evaluate the operational cost risk of BIES. Meanwhile, possible probability distribution uncertainties of PV, electric and thermal loads are all fully considered by a boxlike uncertainty set. Then, the proposed WCVaR model is transformed into a linearized model by the Lagrange duality theory. Finally, the whole energy management model is solved by the CPLEX solver. Case studies and comparative analysis are performed based on a representative BIES. Numerical results show that the proposed model can reduce 33.55% of operational cost, 74.25% of carbon trading cost, and 16.96% of PV curtailment cost for BIES compared with that without GES. In addition, the conditional value at risk (CVaR) with the proposed WCVaR method ($5700.22) is between that with the traditional CVaR method considering multiple operational scenarios ($5684.68) and that with the traditional CVaR method only including the worst operational scenario ($6043.48), which indicates that the proposed WCVaR method can avoid overly-high and overly-low operational cost risks.

Online ADMM for Distributed Optimal Power Flow via Lagrangian Duality

At present, the power system has the characteristics of mutual independence but interconnection, and the interconnection between the various subsystems brings certain challenges to the distributed computing of the power grid. In addition, a substantial amount of naturally uncertain renewable resources are incorporated into the power system, which will impose volatile dynamics on the grid. In this paper, an alternating direction multiplier method (ADMM) is proposed for the power system with real-time renewables to tackle the online optimal power flow (OPF) problem. Due to the adoption of the Lagrangian duality, the proposed distributed ADMM scheme utilizes consensus ADMM to solve the dual OPF problem, which only discloses boundary coupling via the Lagrangian multiplier and further reduces the amount of information communication. Given the natural uncertainty of distributed energy resources (DER), the algorithm avoids the double-loop implementation or the uncertainty of traditional distributed methods of using the boundary information as equality constraints caused by dynamic DER. It is thus capable of providing a provable performance guarantee and is inherently developed to cope with the dynamic OPF problem with renewables in an online fashion. Taking the IEEE 30-bus system as a test feeder, the simulation results verify the efficiency and robustness of the proposed algorithms in solving both the static and dynamic OPF problems; in addition, the online method can effectively avoid the violent fluctuations of the conventional generator output copying with renewables rapid variation in comparison with the offline algorithms.

How Thin and Efficient Can a Metasurface Reflector Be? Universal Bounds on Reflection for Any Direction and Polarization

AbstractLight reflection plays a crucial role in a number of modern technologies. In this paper, analytical expressions for maximal reflected power in any direction and for any polarization are given for generic planar structures made of a single material represented by a complex scalar susceptibility. The problem of optimal light‐matter interactions to maximize reflection is formulated as the solution of an optimization problem in terms of the induced currents, subject to energy conservation and passivity, which admits a global upper bound by using Lagrangian duality. The derived upper bounds apply to a broad range of planar structures, including metasurfaces, gratings, homogenized films, photonic crystal slabs, and more generally, any inhomogeneous planar structure irrespective of its geometrical details. These bounds also set the limit on the minimum possible thickness, for a given lossy material, to achieve a desired reflectance. Moreover, the results allow the discovery of parameter regions where large improvements in the efficiency of a reflective structure are possible compared to existing designs. Examples are given of the implications of these findings for the design of superior and compact reflective components made of real, imperfect (i.e., lossy) materials, such as ultra‐thin and efficient gratings, polarization converters, and light‐weight mirrors for solar/laser sails.

IRS-Aided Uplink Security Enhancement via Energy-Harvesting Jammer

In this paper, we investigate the security enhancement by combining intelligent reflecting surface (IRS) and energy harvesting (EH) jammer for the uplink transmission. Specifically, we propose an IRS-aided secure scheme for the uplink transmission via an EH jammer, to fight against the malicious eavesdropper. The proposed scheme can be divided into an energy transfer (ET) phase and an information transmission (IT) phase. In the first phase, the friendly EH jammer harvests energy from the base station (BS) aided by IRS. We maximize the harvested energy of jammer by obtaining the closed-form solution to the phase-shift matrix of IRS. In the second phase, the user transmits confidential information to the BS while the jamming is generated to confuse the eavesdropper without affecting the legitimate transmission. The phase-shift matrix of IRS and time switching factor are jointly optimized to maximize the secrecy rate. To tackle the non-convex problem, we first decompose it into two sub-problems. The one of IRS can be approximated to convex with fixed time switching factor. Then, the time switching factor can be solved by Lagrange duality. Thus, the solution to the original problem can be obtained by alternately optimizing these two sub-problems. Simulation results show that the proposed Jammer-IRS assisted secure transmission scheme can significantly enhance the uplink security.

Upper bounds on focusing efficiency.

Upper bounds on the focusing efficiency of aperture fields and lens systems are formulated using integral equation representations of Maxwell's equations and Lagrangian duality. Two forms of focusing efficiency are considered based on lens exit plane fields and optimal polarization currents within lens design regions of prescribed shape and available materials. Bounds are compared against the performance of classical prescriptions of ideal lens aperture fields, hyperbolic lens designs, and lenses produced by inverse design. Results demonstrate that, without regularization, focusing efficiency based solely on lens exit plane fields is unbounded, similar to the problem of unbounded antenna directivity. Additionally, results considering extruded two-dimensional dielectric geometries driven by out-of-plane electric fields for the calculation of bounds and inverse design demonstrate that aperture fields based on time-reversal do not necessarily yield optimal lens focusing efficiency, particularly in the case of near-field (high numerical aperture) focusing.

Joint Beamforming Aided Over-the-Air Computation Systems Relying on Both BS-Side and User-Side Reconfigurable Intelligent Surfaces

Over-the-air computation (AirComp) has received substantial attention, given its ability to aggregate massive amounts of data from distributed wireless devices (WDs). However, the computation accuracy at the fusion center (FC) may be severely affected by receiving data corrupted by the poor channel conditions. To mitigate this issue, we consider the employment of reconfigurable intelligent surfaces (RISs) in the AirComp system considered for improving the quality of received data, and hence improve the computation accuracy. However, most previous contributions on RIS-assisted AirComp systems only employ a single RIS in the resultant single-RIS-assisted (SRIS-assisted) AirComp systems. We develop this concept further for mitigating the deleterious channel effects by conceiving a double-RIS-assisted (DRIS-assisted) AirComp system, where one of the RISs is located near the WDs and the other in the vicinity of the FC. We theoretically prove that the DRIS-assisted AirComp system outperforms its SRIS-assisted counterpart in terms of the resultant computation mean-squared-error (MSE). Furthermore, we propose a pair of algorithms for jointly optimizing the transmit power at the WDs, the receive beamforming vector at the FC, and the passive beamforming matrices at the RISs for minimizing the computational MSE. Specifically, the transmit power is updated by exploiting the Lagrange duality method, while the receive beamforming vector is optimized by utilizing the first-order optimality condition. Furthermore, a pair of techniques are developed for optimizing the passive beamforming matrices at the RISs based on semidefinite relaxation (SDR) and penalty-duality-decomposition (PDD), respectively. Both the complexity and the convergence of the proposed algorithms are analyzed. Finally, simulation results are provided for quantifying the overall performance of the resultant DRIS-assisted AirComp system.

Proactive Eavesdropping via Jamming Over Short Packet Suspicious Communications With Finite Blocklength

Short packet communications play key roles in the Internet-of-Things. Conventional Shannon’s coding theorem is not applicable for short packet communications, and the achievable rate in the finite blocklength regime is related to the blocklength and the decoding error probability. Contrary to conventional physical layer security, wireless information surveillance assumes that the communication users are suspicious users and the eavesdroppers are legitimate monitors, and proactive eavesdropping via jamming has been proposed to improve the eavesdropping performance. Existing works on proactive eavesdropping ignored the scenario when the suspicious users adopt the short packet communications. Thus, this paper tries to develop effective proactive eavesdropping schemes suitable for the short packet suspicious communications. Under the truncated channel inversion power allocation and the modified constant power allocation policies at the suspicious source, the problems of jamming power allocation for maximizing the monitoring success probability subject to the average transmit power constraint at the monitor are investigated. Based on the Dinkelbach-type algorithm, the Lagrange duality method and a novel tractable approximate expression for the decoding error probability, we derive suboptimal solutions to the problems. Simulation results demonstrate the effectiveness of the proposed solutions compared to various benchmark algorithms in existing literature.

Trace expressions and associated limits for nonequilibrium Casimir torque

We exploit fluctuational electrodynamics to present trace expressions for the torque experienced by arbitrary objects in a passive, non-absorbing, rotationally invariant background environment. Specializing to a single object, this formalism, together with recently developed techniques for calculating bounds via Lagrange duality, is then used to derive limits on the maximum Casimir torque that a single object with an isotropic electric susceptibility can experience when out of equilibrium with its surrounding environment. The maximum torque achievable at any wavelength is shown to scale in proportion to body volumes in both subwavelength (quasistatics) and macroscopic (ray optics) settings, and come within an order of magnitude of achievable torques on topology optimized bodies. Finally, we discuss how to extend the formalism to multiple bodies, deriving expressions for the torque experienced by two subwavelength particles in proximity to one another.

Duality for composite optimization problem within the framework of abstract convexity

ABSTRACT We study conjugate and Lagrange dualities for composite optimization problems within the framework of abstract convexity. We provide conditions for zero duality gap in conjugate duality. For Lagrange duality, intersection property is applied to obtain zero duality gap. Connection between Lagrange dual and conjugate dual is also established. Examples related to convex and weakly convex functions are given.

Lagrangian Dual Decision Rules for Multistage Stochastic Mixed-Integer Programming

On Decision Rules for Multistage Stochastic Programs with Mixed-Integer Decisions Multistage stochastic programming is a field of stochastic optimization for addressing sequential decision-making problems defined over a stochastic process with a given probability distribution. The solution to such a problem is a decision rule (policy) that maps the history of observations to the decisions. Design of the decision rules in the presence of mixed-integer decisions is quite challenging. In “Lagrangian Dual Decision Rules for Multistage Stochastic Mixed-Integer Programming,” Daryalal, Bodur, and Luedtke introduce Lagrangian dual decision rules, where linear decision rules are applied to dual multipliers associated with Lagrangian duals of a multistage stochastic mixed-integer programming (MSMIP) model. The restricted decisions are then used in the development of new primal- and dual-bounding methods. This yields a new general-purpose approximation approach for MSMIP, free of strong assumptions made in the literature, such as stagewise independence or existence of a tractable-sized scenario-tree representation.

RIS-Assisted Receive Quadrature Space-Shift Keying: A New Paradigm and Performance Analysis

Reconfigurable intelligent surfaces (RISs) represent a promising candidate for sixth-generation (6G) wireless networks, as the RIS technology provides a new solution to control the propagation channel in order to improve the efficiency of a wireless link through enhancing the received signal power. In this paper, we propose RIS-assisted receive quadrature space shift keying (RIS-RQSSK), which enhances the spectral efficiency of an RIS-based index modulation (IM) system by using the real and imaginary dimensions independently for the purpose of IM. Therefore, the error rate performance of the system is improved as all RIS elements reflect the incident transmit signal toward both selected receive antennas. At the receiver, a low-complexity but effective greedy detector (GD) can be employed which determines the maximum energy per dimension at the receive antennas. A max-min optimization problem is defined to maximize the received signal-to-noise ratio (SNR) components at both selected receive antennas; an analytical solution is provided based on Lagrange duality. In particular, the multi-variable optimization problem is shown to reduce to the solution of a single-variable equation, which results in a very simple design procedure. In addition, we investigate the average bit error probability (ABEP) of the proposed RIS-RQSSK system and derive a closed-form approximate upper bound on the ABEP. We also provide extensive numerical simulations to validate our derivations. Numerical results show that the proposed RIS-RQSSK scheme substantially outperforms recent prominent benchmark schemes. This enhancement considerably increases with an increasing number of receive antennas.

Joint Antenna and Device Scheduling in Full-Duplex MIMO Wireless-Powered Communication Networks

In this article, we study a joint antenna and Internet of Things device (ID) scheduling problem in the full-duplex (FD) multiple-input–multiple-output (MIMO) wireless-powered communication network (WPCN) over time-varying fading channels. We first formulate an optimization problem to maximize the average sum rate of IDs while satisfying their minimum average data rate requirements by jointly scheduling ID selection for uplink data transmission, antenna switching, and beamforming. To deal with the problem, we propose a scheduling algorithm based on Lagrangian duality and the stochastic optimization theory. The proposed scheduling algorithm necessitates solving per-time-slot problems, each of which aims at maximizing the weighted sum of the selected ID’s uplink data rate and the nonselected IDs’ harvested power from the downlink by jointly optimizing ID selection, antenna switching between uplink and downlink, and beamforming at that time slot. To solve the per-time-slot problem, we develop a joint ID selection, antenna switching, and beamforming (Joint-IAB) algorithm based on the block coordinate descent (BCD) and successive convex approximation (SCA) methods. Through simulation, we demonstrate that our scheduling algorithm with the proposed Joint-IAB algorithm provides better performance than the other scheduling algorithms while well satisfying the given minimum average data rate requirements of IDs.

Cellular-Connected Multi-UAV MEC Networks: An Online Stochastic Optimization Approach

In this paper, we consider a mobile edge computing (MEC) network where multiple cellular-connected unmanned aerial vehicles (UAVs) can offload their computation tasks to multiple ground base stations (GBSs). In practice, the UAVs are generally unable to master stochastic information of task arrival and channel changes in advance, which may cause a severe issue in terms of energy consumption. Therefore, we formulate a stochastic optimization problem with the goal of minimizing the average weighted sum energy consumption, by jointly optimizing UAV-GBS associations, communication and computation resource allocation, and three-dimensional (3D) UAV trajectories, during which a velocity-triggered penalty term (VTPT) is designed to suppress a large amount of the energy consumption of the UAVs. To handle the stochastic problem, we propose an online resource allocation and trajectory optimization algorithm with outer and inner structures. The outer structure transforms the original problem to a deterministic one by applying the Lyapunov-based optimization framework. The inner structure solves the obtained deterministic problem via the Lagrange duality method and the successive convex approximation technique, based on the block coordinate descent framework. Numerical results demonstrate that: 1) VTPT dramatically decreases the UAVs’ energy consumption, and 2) the proposed algorithm not only reduces the energy consumption but also ensures the computation queue stability compared with other benchmark schemes.

Integrated optimization model for hierarchical service network design and passenger assignment in an urban rail transit network: A Lagrangian duality reformulation and an iterative layered optimization framework based on forward-passing and backpropagation

This study solves an integrated operational problem regarding hierarchical service network design and passenger assignment for urban rail transit systems. We propose an innovative nonlinear programming model for determining the number of stocking trains at each depot, number of operating trains on each line, and line-based service frequency and capacity. Given a certain passenger demand matrix, this model simultaneously determines the system-optimal path flow while assigning passengers to lines to minimize the passenger total travel cost. The proposed nonlinear programming model is then reformulated based on Lagrangian duality as two resource allocation sub-problems represented as artificial neural networks. The forward pass of the train flow sequentially assigns train resources to candidate depots and lines, and the forward pass of the passenger flow sequentially assigns the passenger demand to candidate paths and links. The solution can be improved by backpropagation of the first-order gradients and re-assignment of the train resources and passenger demand with updated weights between different layers under the proposed layered optimization framework. A comparative analysis indicates that the proposed solution approach can obtain an approximate optimal solution for the integrated optimization model, thereby providing an optimized operational hierarchical service plan and system-optimal passenger assignment results. The proposed methodology and solution approach are evaluated on a simple network case and Beijing Metro Network case.

α-Fairness-Maximizing User Association in Energy-Constrained Small Cell Networks

Renewable energy source (RES)-powered base stations have received tremendous research interest in recent years because they can expand network coverage without building a power grid. This paper proposes a novel user association (UA), resource allocation (RA), and dynamic power control (PC) scheme to maximize the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> -fairness in RES-assisted small cell networks. The <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> -fairness is a general notion that flexibly adjusts the balance between the throughput, proportional fairness, and max-min fairness according to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> . Nevertheless, none of the existing studies has proposed UA, RA, and PC to maximize the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> -fairness due to its NP-hardness. Furthermore, fixed-policy-based PC designs cannot consider time-varying environments (e.g., energy harvesting models and wireless channels) of the RES-assisted networks. We first provide a Lagrangian duality-based algorithm to solve the UA and RA problem for a fixed PC. Next, we propose a dynamic PC scheme based on deep reinforcement learning (DRL) that chooses the best PC considering the time-varying environments. However, because the UA and RA algorithm executed in each step of the dynamic PC requires a long computation time, we aim to accelerate the computation of the UA and RA with DRL. Inspired by the Lagrangian duality, we design a DRL-based UA and RA with a low-dimensional continuous variable by relaxing the UA variable, the cardinality of which increases exponentially with the number of base stations and users. The simulation results show that the proposed scheme achieves a 100 times shorter computation time than the optimization-based schemes by computing only two neural networks. In particular, although there have been numerous studies on the proportional fairness maximization, the proposed scheme outperforms the optimization-based schemes in the throughput, proportional fairness, and max-min fairness metrics.

Energy-Efficient beyond 5G Multiple Access Technique with Simultaneous Wireless Information and Power Transfer for the Factory of the Future

In the Industrial Internet of Things (IIoT), non-orthogonal multiple access (NOMA) has emerged as a viable multiple access method due to its superior efficiency. In this paper, a new power allocation technique for NOMA-enabled IIoT devices is presented with trade-offs between increasing energy efficiency and decreasing power consumption. We present a joint optimization of transmission rate and energy harvesting in simultaneous wireless information and power transfer (SWIPT) NOMA-enabled IIoT devices. With the power splitting (PS) approach, we examine how to improve overall transmission rate and harvested energy, simultaneously, while fulfilling the minimum rate and harvested energy needs of each IIoT device in a SWIPT-enabled NOMA system. An objective function is established by adding transmission rates obtained from information decoding and the transformed throughput from energy harvesting. The combination of management approaches with Industry 4.0 technology provides a viable strategy to decrease industrial production’s energy use. Several performance metrics may be utilized to study manufacturing process optimization. The efficiency of production equipment may be measured by looking at the overall effectiveness (OE) of the equipment in use. We divide the non-convex optimization problem into two sub-problems, based on the Lagrangian duality method, and solve them to find the optimal solution for the non-convex problem. The approach is validated based on physical layer parameter settings that represent potential factory of the future scenarios. Simulation results confirm the effectiveness of the presented method in a SWIPT-enabled NOMA system, provide considerable performance gains over the classic rate maximization strategy, and demonstrate the energy efficiency of the presented method compared with the conventional system. The results show huge potential of our solutions to reduce the future huge energy demand related to factory automation.

Joint Subcarrier and Transmission Power Allocation in OFDMA-Based WPT System for Mobile-Edge Computing in IoT Environment

Mobile-edge computing (MEC) is expected to play an important role in the next-generation of Internet-of-Things (IoT) services with artificial intelligence (AI) by providing the sustainable computation capability of resource-constrained IoT devices. Since the finite battery lifetime has been a longstanding challenge of the MEC system for IoT services, the wireless power transfer (WPT) technology has been recently developed for the MEC system in order to support the perpetual operation of IoT devices. In this article, we introduce two resource allocation problems for OFDMA-based WPT-MEC systems: 1) a max–min energy fairness (MMEF) problem and 2) a power sum maximization (PSM) problem. These problems ensure high-performance computations for AI-based applications, where the network reliability and the tremendous power consumption may be required. Moreover, we incorporate a logarithmic nonlinear energy harvesting (EH) model into our formulated problems, which lead to nonconvex mixed-integer nonlinear programming (MINLP) problems. In order to resolve these NP-hard problems, we convert the proposed problems into their equivalent convex forms by applying the continuous relaxation method. The near-optimal solutions are thereby obtained in closed-form expressions by leveraging the Lagrangian duality method for each relaxed problem. Numerical results are presented to validate the merits of the proposed algorithms of MMEF&PSM over the alternative benchmark algorithm and provide significant insights on the effects of the key system parameters, including the number of IoT devices and power transmission subcarriers.