Concave Optimization Research Articles

Online Multi-Resource Allocation for Network Slicing in 5G with Distributed Algorithms

With the increasing scale of mobile cellular network applications, 5G mobile network infrastructure provides customizable services to users in the form of network slices. How to effectively allocate existing resources in real dynamic networks with time-varying network utility is a key issue that previous work did not consider. This paper first initializes the multi-resource allocation problem of network slicing in an online manner, where the utility function is set to change over time. Therefore, we propose Metis, an online network slicing resource allocation framework that combines the time-varying nature of the network utility function given bandwidth and processing power constraints with the requirement of virtual network function isolation. The goal is to maximize the cumulative network utility in the long term and specify multiple resource allocation problems by utilizing concave optimization methods. In addition, a distributed algorithm based on the online alternating direction method of multipliers with regret optimization has been developed to achieve optimal resource allocation. Our mathematical analysis proves that Metis can provably converge to the optimal solution and the result of experiments demonstrates a steady state behavior of Metis, which converges in dynamic network settings.

Regularized online exponentially concave optimization

In this paper, we investigate regularized online exponentially concave (abbr. exp-concave) optimization, in which each loss function consists of a time-varying exp-concave function and a fixed convex regularization. If the whole loss function is exp-concave, a classical method called online Newton step (ONS) enjoys an O(dlogT) regret bound, where d is the dimensionality and T is the time horizon. However, in the regularized setting, the sum of an exp-concave function and a convex regularization is not necessarily an exp-concave function, which implies that ONS is not applicable. To address this problem, we propose the proximal online Newton step (ProxONS), and show that it can attain the same O(dlogT) regret bound for any convex regularization. The main idea is to first perform an iteration of ONS with the exp-concave part in each loss function and then perform a proximal mapping with the regularization part. Furthermore, we demonstrate that by utilizing the standard online-to-batch conversion, our ProxONS can be extended to solve stochastic optimization with a regularized exp-concave objective, and enjoy an O(dlogT/T) convergence rate with high probability. Experimental results on two real datasets verify the effectiveness of our ProxONS.

On the equivalence between Value-at-Risk- and Expected Shortfall-based risk measures in non-concave optimization

We study a non-concave optimization problem in which an insurance company maximizes the expected utility of the surplus under a risk-based regulatory constraint. The non-concavity does not stem from the utility function, but from non-linear functions related to the terminal wealth characterizing the surplus. For this problem, we consider four different prevalent risk constraints (Expected Shortfall, Expected Discounted Shortfall, Value-at-Risk, and Average Value-at-Risk), and investigate their effects on the optimal solution. Our main contributions are in obtaining an analytical solution under each of the four risk constraints in the form of the optimal terminal wealth. We show that the four risk constraints lead to the same optimal solution, which differs from previous conclusions obtained from the corresponding concave optimization problem under a risk constraint. Compared with the benchmark unconstrained utility maximization problem, all the four risk constraints effectively and equivalently reduce the set of zero terminal wealth, but do not fully eliminate this set, indicating the success and failure of the respective financial regulations.1

Optimal asset allocation for DC pension subject to allocation and terminal wealth constraints under a remuneration scheme

. We investigate the optimal investment problem faced by a defined contribution (DC) pension fund manager under simultaneous allocation and expected shortfall (ES) constraints. Under a non concave utility, a Value-at-Risk (VaR) constraint does not lead to the full prevention of moral hazard. As a widely employed risk management tool, whether an ES constraint can provide a more effective protection than a VaR constraint has been a focus point of research. We apply a dual control approach and a concavification technique to solve the ES-constrained optimization problem for a DC pension plan under an incentive scheme and derive the closed-form representations of the optimal wealth and portfolio processes. Furthermore, we compare the effect of an ES constraint on the optimal investment behavior with that under a VaR constraint in the presence of an option-like scheme for the DC pension members. Theoretical and numerical results show that for a relatively low protection level, a joint VaR and an ES constraints induce the same structure of the optimal solution, which implies that for a non concave optimization problem, the ES-based risk management has lost its advantage over the VaR-based risk management. Therefore, it needs to design a more efficient risk measure to improve the risk management for a DC pension plan.

The Effect of Supply Uncertainty on Dynamic Procurement and Pricing Strategies Under Lost Sales

The uncertainty in the supplier’s material flows has become a norm rather than an exception in supply chains. The supply uncertainty can result in unexpected inventory shortfall, amplifying lost sales. However, the design of inventory replenishment and product pricing policy to mitigate both supply uncertainty and demand loss remains unexplored. This is because the resulting dynamic planning problem is highly nonconcave and thus intractable. To address this challenge, we propose an approach that focuses on a class of intuitively appealing and practically plausible policies. Specifically, as the level of on-hand inventory increases, we expect an increased amount of demand fulfillment and a decreased product price. Applying the notions of stochastic functions, we show that, under general conditions of the stochastic supply and demand functions, the dynamic planning problem becomes a concave optimization problem over the restricted policy class. We further reduce the set of candidate policies to a refined class by excluding the dominated policies. A refined policy can be easily computed, and appropriately selected refined policies produce close-to-optimal profits. These developments allow us to evaluate the consequences of demand loss. In particular, demand retention through backordering can be beneficial when overstocking is costly in relation to understocking, but the benefit is insensitive to supply and demand uncertainties. Moreover, inventory-based dynamic pricing is more valuable in mitigating supply risk under lost sales than under backordering.

Radio resource allocation for energy efficiency maximization in satellite–terrestrial integrated networks

Satellite–Terrestrial Integrated Networks (STIN), where a satellite access network liaising with terrestrial networks, is useful not only in proffering seamless coverage but also in improving the backhaul capacity for heavy traffic/dense network scenarios. Therefore, an energy-efficient STIN is envisioned to be a valued gap filler both in public safety networks and in the provision of high-speed data services with ubiquitous coverage in remote areas. STIN necessitate admission control, user association, optimal power distribution and spectrum resource allocation to attain the desired quality-of-service standards. This paper investigates joint admission control, user association and power distribution for ensuring fairness while associating users in STIN and fairness in the allocation of spectrum resources to associated users in STIN with an overall objective to maximize the energy efficiency of STIN. The reviewed problem is a concave fractional programming problem which by utilizing Charnes–Cooper transformation is converted into a concave optimization problem. Subsequently, the concave optimization problem is resolved via the proposed outer approximation algorithm. The performance of the ϵ-optimum solution is extensively evaluated via the execution of different system parameters including number of users, user association, user fairness and resource block fairness.

Array design for mm-wave UAV-assisted cooperative communications with simultaneous wireless information and power transmission

High performance UAV-assisted communications system using simultaneous wireless information and power transmission (SWIPT) in mm-wave band is presented. UAV is a moving relay powered from a ground source through a power-splitting mechanism. In mm-wave band we utilize antenna array to increase the antenna gain while keeping array size small and practical. The radiation pattern of the UAV antenna is continuously adjusted to peak towards the source and destination. Two array geometries, a line and a cross, are designed for UAV antenna. We achieve near optimal pattern, utilizing innovative low power switches instead of phase shifters which are high power consuming components and using them here defies the purpose. We maximize the end-to-end cooperative throughput by optimizing the UAV power profile, power-splitting ratio profile, antenna weights (0, 1), and UAV trajectory for amplify-and-forward (AF) protocol. We consider two cases. Case1: UAV transmits and receives data simultaneously along two predefined trajectories. The antenna weights for line and cross arrays are optimized utilizing genetic algorithm. The power profile and, power-splitting ratio profile are also optimized using the penalty method. Case2: UAV accumulates the data and power along an optimal trajectory until it reaches the vicinity of target, when it transmits data at high bit rate. Here we define the optimization of parameters mentioned in Case1, while at an optimal point along the trajectory, as sub-problem1, and finding the next optimal point as sub-problem2. Sub-problem1 is solved using the genetic algorithm and dual decomposition method. Sub-problem2 is then solved using successive concave optimization. The overall problem, i.e. cooperative throughput, is solved by reciprocal iteration over the two sub problems. The simulation results show the proposed mm-wave band cross array antenna and switches can overcome the high frequency propagation losses, hence, achieving higher power harvest and data rates. The achieved higher data throughput outperforms the conventional single antenna low frequency systems.

Full Duplex Relaying with Intelligent Reflecting Surface: Joint Beamforming and Phase Adjustment

In this paper, an optimization algorithm is proposed for the system, where a decode-and-forward (DaF) protocol based full duplex relay (FDR) and an intelligent reflecting surface (IRS) work together to deliver message signals from a source to a destination. The transmission is carried out in a single time frame due to the full duplex nature of the FDR and IRS. The joint optimization of the beamformers for the multiple antenna sets at the source along with the FDR antenna sets and the phase adjustment of the reflecting elements at the IRS is proposed. Though separate optimizations of the beamformers and the phase adjustment were available previously, the joint one is more challenging due to the increased number of signal paths and the impacts from these new paths. To accommodate conflicting requirements from different paths, the reflecting elements of the IRS are cleverly grouped into the corresponding number of sets, where the requirements are met separately through the optimizations of the phase values of the corresponding group. For the beamformer optimization, we parameterize the beamforming vectors, which results in two concave optimization problems for the source and for the FDR relay. These optimization sets are implemented in an alternating manner to reach a reasonable solution. Simulation results are provided to illustrate the impacts of the proposed algorithm in various implementation scenarios and parameter sets. These results identify the gains the proposed algorithm provides and the scenarios where these gains are expected.

Dual Ascent Algorithm-Based Improved Droop Control for Efficient Operation of AC Microgrid

In this work, the loss (including wire loss and converter loss) of island three-phase AC microgrid is modeled as a quadratic function of the current distribution coefficient, that is, a concave function with equality and inequality constraints. On the basis of the concave optimization principle, the optimal current distribution coefficient of the distributed energy unit (DEU) is calculated online by the double ascent optimization method (DAOM) to minimize the distribution loss. It is proven that the concave function with multi-variables can be optimized by the DAOM. Using the average reactive power distribution scheme, the optimal active power distribution coefficient with the minimum distribution loss of the AC microgrid can be obtained in real time. In addition, given the high R/X ratio in the short-distance AC microgrid, the active power–frequency (P-ω) droop control and reactive power–voltage amplitude (Q-E) droop control are not suitable for power distribution among DEUs. Thus, an advanced strategy comprising active power–voltage amplitude (P-E) droop control and reactive power-frequency (Q-ω) droop control is proposed to dispatch the output active powers and reactive powers of DEUs. Simulation examples are provided to verify the convexity of the proposed model and the effectiveness of the control strategy.

Energy Efficient Power Control for Cognitive Multibeam-Satellite Terrestrial Networks With Poisson Distributed Users

With great potential to alleviate spectrum scarcity in the next generation wireless communication, cognitive satellite terrestrial networks (CSTNs) have attracted considerable attention recently. To realize efficient spectrum sharing between satellite and terrestrial networks, in this paper, we propose an energy efficient power control scheme for CSTNs, where a multibeam satellite network acts as the secondary system and the cellular network acts as the primary system. Especially, satellite users and primary base stations are spatially Poisson distributed and the channel state information (CSI) of satellite interference link is assumed to be outdated. We firstly analyze the statistical characteristic of the terrestrial aggregate interference with probabilistic stochastic geometry. On this basis, to guarantee reliable communications for both networks, we formulate the power control scheme as an energy efficiency maximization problem subjected to interference power constraints imposed by terrestrial communications and outage constraints of satellite communications. Further, by employing the Dinkelbach and Lagrange duality method, we solve the nonlinear concave optimization problem and derive the optimal solution of the transmit power. Finally, simulation results demonstrate the validity of the theoretical results, reveal the trade-off between the performance of two networks, present the impacts of aggregate interference on the performance of satellite networks, and show the performance superiority of considering outdated CSI compared with perfect CSI assumption in terms of interference violation probability.

Joint user association and energy efficiency maximization in beyond 5G heterogeneous networks

SummaryHeterogeneous networks (HetNets) seem to be the future of data networks supported by macrocells, small cells, relays, and D2D communication. The main purpose of HetNets is to facilitate maximum users while keeping energy efficiency (EE) to its peak. In this research work, we have formulated joint users association and EE maximization problem for HetNets, where our main goal is to increase the number of users associated with HetNets while keeping EE to its maximum. Formulated problem is a concave fractional problem in nature. We have used Charnes–Cooper Transformation to convert it to the concave optimization problem. We have used the (MADS) algorithm to solve the formulated optimization problem. Results have been analyzed after extensive simulations. Performance of MADS algorithm have been shown with respect to different system parameters, that is, the number of users associated, the minimum required data rate, and joint maximization of users associated and EE.

Energy Efficient Resource Allocation for H-NOMA Assisted B5G HetNets

The resource allocation solution offered based on Non-orthogonal multiple access (NOMA) and orthogonal multiple access (OMA) schemes are sub-optimal to address the challenging quality of service (QoS) and higher data rate viz-a-viz energy efficiency (EE) requirements in 5th generation (5G) cellular networks. In this work, we maximize the EE using user equipment (UE) clustering (UE-C) with downlink hybrid NOMA (H-NOMA) assisted beyond 5G (B5G) HetNets. We formulate an optimization problem incorporating UE admission in a cluster, UE association with a base station (BS), and power allocation assisted by H-NOMA, i.e., OMA and NOMA schemes in the macro base station (MBS) only and heterogeneous networks (HetNets) environments. The problem formulated is a type of non-linear concave fractional programming (CFP) problem. The Charnes-Cooper transformation (CCT) is applied to the formulated non-linear CFP problem to convert it into a concave optimization, i.e., mixed-integer non-linear programming (MINLP) problem. A two-phase ϵ-optimal outer approximation algorithm (OAA) is used to solve the MINLP problem. The simulation results show that H-NOMA with HetNets outperforms H-NOMA with MBS only in terms of UE admission, UE association, throughput, and EE.

Параметрическая регуляризация линейно-квадратичной задачи на множестве кусочно-линейных управлений

A linear-quadratic problem with arbitrary matrices in the functional and multidimensional control with convex constraint is considered. Acceptable controls are piecewise linear vector functions within an uneven grid of possible corner points. The reduction of the optimal control problem into a finite-dimensional format is carried out using vector formalization of the linear spline construction and block matrices together with the corresponding operations. The possibility of influencing the functional in the original problem is provided by using parameters with quadratic forms. The choice of these parameters is focused on the regularization of the functional in the sense of providing it with the properties of convexity or concavity at the level of a finite-dimensional model. The conditions for the choice of parameters are in the nature of inequalities with respect to the extreme eigenvalues of the block matrices forming the objective function. The corresponding convex or concave optimization problems can be solved in a finite number of iterations.

An optimal energy harvesting scheme for simultaneous lightwave information and power transfer over multi-layer turbulence-induced underwater channel

In this paper, we propose an optimal and energy-efficient time-splitting based simultaneous light-wave information and power transfer for underwater optical wireless communications. We consider a multi-layer turbulence-induced underwater optical channel with each layer experiencing independent and non-identically distributed oceanic turbulence. A concave optimization problem to maximize the energy harvesting for a target data rate is formulated for two cases, that is, the deterministic and the stochastic energy harvesting rate. Specifically, the analytical expression of optimal time-splitting ratio is derived. Furthermore, it is shown that the value of time-splitting ratio less than 0.5 is preferred for an efficient performance. Simulation results validate the analysis and demonstrate that the proposed scheme significantly improves the data rate and the energy harvesting for underwater optical communications.

Resource allocation for energy efficiency optimization in uplink–downlink decoupled 5G heterogeneous networks

SummaryHeterogeneous networks (HetNets) are a practical solution for traffic offloading from a high powered base station (HBS) to a low powered base station (LBS). In the HetNets, uplink (UL)–downlink (DL) decoupled access (UDDa) strategy is an optimal solution that ensures cell association, independently, in the UL and DL. This strategy offloads HBS cell edge mobile device (MD) to the nearby LBS in the UL. However, energy‐efficient cell association for traffic offloading from HBS to LBS, relay, or device to device (D2D) in the UL employing UDDa strategy has not been explored in the past work. We formulate mathematical models which ensure energy‐efficient cell association, power allocation, and traffic offloading employing traditional UL‐DL coupled access (UDCa) and UDDa strategies in the HetNets. The formulated problems are concave fractional programming (CFP) problems. The CFP is changed to a concave optimization problem using Charnes–Cooper transformation (CCT). The transformed problems are solved using an outer approximation algorithm to obtain optimal solution. Simulation results show the effectiveness of UDDa strategy over UDCa strategy in terms of MDs association, traffic offloading in the UL and DL, interference mitigation in the UL, data rate, and energy efficiency in the HetNets.

Using concave optimization methods for inexact quadratic programming problems with an application to waste management

Quadratic programming is potentially capable of strategic decision making in real world problems. However, practical problems rarely conform to crisp parameters, and hence the prospects of these problems with inexact parameters are inevitably higher. The existing studies regarding public welfare schemes/ organizations reveal that their objectives end up as minimization of cost functions and are governed by linear or concave quadratic programming problems. The present study proposes a method that can be applied to concave type quadratic objective function subject to linear constraints with inexact parameters. A comparison is also drawn with existing methods to establish its simplicity and efficiency. Further, a numerical example is illustrated, and finally, a waste management problem is formulated and solved using the proposed method.

Dynamic Transmission Rate Control for Multi‐Interface IoT Devices: A Stochastic Optimization Framework

Recent advances in the Internet of Things (IoT) technologies have enabled ubiquitous smart devices to sense and process various kinds of data. However, these innovations also raise the concern of efficient data transmission. Tackling the above issue is nontrivial since the resource constraints and environmental randomness in IoT require a lightweight transmission scheme while guaranteeing system stability. In this paper, we formulate the transmission scheduling problem of multi‐interface IoT devices as a concave optimization, aimed at accommodating the randomness of the IoT environment within the network capacity. By applying the Lyapunov optimization technique, we divide the stochastic problem into a series of low‐complex subproblems, which can be individually solved per time slot, and develop a dynamical control algorithm that does not require a priori knowledge such as link states. Theoretical analysis shows that our algorithms nicely bound the average queue length and are asymptotically optimal. Finally, extensive simulation results verify the theoretical conclusions and validate the effectiveness of the proposed algorithm.

Delay Consensus Margin of First-Order Multiagent Systems With Undirected Graphs and PD Protocols

In this article, we study robust consensus problems for continuous-time first-order multiagent systems (MASs) with time delays. We assume that the agents input is subject to an uncertain constant delay, which may arise due to interagent communication or additionally, also by self-delay in the agent dynamics. We consider dynamic feedback control protocol in the form of proportional and derivative (PD) control, and seek to determine the delay consensus margin (DCM) achievable by PD feedback protocols, whereas the DCM is a robustness measure that defines the maximal range of delay within which robust consensus can be achieved despite the uncertainty in the delay. With an undirected graph, we show that the DCM can be determined exactly by solving a unimodal concave optimization problem, which is one of univariate convex optimization and can be solved using convex optimization or gradient-based numerical methods. The results show how unstable agent dynamics and graph connectivity may limit the range of delay tolerable, so that consensus can or cannot be maintained in the presence of delay variations.

Sparse FIR Filter Design With k-Max Sparsity and Peak Error Constraints

FIR filters have boosted the development of digital signal processing, beamformers, and so on, due to their stability and low coefficient sensitivity. Generally, the design of FIR filters follows two main principles, i.e., the specification on the response error and the low implementation complexity. In this brief, we describe the sparsity of the filter coefficients using the k-maximum function, which equals to <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> -norm under mild conditions and has no restriction on the magnitude of nonzero coefficients. In order to avoid possible violation of specifications on response errors caused by frequency discretization, we estimate the frequencies at which the magnitude of the response error is maximized when constructing linear problems in the proposed algorithm. To address the nonlinearity and nonconvexity of the resulted optimization problem, we transform it into a piecewise linear concave optimization (PLCO) problem. Considering the fact that a PLCO problems is reduced to a linear programming (LP) problem locally, we outline an iterative algorithm by solving a series of LP problems and provide a brief complexity analysis. Numerical experiments on the propose method and some state-of-the-art methods are performed, the result of which shows the excellent performance of the propose method on balancing the sparsity and computational efficiency.

Hierarchical Resource Allocation in Multi-Service Wireless Networks With Wireless Network Virtualization

To balance the contradiction between the rapid growth of data service demands and the limited spectrum resources, wireless network virtualization (WNV) has been proposed as a promising technology by isolating and sharing wireless resources among different virtual networks in the future wireless networks. In this paper, a two-dimension-time-scale hierarchical resource allocation scheme is proposed in the multi-service wireless virtualized network, which consists of three 5G generic scenarios. The resource slicing problem is decomposed into two time scales including large time period for inter-slice resource pre-allocation and small time slot for intra-slice resource scheduling. In large time period, the inter-slice resource pre-allocation problem is formulated as a multi-objective optimization problem (MOOP) by modeling the packets arriving and serving process of each slice as a queueing system. While in small time slot, the resource block (RB) and power scheduling in each slice is formulated as a stochastic optimization problem considering dynamic traffic arrivals and time-varying channel conditions, which is aimed at optimizing the overall performance subject to various quality of service (QoS) requirements such as network stability, delay, reliability, transmission rate and power constraints. The stochastic optimization problem can be transformed into a delay-aware optimization problem by applying Lyapunov optimization technique, and be solved by the proposed algorithm consisting of a heuristic algorithm and a concave optimization algorithm. The simulation results show that the proposed schemes are close to the optimal solution with a lower complexity, which can also achieve a performance-delay tradeoff related to the control factor.